id	sid	tid	token	lemma	pos
cte-761	1	1	enhancing	enhance	VERB
cte-761	1	2	mathematical	mathematical	ADJ
cte-761	1	3	modelling	modelling	NOUN
cte-761	1	4	education	education	NOUN
cte-761	1	5	at	at	ADP
cte-761	1	6	agricultural	agricultural	ADJ
cte-761	1	7	universities	university	NOUN
cte-761	1	8	:	:	PUNCT
cte-761	1	9	a	a	DET
cte-761	1	10	comparative	comparative	ADJ
cte-761	1	11	study	study	NOUN
cte-761	1	12	of	of	ADP
cte-761	1	13	dynamic	dynamic	ADJ
cte-761	1	14	vector	vector	NOUN
cte-761	1	15	diagrams	diagram	NOUN
cte-761	1	16	using	use	VERB
cte-761	1	17	geogebra	geogebra	NOUN
cte-761	1	18	cte	cte	NOUN
cte-761	1	19	workshop	workshop	NOUN
cte-761	1	20	proceedings	proceeding	NOUN
cte-761	1	21	,	,	PUNCT
cte-761	1	22	2025	2025	NUM
cte-761	1	23	,	,	PUNCT
cte-761	1	24	vol	vol	NOUN
cte-761	1	25	.	.	PROPN
cte-761	1	26	12	12	NUM
cte-761	1	27	,	,	PUNCT
cte-761	1	28	pp	pp	ADJ
cte-761	1	29	.	.	PUNCT
cte-761	2	1	235–252	235–252	NUM
cte-761	3	1	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PRON
cte-761	3	2	enhancing	enhance	VERB
cte-761	3	3	mathematical	mathematical	ADJ
cte-761	3	4	modelling	modelling	NOUN
cte-761	3	5	education	education	NOUN
cte-761	3	6	at	at	ADP
cte-761	3	7	agricultural	agricultural	ADJ
cte-761	3	8	universities	university	NOUN
cte-761	3	9	:	:	PUNCT
cte-761	3	10	a	a	DET
cte-761	3	11	comparative	comparative	ADJ
cte-761	3	12	study	study	NOUN
cte-761	3	13	of	of	ADP
cte-761	3	14	dynamic	dynamic	ADJ
cte-761	3	15	vector	vector	NOUN
cte-761	3	16	diagrams	diagram	NOUN
cte-761	3	17	using	use	VERB
cte-761	3	18	geogebra	geogebra	NOUN
cte-761	3	19	leonid	leonid	PROPN
cte-761	3	20	o.	o.	PROPN
cte-761	3	21	flehantov	flehantov	PROPN
cte-761	3	22	,	,	PUNCT
cte-761	3	23	yuliia	yuliia	PROPN
cte-761	3	24	i.	i.	PROPN
cte-761	3	25	ovsiienko	ovsiienko	NOUN
cte-761	3	26	and	and	CCONJ
cte-761	3	27	anatolii	anatolii	NOUN
cte-761	3	28	v.	v.	ADP
cte-761	3	29	antonets	antonet	NOUN
cte-761	3	30	poltava	poltava	PROPN
cte-761	3	31	state	state	PROPN
cte-761	3	32	agrarian	agrarian	PROPN
cte-761	3	33	academy	academy	PROPN
cte-761	3	34	,	,	PUNCT
cte-761	3	35	1/3	1/3	NUM
cte-761	3	36	skovorody	skovorody	NOUN
cte-761	3	37	str	str	PROPN
cte-761	3	38	.	.	PUNCT
cte-761	3	39	,	,	PUNCT
cte-761	3	40	poltava	poltava	PROPN
cte-761	3	41	,	,	PUNCT
cte-761	3	42	36000	36000	NUM
cte-761	3	43	,	,	PUNCT
cte-761	3	44	ukraine	ukraine	PROPN
cte-761	3	45	abstract	abstract	NOUN
cte-761	3	46	.	.	PUNCT
cte-761	4	1	this	this	DET
cte-761	4	2	research	research	NOUN
cte-761	4	3	investigates	investigate	VERB
cte-761	4	4	the	the	DET
cte-761	4	5	effectiveness	effectiveness	NOUN
cte-761	4	6	of	of	ADP
cte-761	4	7	visualisation	visualisation	NOUN
cte-761	4	8	techniques	technique	NOUN
cte-761	4	9	in	in	ADP
cte-761	4	10	teaching	teach	VERB
cte-761	4	11	mathematical	mathematical	ADJ
cte-761	4	12	modelling	modelling	NOUN
cte-761	4	13	fundamentals	fundamental	NOUN
cte-761	4	14	to	to	ADP
cte-761	4	15	agricultural	agricultural	ADJ
cte-761	4	16	university	university	NOUN
cte-761	4	17	students	student	NOUN
cte-761	4	18	.	.	PUNCT
cte-761	5	1	we	we	PRON
cte-761	5	2	examine	examine	VERB
cte-761	5	3	the	the	DET
cte-761	5	4	hypothesis	hypothesis	NOUN
cte-761	5	5	that	that	PRON
cte-761	5	6	dynamic	dynamic	ADJ
cte-761	5	7	vector	vector	NOUN
cte-761	5	8	diagrams	diagram	NOUN
cte-761	5	9	representing	represent	VERB
cte-761	5	10	mechanical	mechanical	ADJ
cte-761	5	11	motion	motion	NOUN
cte-761	5	12	characteristics	characteristic	NOUN
cte-761	5	13	(	(	PUNCT
cte-761	5	14	velocity	velocity	NOUN
cte-761	5	15	,	,	PUNCT
cte-761	5	16	acceleration	acceleration	NOUN
cte-761	5	17	,	,	PUNCT
cte-761	5	18	and	and	CCONJ
cte-761	5	19	force	force	NOUN
cte-761	5	20	)	)	PUNCT
cte-761	5	21	enhance	enhance	VERB
cte-761	5	22	student	student	NOUN
cte-761	5	23	learning	learn	VERB
cte-761	5	24	outcomes	outcome	NOUN
cte-761	5	25	.	.	PUNCT
cte-761	6	1	the	the	DET
cte-761	6	2	study	study	NOUN
cte-761	6	3	compares	compare	VERB
cte-761	6	4	three	three	NUM
cte-761	6	5	instructional	instructional	ADJ
cte-761	6	6	approaches	approach	NOUN
cte-761	6	7	:	:	PUNCT
cte-761	6	8	using	use	VERB
cte-761	6	9	excel	excel	NOUN
cte-761	6	10	spreadsheets	spreadsheet	NOUN
cte-761	6	11	,	,	PUNCT
cte-761	6	12	utilising	utilise	VERB
cte-761	6	13	geogebra	geogebra	NOUN
cte-761	6	14	dynamic	dynamic	ADJ
cte-761	6	15	geometry	geometry	NOUN
cte-761	6	16	software	software	NOUN
cte-761	6	17	,	,	PUNCT
cte-761	6	18	and	and	CCONJ
cte-761	6	19	employing	employ	VERB
cte-761	6	20	both	both	DET
cte-761	6	21	tools	tool	NOUN
cte-761	6	22	simultaneously	simultaneously	ADV
cte-761	6	23	.	.	PUNCT
cte-761	7	1	our	our	PRON
cte-761	7	2	methodology	methodology	NOUN
cte-761	7	3	involved	involve	VERB
cte-761	7	4	167	167	NUM
cte-761	7	5	engineering	engineering	NOUN
cte-761	7	6	students	student	NOUN
cte-761	7	7	divided	divide	VERB
cte-761	7	8	into	into	ADP
cte-761	7	9	three	three	NUM
cte-761	7	10	homogeneous	homogeneous	ADJ
cte-761	7	11	groups	group	NOUN
cte-761	7	12	,	,	PUNCT
cte-761	7	13	each	each	PRON
cte-761	7	14	completing	complete	VERB
cte-761	7	15	identical	identical	ADJ
cte-761	7	16	modelling	modelling	NOUN
cte-761	7	17	tasks	task	NOUN
cte-761	7	18	concerning	concern	VERB
cte-761	7	19	projectile	projectile	NOUN
cte-761	7	20	motion	motion	NOUN
cte-761	7	21	under	under	ADP
cte-761	7	22	various	various	ADJ
cte-761	7	23	conditions	condition	NOUN
cte-761	7	24	.	.	PUNCT
cte-761	8	1	results	result	NOUN
cte-761	8	2	demonstrate	demonstrate	VERB
cte-761	8	3	that	that	SCONJ
cte-761	8	4	students	student	NOUN
cte-761	8	5	who	who	PRON
cte-761	8	6	simultaneously	simultaneously	ADV
cte-761	8	7	employed	employ	VERB
cte-761	8	8	excel	excel	NOUN
cte-761	8	9	and	and	CCONJ
cte-761	8	10	geogebra	geogebra	NOUN
cte-761	8	11	with	with	ADP
cte-761	8	12	dynamic	dynamic	ADJ
cte-761	8	13	vector	vector	NOUN
cte-761	8	14	diagram	diagram	NOUN
cte-761	8	15	visualisation	visualisation	NOUN
cte-761	8	16	achieved	achieve	VERB
cte-761	8	17	significantly	significantly	ADV
cte-761	8	18	higher	high	ADJ
cte-761	8	19	academic	academic	ADJ
cte-761	8	20	performance	performance	NOUN
cte-761	8	21	(	(	PUNCT
cte-761	8	22	mean	mean	VERB
cte-761	8	23	score	score	NOUN
cte-761	8	24	78.25	78.25	NUM
cte-761	8	25	)	)	PUNCT
cte-761	8	26	compared	compare	VERB
cte-761	8	27	to	to	ADP
cte-761	8	28	those	those	PRON
cte-761	8	29	using	use	VERB
cte-761	8	30	either	either	CCONJ
cte-761	8	31	excel	excel	VERB
cte-761	8	32	(	(	PUNCT
cte-761	8	33	73.36	73.36	NUM
cte-761	8	34	)	)	PUNCT
cte-761	8	35	or	or	CCONJ
cte-761	8	36	geogebra	geogebra	NOUN
cte-761	8	37	(	(	PUNCT
cte-761	8	38	73.85	73.85	NUM
cte-761	8	39	)	)	PUNCT
cte-761	8	40	exclusively	exclusively	ADV
cte-761	8	41	.	.	PUNCT
cte-761	9	1	statistical	statistical	ADJ
cte-761	9	2	analysis	analysis	NOUN
cte-761	9	3	through	through	ADP
cte-761	9	4	anova	anova	PROPN
cte-761	9	5	confirms	confirm	VERB
cte-761	9	6	these	these	DET
cte-761	9	7	differences	difference	NOUN
cte-761	9	8	are	be	AUX
cte-761	9	9	significant	significant	ADJ
cte-761	9	10	(	(	PUNCT
cte-761	9	11	p=0.010613	p=0.010613	PROPN
cte-761	9	12	)	)	PUNCT
cte-761	9	13	.	.	PUNCT
cte-761	10	1	we	we	PRON
cte-761	10	2	observed	observe	VERB
cte-761	10	3	that	that	PRON
cte-761	10	4	excel	excel	VERB
cte-761	10	5	users	user	NOUN
cte-761	10	6	demonstrated	demonstrate	VERB
cte-761	10	7	stronger	strong	ADJ
cte-761	10	8	quantitative	quantitative	ADJ
cte-761	10	9	analytical	analytical	ADJ
cte-761	10	10	skills	skill	NOUN
cte-761	10	11	,	,	PUNCT
cte-761	10	12	while	while	SCONJ
cte-761	10	13	geogebra	geogebra	NOUN
cte-761	10	14	users	user	NOUN
cte-761	10	15	excelled	excel	VERB
cte-761	10	16	in	in	ADP
cte-761	10	17	qualitative	qualitative	ADJ
cte-761	10	18	assessment	assessment	NOUN
cte-761	10	19	tasks	task	NOUN
cte-761	10	20	.	.	PUNCT
cte-761	11	1	this	this	DET
cte-761	11	2	research	research	NOUN
cte-761	11	3	extends	extend	VERB
cte-761	11	4	previous	previous	ADJ
cte-761	11	5	findings	finding	NOUN
cte-761	11	6	on	on	ADP
cte-761	11	7	visualisation	visualisation	NOUN
cte-761	11	8	in	in	ADP
cte-761	11	9	mathematical	mathematical	ADJ
cte-761	11	10	education	education	NOUN
cte-761	11	11	and	and	CCONJ
cte-761	11	12	provides	provide	VERB
cte-761	11	13	practical	practical	ADJ
cte-761	11	14	insights	insight	NOUN
cte-761	11	15	for	for	ADP
cte-761	11	16	enhancing	enhance	VERB
cte-761	11	17	stem	stem	NOUN
cte-761	11	18	education	education	NOUN
cte-761	11	19	in	in	ADP
cte-761	11	20	agricultural	agricultural	ADJ
cte-761	11	21	universities	university	NOUN
cte-761	11	22	through	through	ADP
cte-761	11	23	appropriate	appropriate	ADJ
cte-761	11	24	technology	technology	NOUN
cte-761	11	25	integration	integration	NOUN
cte-761	11	26	.	.	PUNCT
cte-761	12	1	keywords	keyword	NOUN
cte-761	12	2	:	:	PUNCT
cte-761	12	3	mathematical	mathematical	ADJ
cte-761	12	4	modelling	modelling	NOUN
cte-761	12	5	education	education	NOUN
cte-761	12	6	,	,	PUNCT
cte-761	12	7	agricultural	agricultural	ADJ
cte-761	12	8	engineering	engineering	NOUN
cte-761	12	9	education	education	NOUN
cte-761	12	10	,	,	PUNCT
cte-761	12	11	visualisation	visualisation	NOUN
cte-761	12	12	techniques	technique	NOUN
cte-761	12	13	,	,	PUNCT
cte-761	12	14	dynamic	dynamic	ADJ
cte-761	12	15	vector	vector	NOUN
cte-761	12	16	diagrams	diagram	NOUN
cte-761	12	17	,	,	PUNCT
cte-761	12	18	geogebra	geogebra	NOUN
cte-761	12	19	,	,	PUNCT
cte-761	12	20	excel	excel	NOUN
cte-761	12	21	,	,	PUNCT
cte-761	12	22	educational	educational	ADJ
cte-761	12	23	technology	technology	NOUN
cte-761	12	24	,	,	PUNCT
cte-761	12	25	projectile	projectile	NOUN
cte-761	12	26	motion	motion	NOUN
cte-761	12	27	modelling	modelling	NOUN
cte-761	12	28	,	,	PUNCT
cte-761	12	29	stem	stem	NOUN
cte-761	12	30	pedagogy	pedagogy	NOUN
cte-761	12	31	,	,	PUNCT
cte-761	12	32	comparative	comparative	ADJ
cte-761	12	33	instructional	instructional	ADJ
cte-761	12	34	methods	method	NOUN
cte-761	12	35	1	1	NUM
cte-761	12	36	.	.	PUNCT
cte-761	13	1	introduction	introduction	NOUN
cte-761	13	2	contemporary	contemporary	ADJ
cte-761	13	3	education	education	NOUN
cte-761	13	4	is	be	AUX
cte-761	13	5	witnessing	witness	VERB
cte-761	13	6	profound	profound	ADJ
cte-761	13	7	technological	technological	ADJ
cte-761	13	8	transformations	transformation	NOUN
cte-761	13	9	that	that	PRON
cte-761	13	10	reshape	reshape	VERB
cte-761	13	11	traditional	traditional	ADJ
cte-761	13	12	teaching	teaching	NOUN
cte-761	13	13	methodologies	methodology	NOUN
cte-761	13	14	.	.	PUNCT
cte-761	14	1	one	one	NUM
cte-761	14	2	significant	significant	ADJ
cte-761	14	3	shift	shift	NOUN
cte-761	14	4	involves	involve	VERB
cte-761	14	5	the	the	DET
cte-761	14	6	transition	transition	NOUN
cte-761	14	7	from	from	ADP
cte-761	14	8	predominantly	predominantly	ADV
cte-761	14	9	verbal	verbal	ADJ
cte-761	14	10	communication	communication	NOUN
cte-761	14	11	channels	channel	NOUN
cte-761	14	12	in	in	ADP
cte-761	14	13	learning	learn	VERB
cte-761	14	14	towards	towards	ADP
cte-761	14	15	visuallyoriented	visuallyoriente	VERB
cte-761	14	16	approaches	approach	NOUN
cte-761	14	17	.	.	PUNCT
cte-761	15	1	while	while	SCONJ
cte-761	15	2	researchers	researcher	NOUN
cte-761	15	3	continue	continue	VERB
cte-761	15	4	to	to	PART
cte-761	15	5	investigate	investigate	VERB
cte-761	15	6	this	this	DET
cte-761	15	7	phenomenon	phenomenon	NOUN
cte-761	15	8	,	,	PUNCT
cte-761	15	9	today	today	NOUN
cte-761	15	10	’s	’s	PART
cte-761	15	11	students	student	NOUN
cte-761	15	12	have	have	AUX
cte-761	15	13	already	already	ADV
cte-761	15	14	embraced	embrace	VERB
cte-761	15	15	visual	visual	ADJ
cte-761	15	16	learning	learning	NOUN
cte-761	15	17	methods	method	NOUN
cte-761	15	18	.	.	PUNCT
cte-761	16	1	the	the	DET
cte-761	16	2	pedagogical	pedagogical	ADJ
cte-761	16	3	approach	approach	NOUN
cte-761	16	4	this	this	DET
cte-761	16	5	study	study	NOUN
cte-761	16	6	examines	examine	NOUN
cte-761	16	7	aligns	align	VERB
cte-761	16	8	with	with	ADP
cte-761	16	9	the	the	DET
cte-761	16	10	ancient	ancient	ADJ
cte-761	16	11	wisdom	wisdom	NOUN
cte-761	16	12	:	:	PUNCT
cte-761	16	13	“	"	PUNCT
cte-761	16	14	i	i	PRON
cte-761	16	15	hear	hear	VERB
cte-761	16	16	and	and	CCONJ
cte-761	16	17	forget	forget	VERB
cte-761	16	18	.	.	PUNCT
cte-761	17	1	i	i	PRON
cte-761	17	2	see	see	VERB
cte-761	17	3	and	and	CCONJ
cte-761	17	4	remember	remember	VERB
cte-761	17	5	.	.	PUNCT
cte-761	18	1	i	i	PRON
cte-761	18	2	do	do	VERB
cte-761	18	3	and	and	CCONJ
cte-761	18	4	understand	understand	VERB
cte-761	18	5	”	"	PUNCT
cte-761	19	1	[	[	X
cte-761	19	2	13	13	NUM
cte-761	19	3	]	]	PUNCT
cte-761	19	4	.	.	PUNCT
cte-761	20	1	we	we	PRON
cte-761	20	2	investigate	investigate	VERB
cte-761	20	3	the	the	DET
cte-761	20	4	impact	impact	NOUN
cte-761	20	5	of	of	ADP
cte-761	20	6	visualisation	visualisation	NOUN
cte-761	20	7	techniques	technique	NOUN
cte-761	20	8	on	on	ADP
cte-761	20	9	teaching	teach	VERB
cte-761	20	10	the	the	DET
cte-761	20	11	fundamentals	fundamental	NOUN
cte-761	20	12	of	of	ADP
cte-761	20	13	mathematical	mathematical	ADJ
cte-761	20	14	modelling	modelling	NOUN
cte-761	20	15	(	(	PUNCT
cte-761	20	16	bmm	bmm	PROPN
cte-761	20	17	)	)	PUNCT
cte-761	20	18	using	use	VERB
cte-761	20	19	geogebra	geogebra	NOUN
cte-761	20	20	,	,	PUNCT
cte-761	20	21	a	a	DET
cte-761	20	22	dynamic	dynamic	ADJ
cte-761	20	23	geometry	geometry	NOUN
cte-761	20	24	system	system	NOUN
cte-761	20	25	[	[	X
cte-761	20	26	9	9	NUM
cte-761	20	27	]	]	PUNCT
cte-761	20	28	.	.	PUNCT
cte-761	21	1	this	this	DET
cte-761	21	2	research	research	NOUN
cte-761	21	3	builds	build	VERB
cte-761	21	4	upon	upon	SCONJ
cte-761	21	5	established	establish	VERB
cte-761	21	6	methodological	methodological	ADJ
cte-761	21	7	principles	principle	NOUN
cte-761	21	8	for	for	ADP
cte-761	21	9	integrating	integrate	VERB
cte-761	21	10	information	information	NOUN
cte-761	21	11	and	and	CCONJ
cte-761	21	12	communication	communication	NOUN
cte-761	21	13	technologies	technology	NOUN
cte-761	21	14	(	(	PUNCT
cte-761	21	15	ict	ict	PROPN
cte-761	21	16	)	)	PUNCT
cte-761	21	17	into	into	ADP
cte-761	21	18	university	university	NOUN
cte-761	21	19	-	-	PUNCT
cte-761	21	20	level	level	NOUN
cte-761	21	21	mathematical	mathematical	ADJ
cte-761	21	22	education	education	NOUN
cte-761	21	23	[	[	X
cte-761	21	24	1	1	NUM
cte-761	21	25	,	,	PUNCT
cte-761	21	26	12	12	NUM
cte-761	21	27	]	]	PUNCT
cte-761	21	28	.	.	PUNCT
cte-761	22	1	teaching	teach	VERB
cte-761	22	2	mathematical	mathematical	ADJ
cte-761	22	3	modelling	modelling	NOUN
cte-761	22	4	constitutes	constitute	VERB
cte-761	22	5	an	an	DET
cte-761	22	6	essential	essential	ADJ
cte-761	22	7	component	component	NOUN
cte-761	22	8	in	in	ADP
cte-761	22	9	training	train	VERB
cte-761	22	10	�	�	PROPN
cte-761	22	11	0000	0000	NUM
cte-761	22	12	-	-	PUNCT
cte-761	22	13	0002	0002	NUM
cte-761	22	14	-	-	PUNCT
cte-761	22	15	4689	4689	NUM
cte-761	22	16	-	-	SYM
cte-761	22	17	1457	1457	NUM
cte-761	22	18	(	(	PUNCT
cte-761	22	19	l.	l.	PROPN
cte-761	22	20	o.	o.	PROPN
cte-761	22	21	flehantov	flehantov	PROPN
cte-761	22	22	)	)	PUNCT
cte-761	22	23	;	;	PUNCT
cte-761	22	24	0000	0000	NUM
cte-761	22	25	-	-	PUNCT
cte-761	22	26	0002	0002	NUM
cte-761	22	27	-	-	PUNCT
cte-761	22	28	4873	4873	NUM
cte-761	22	29	-	-	SYM
cte-761	22	30	9061	9061	NUM
cte-761	22	31	(	(	PUNCT
cte-761	22	32	y.	y.	PROPN
cte-761	22	33	i.	i.	PROPN
cte-761	22	34	ovsiienko	ovsiienko	PROPN
cte-761	22	35	)	)	PUNCT
cte-761	22	36	;	;	PUNCT
cte-761	22	37	0000	0000	NUM
cte-761	22	38	-	-	PUNCT
cte-761	22	39	0002	0002	NUM
cte-761	22	40	-	-	PUNCT
cte-761	22	41	2332	2332	NUM
cte-761	22	42	-	-	SYM
cte-761	22	43	6711	6711	NUM
cte-761	22	44	(	(	PUNCT
cte-761	22	45	a.	a.	NOUN
cte-761	22	46	v.	v.	ADP
cte-761	22	47	antonets	antonet	NOUN
cte-761	22	48	)	)	PUNCT
cte-761	22	49	#	#	NOUN
cte-761	22	50	leonid.flegantov@pdaa.edu.ua	leonid.flegantov@pdaa.edu.ua	NOUN
cte-761	22	51	(	(	PUNCT
cte-761	22	52	l.	l.	PROPN
cte-761	22	53	o.	o.	PROPN
cte-761	22	54	flehantov	flehantov	PROPN
cte-761	22	55	)	)	PUNCT
cte-761	22	56	;	;	PUNCT
cte-761	22	57	iuliia.ovsiienko@pdaa.edu.ua	iuliia.ovsiienko@pdaa.edu.ua	NOUN
cte-761	22	58	(	(	PUNCT
cte-761	22	59	y.	y.	PROPN
cte-761	22	60	i.	i.	PROPN
cte-761	22	61	ovsiienko	ovsiienko	PROPN
cte-761	22	62	)	)	PUNCT
cte-761	22	63	;	;	PUNCT
cte-761	23	1	anatolii.antonets@pdaa.edu.ua	anatolii.antonets@pdaa.edu.ua	PROPN
cte-761	23	2	(	(	PUNCT
cte-761	23	3	a.	a.	PROPN
cte-761	23	4	v.	v.	ADP
cte-761	23	5	antonets	antonet	NOUN
cte-761	23	6	)	)	PUNCT
cte-761	24	1	~	~	PUNCT
cte-761	24	2	https://www.pdaa.edu.ua/people/flegantov-leonid-oleksiyovich	https://www.pdaa.edu.ua/people/flegantov-leonid-oleksiyovich	INTJ
cte-761	24	3	(	(	PUNCT
cte-761	24	4	l.	l.	PROPN
cte-761	24	5	o.	o.	PROPN
cte-761	24	6	flehantov	flehantov	PROPN
cte-761	24	7	)	)	PUNCT
cte-761	24	8	;	;	PUNCT
cte-761	24	9	https://www.pdaa.edu.ua/people/ovsiyenko-yuliya-ivanivna	https://www.pdaa.edu.ua/people/ovsiyenko-yuliya-ivanivna	NOUN
cte-761	24	10	(	(	PUNCT
cte-761	24	11	y.	y.	PROPN
cte-761	24	12	i.	i.	PROPN
cte-761	24	13	ovsiienko	ovsiienko	PROPN
cte-761	24	14	)	)	PUNCT
cte-761	24	15	;	;	PUNCT
cte-761	24	16	https://www.pdaa.edu.ua/people/antonec-anatoliy-viktorovich	https://www.pdaa.edu.ua/people/antonec-anatoliy-viktorovich	PROPN
cte-761	24	17	(	(	PUNCT
cte-761	24	18	a.	a.	PROPN
cte-761	24	19	v.	v.	ADP
cte-761	24	20	antonets	antonet	NOUN
cte-761	24	21	)	)	PUNCT
cte-761	24	22	cte	cte	NOUN
cte-761	24	23	workshop	workshop	NOUN
cte-761	24	24	proceedings	proceeding	NOUN
cte-761	24	25	©	©	ADP
cte-761	24	26	copyright	copyright	NOUN
cte-761	24	27	for	for	ADP
cte-761	24	28	this	this	DET
cte-761	24	29	article	article	NOUN
cte-761	24	30	by	by	ADP
cte-761	24	31	its	its	PRON
cte-761	24	32	authors	author	NOUN
cte-761	24	33	,	,	PUNCT
cte-761	24	34	published	publish	VERB
cte-761	24	35	by	by	ADP
cte-761	24	36	the	the	DET
cte-761	24	37	academy	academy	NOUN
cte-761	24	38	of	of	ADP
cte-761	24	39	cognitive	cognitive	ADJ
cte-761	24	40	and	and	CCONJ
cte-761	24	41	natural	natural	ADJ
cte-761	24	42	sciences	science	NOUN
cte-761	24	43	.	.	PUNCT
cte-761	25	1	this	this	PRON
cte-761	25	2	is	be	AUX
cte-761	25	3	an	an	DET
cte-761	25	4	open	open	ADJ
cte-761	25	5	access	access	NOUN
cte-761	25	6	article	article	NOUN
cte-761	25	7	distributed	distribute	VERB
cte-761	25	8	under	under	ADP
cte-761	25	9	the	the	DET
cte-761	25	10	terms	term	NOUN
cte-761	25	11	of	of	ADP
cte-761	25	12	the	the	DET
cte-761	25	13	creative	creative	ADJ
cte-761	25	14	commons	common	NOUN
cte-761	25	15	license	license	NOUN
cte-761	25	16	attribution	attribution	NOUN
cte-761	25	17	4.0	4.0	NUM
cte-761	25	18	international	international	NOUN
cte-761	25	19	(	(	PUNCT
cte-761	25	20	cc	cc	NOUN
cte-761	25	21	by	by	ADP
cte-761	25	22	4.0	4.0	NUM
cte-761	25	23	)	)	PUNCT
cte-761	25	24	,	,	PUNCT
cte-761	25	25	which	which	PRON
cte-761	25	26	permits	permit	VERB
cte-761	25	27	unrestricted	unrestricted	ADJ
cte-761	25	28	use	use	NOUN
cte-761	25	29	,	,	PUNCT
cte-761	25	30	distribution	distribution	NOUN
cte-761	25	31	,	,	PUNCT
cte-761	25	32	and	and	CCONJ
cte-761	25	33	reproduction	reproduction	NOUN
cte-761	25	34	in	in	ADP
cte-761	25	35	any	any	DET
cte-761	25	36	medium	medium	NOUN
cte-761	25	37	,	,	PUNCT
cte-761	25	38	provided	provide	VERB
cte-761	25	39	the	the	DET
cte-761	25	40	original	original	ADJ
cte-761	25	41	work	work	NOUN
cte-761	25	42	is	be	AUX
cte-761	25	43	properly	properly	ADV
cte-761	25	44	cited	cite	VERB
cte-761	25	45	.	.	PUNCT
cte-761	26	1	235	235	NUM
cte-761	27	1	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	27	2	https://orcid.org/0000-0002-4689-1457	https://orcid.org/0000-0002-4689-1457	NOUN
cte-761	27	3	https://orcid.org/0000-0002-4873-9061	https://orcid.org/0000-0002-4873-9061	NOUN
cte-761	27	4	https://orcid.org/0000-0002-2332-6711	https://orcid.org/0000-0002-2332-6711	PROPN
cte-761	27	5	mailto:leonid.flegantov@pdaa.edu.ua	mailto:leonid.flegantov@pdaa.edu.ua	PROPN
cte-761	27	6	mailto:iuliia.ovsiienko@pdaa.edu.ua	mailto:iuliia.ovsiienko@pdaa.edu.ua	PROPN
cte-761	27	7	mailto:anatolii.antonets@pdaa.edu.ua	mailto:anatolii.antonets@pdaa.edu.ua	PROPN
cte-761	28	1	https://www.pdaa.edu.ua/people/flegantov-leonid-oleksiyovich	https://www.pdaa.edu.ua/people/flegantov-leonid-oleksiyovich	INTJ
cte-761	28	2	https://www.pdaa.edu.ua/people/ovsiyenko-yuliya-ivanivna	https://www.pdaa.edu.ua/people/ovsiyenko-yuliya-ivanivna	VERB
cte-761	28	3	https://www.pdaa.edu.ua/people/antonec-anatoliy-viktorovich	https://www.pdaa.edu.ua/people/antonec-anatoliy-viktorovich	NOUN
cte-761	28	4	https://acnsci.org/cte	https://acnsci.org/cte	NOUN
cte-761	28	5	https://creativecommons.org/licenses/by/4.0/deed.en	https://creativecommons.org/licenses/by/4.0/deed.en	ADV
cte-761	28	6	https://acnsci.org	https://acnsci.org	PROPN
cte-761	28	7	cte	cte	NOUN
cte-761	28	8	workshop	workshop	NOUN
cte-761	28	9	proceedings	proceeding	NOUN
cte-761	28	10	,	,	PUNCT
cte-761	28	11	2025	2025	NUM
cte-761	28	12	,	,	PUNCT
cte-761	28	13	vol	vol	NOUN
cte-761	28	14	.	.	PROPN
cte-761	28	15	12	12	NUM
cte-761	28	16	,	,	PUNCT
cte-761	28	17	pp	pp	ADJ
cte-761	28	18	.	.	PUNCT
cte-761	29	1	235–252	235–252	NUM
cte-761	29	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	ADJ
cte-761	29	3	modern	modern	ADJ
cte-761	29	4	agricultural	agricultural	ADJ
cte-761	29	5	engineers	engineer	NOUN
cte-761	29	6	,	,	PUNCT
cte-761	29	7	providing	provide	VERB
cte-761	29	8	them	they	PRON
cte-761	29	9	with	with	ADP
cte-761	29	10	analytical	analytical	ADJ
cte-761	29	11	skills	skill	NOUN
cte-761	29	12	necessary	necessary	ADJ
cte-761	29	13	for	for	ADP
cte-761	29	14	addressing	address	VERB
cte-761	29	15	complex	complex	ADJ
cte-761	29	16	agricultural	agricultural	ADJ
cte-761	29	17	production	production	NOUN
cte-761	29	18	challenges	challenge	NOUN
cte-761	29	19	.	.	PUNCT
cte-761	30	1	the	the	DET
cte-761	30	2	practical	practical	ADJ
cte-761	30	3	value	value	NOUN
cte-761	30	4	of	of	ADP
cte-761	30	5	this	this	DET
cte-761	30	6	training	training	NOUN
cte-761	30	7	stems	stem	VERB
cte-761	30	8	from	from	ADP
cte-761	30	9	its	its	PRON
cte-761	30	10	effectiveness	effectiveness	NOUN
cte-761	30	11	as	as	ADP
cte-761	30	12	an	an	DET
cte-761	30	13	educational	educational	ADJ
cte-761	30	14	strategy	strategy	NOUN
cte-761	30	15	in	in	ADP
cte-761	30	16	contemporary	contemporary	ADJ
cte-761	30	17	contexts	context	NOUN
cte-761	30	18	[	[	X
cte-761	30	19	2	2	NUM
cte-761	30	20	]	]	PUNCT
cte-761	30	21	.	.	PUNCT
cte-761	31	1	this	this	DET
cte-761	31	2	study	study	NOUN
cte-761	31	3	directly	directly	ADV
cte-761	31	4	extends	extend	VERB
cte-761	31	5	our	our	PRON
cte-761	31	6	earlier	early	ADJ
cte-761	31	7	research	research	NOUN
cte-761	31	8	,	,	PUNCT
cte-761	31	9	which	which	PRON
cte-761	31	10	demonstrated	demonstrate	VERB
cte-761	31	11	that	that	SCONJ
cte-761	31	12	visualising	visualise	VERB
cte-761	31	13	dynamic	dynamic	ADJ
cte-761	31	14	characteristics	characteristic	NOUN
cte-761	31	15	of	of	ADP
cte-761	31	16	mechanical	mechanical	ADJ
cte-761	31	17	processes	process	NOUN
cte-761	31	18	using	use	VERB
cte-761	31	19	geogebra	geogebra	NOUN
cte-761	31	20	improved	improve	VERB
cte-761	31	21	student	student	NOUN
cte-761	31	22	learning	learn	VERB
cte-761	31	23	outcomes	outcome	NOUN
cte-761	31	24	in	in	ADP
cte-761	31	25	mathematical	mathematical	ADJ
cte-761	31	26	modelling	modelling	NOUN
cte-761	31	27	courses	course	NOUN
cte-761	31	28	.	.	PUNCT
cte-761	32	1	our	our	PRON
cte-761	32	2	previous	previous	ADJ
cte-761	32	3	experiment	experiment	NOUN
cte-761	32	4	conducted	conduct	VERB
cte-761	32	5	in	in	ADP
cte-761	32	6	2018	2018	NUM
cte-761	32	7	revealed	reveal	VERB
cte-761	32	8	promising	promising	ADJ
cte-761	32	9	results	result	NOUN
cte-761	32	10	when	when	SCONJ
cte-761	32	11	students	student	NOUN
cte-761	32	12	simultaneously	simultaneously	ADV
cte-761	32	13	used	use	VERB
cte-761	32	14	excel	excel	NOUN
cte-761	32	15	and	and	CCONJ
cte-761	32	16	geogebra	geogebra	NOUN
cte-761	32	17	,	,	PUNCT
cte-761	32	18	achieving	achieve	VERB
cte-761	32	19	an	an	DET
cte-761	32	20	average	average	ADJ
cte-761	32	21	score	score	NOUN
cte-761	32	22	of	of	ADP
cte-761	32	23	77.4	77.4	NUM
cte-761	32	24	compared	compare	VERB
cte-761	32	25	to	to	ADP
cte-761	32	26	students	student	NOUN
cte-761	32	27	using	use	VERB
cte-761	32	28	only	only	ADV
cte-761	32	29	excel	excel	VERB
cte-761	32	30	(	(	PUNCT
cte-761	32	31	73.4	73.4	NUM
cte-761	32	32	)	)	PUNCT
cte-761	32	33	or	or	CCONJ
cte-761	32	34	only	only	ADV
cte-761	32	35	geogebra	geogebra	NOUN
cte-761	32	36	(	(	PUNCT
cte-761	32	37	73.7	73.7	NUM
cte-761	32	38	)	)	PUNCT
cte-761	32	39	.	.	PUNCT
cte-761	33	1	this	this	PRON
cte-761	33	2	suggested	suggest	VERB
cte-761	33	3	that	that	SCONJ
cte-761	33	4	visualisation	visualisation	NOUN
cte-761	33	5	of	of	ADP
cte-761	33	6	modelling	model	VERB
cte-761	33	7	results	result	NOUN
cte-761	33	8	creates	create	VERB
cte-761	33	9	additional	additional	ADJ
cte-761	33	10	conditions	condition	NOUN
cte-761	33	11	for	for	ADP
cte-761	33	12	improving	improve	VERB
cte-761	33	13	students	student	NOUN
cte-761	33	14	’	'	PUNCT
cte-761	33	15	knowledge	knowledge	NOUN
cte-761	33	16	,	,	PUNCT
cte-761	33	17	taking	take	VERB
cte-761	33	18	into	into	ADP
cte-761	33	19	account	account	NOUN
cte-761	33	20	the	the	DET
cte-761	33	21	specifics	specific	NOUN
cte-761	33	22	of	of	ADP
cte-761	33	23	their	their	PRON
cte-761	33	24	professional	professional	ADJ
cte-761	33	25	training	training	NOUN
cte-761	33	26	.	.	PUNCT
cte-761	34	1	the	the	DET
cte-761	34	2	main	main	ADJ
cte-761	34	3	aim	aim	NOUN
cte-761	34	4	of	of	ADP
cte-761	34	5	this	this	DET
cte-761	34	6	study	study	NOUN
cte-761	34	7	is	be	AUX
cte-761	34	8	to	to	PART
cte-761	34	9	further	far	ADV
cte-761	34	10	test	test	VERB
cte-761	34	11	the	the	DET
cte-761	34	12	hypothesis	hypothesis	NOUN
cte-761	34	13	that	that	PRON
cte-761	34	14	visual	visual	ADJ
cte-761	34	15	representation	representation	NOUN
cte-761	34	16	enhances	enhance	VERB
cte-761	34	17	educational	educational	ADJ
cte-761	34	18	achievements	achievement	NOUN
cte-761	34	19	of	of	ADP
cte-761	34	20	agricultural	agricultural	ADJ
cte-761	34	21	university	university	NOUN
cte-761	34	22	students	student	NOUN
cte-761	34	23	when	when	SCONJ
cte-761	34	24	studying	study	VERB
cte-761	34	25	mathematical	mathematical	ADJ
cte-761	34	26	modelling	modelling	NOUN
cte-761	34	27	fundamentals	fundamental	NOUN
cte-761	34	28	.	.	PUNCT
cte-761	35	1	specifically	specifically	ADV
cte-761	35	2	,	,	PUNCT
cte-761	35	3	we	we	PRON
cte-761	35	4	investigate	investigate	VERB
cte-761	35	5	the	the	DET
cte-761	35	6	effectiveness	effectiveness	NOUN
cte-761	35	7	of	of	ADP
cte-761	35	8	dynamic	dynamic	ADJ
cte-761	35	9	vector	vector	NOUN
cte-761	35	10	diagrams	diagram	NOUN
cte-761	35	11	as	as	ADP
cte-761	35	12	a	a	DET
cte-761	35	13	visualisation	visualisation	NOUN
cte-761	35	14	method	method	NOUN
cte-761	35	15	for	for	ADP
cte-761	35	16	depicting	depict	VERB
cte-761	35	17	mechanical	mechanical	ADJ
cte-761	35	18	movement	movement	NOUN
cte-761	35	19	characteristics	characteristic	NOUN
cte-761	35	20	.	.	PUNCT
cte-761	36	1	we	we	PRON
cte-761	36	2	describe	describe	VERB
cte-761	36	3	the	the	DET
cte-761	36	4	methodology	methodology	NOUN
cte-761	36	5	employed	employ	VERB
cte-761	36	6	,	,	PUNCT
cte-761	36	7	compare	compare	VERB
cte-761	36	8	the	the	DET
cte-761	36	9	current	current	ADJ
cte-761	36	10	research	research	NOUN
cte-761	36	11	results	result	NOUN
cte-761	36	12	with	with	ADP
cte-761	36	13	previous	previous	ADJ
cte-761	36	14	findings	finding	NOUN
cte-761	36	15	,	,	PUNCT
cte-761	36	16	and	and	CCONJ
cte-761	36	17	evaluate	evaluate	VERB
cte-761	36	18	student	student	NOUN
cte-761	36	19	educational	educational	ADJ
cte-761	36	20	achievements	achievement	NOUN
cte-761	36	21	using	use	VERB
cte-761	36	22	different	different	ADJ
cte-761	36	23	software	software	NOUN
cte-761	36	24	tools	tool	NOUN
cte-761	36	25	.	.	PUNCT
cte-761	37	1	2	2	X
cte-761	37	2	.	.	X
cte-761	37	3	experiment	experiment	NOUN
cte-761	37	4	description	description	NOUN
cte-761	37	5	2.1	2.1	NUM
cte-761	37	6	.	.	PUNCT
cte-761	38	1	general	general	ADJ
cte-761	38	2	design	design	NOUN
cte-761	38	3	this	this	DET
cte-761	38	4	study	study	NOUN
cte-761	38	5	presents	present	VERB
cte-761	38	6	results	result	NOUN
cte-761	38	7	from	from	ADP
cte-761	38	8	a	a	DET
cte-761	38	9	pedagogical	pedagogical	ADJ
cte-761	38	10	experiment	experiment	NOUN
cte-761	38	11	conducted	conduct	VERB
cte-761	38	12	between	between	ADP
cte-761	38	13	september	september	PROPN
cte-761	38	14	-	-	PUNCT
cte-761	38	15	november	november	PROPN
cte-761	38	16	2019	2019	NUM
cte-761	38	17	and	and	CCONJ
cte-761	38	18	september	september	PROPN
cte-761	38	19	-	-	PUNCT
cte-761	38	20	november	november	PROPN
cte-761	38	21	2020	2020	NUM
cte-761	38	22	.	.	PUNCT
cte-761	39	1	the	the	DET
cte-761	39	2	experiment	experiment	NOUN
cte-761	39	3	involved	involve	VERB
cte-761	39	4	167	167	NUM
cte-761	39	5	students	student	NOUN
cte-761	39	6	from	from	ADP
cte-761	39	7	the	the	DET
cte-761	39	8	faculty	faculty	NOUN
cte-761	39	9	of	of	ADP
cte-761	39	10	engineering	engineering	NOUN
cte-761	39	11	and	and	CCONJ
cte-761	39	12	technology	technology	NOUN
cte-761	39	13	at	at	ADP
cte-761	39	14	poltava	poltava	PROPN
cte-761	39	15	state	state	PROPN
cte-761	39	16	agrarian	agrarian	PROPN
cte-761	39	17	academy	academy	PROPN
cte-761	39	18	.	.	PUNCT
cte-761	40	1	this	this	DET
cte-761	40	2	sample	sample	NOUN
cte-761	40	3	size	size	NOUN
cte-761	40	4	enabled	enable	VERB
cte-761	40	5	us	we	PRON
cte-761	40	6	to	to	PART
cte-761	40	7	establish	establish	VERB
cte-761	40	8	significant	significant	ADJ
cte-761	40	9	differences	difference	NOUN
cte-761	40	10	between	between	ADP
cte-761	40	11	group	group	NOUN
cte-761	40	12	means	mean	VERB
cte-761	40	13	at	at	ADP
cte-761	40	14	the	the	DET
cte-761	40	15	level	level	NOUN
cte-761	40	16	of	of	ADP
cte-761	40	17	1	1	NUM
cte-761	40	18	point	point	NOUN
cte-761	40	19	using	use	VERB
cte-761	40	20	anova	anova	PROPN
cte-761	40	21	,	,	PUNCT
cte-761	40	22	with	with	ADP
cte-761	40	23	a	a	DET
cte-761	40	24	significance	significance	NOUN
cte-761	40	25	level	level	NOUN
cte-761	40	26	of	of	ADP
cte-761	40	27	0.05	0.05	NUM
cte-761	40	28	,	,	PUNCT
cte-761	40	29	three	three	NUM
cte-761	40	30	comparison	comparison	NOUN
cte-761	40	31	groups	group	NOUN
cte-761	40	32	,	,	PUNCT
cte-761	40	33	and	and	CCONJ
cte-761	40	34	a	a	DET
cte-761	40	35	statistical	statistical	ADJ
cte-761	40	36	power	power	NOUN
cte-761	40	37	of	of	ADP
cte-761	40	38	80	80	NUM
cte-761	40	39	%	%	NOUN
cte-761	40	40	.	.	PUNCT
cte-761	41	1	at	at	ADP
cte-761	41	2	the	the	DET
cte-761	41	3	experiment	experiment	NOUN
cte-761	41	4	’s	’s	PART
cte-761	41	5	commencement	commencement	NOUN
cte-761	41	6	,	,	PUNCT
cte-761	41	7	students	student	NOUN
cte-761	41	8	were	be	AUX
cte-761	41	9	categorised	categorise	VERB
cte-761	41	10	into	into	ADP
cte-761	41	11	three	three	NUM
cte-761	41	12	groups	group	NOUN
cte-761	41	13	(	(	PUNCT
cte-761	41	14	e	e	NOUN
cte-761	41	15	,	,	PUNCT
cte-761	41	16	g	g	NOUN
cte-761	41	17	,	,	PUNCT
cte-761	41	18	and	and	CCONJ
cte-761	41	19	eg	eg	NOUN
cte-761	41	20	)	)	PUNCT
cte-761	41	21	using	use	VERB
cte-761	41	22	a	a	DET
cte-761	41	23	methodology	methodology	NOUN
cte-761	41	24	that	that	PRON
cte-761	41	25	ensured	ensure	VERB
cte-761	41	26	initial	initial	ADJ
cte-761	41	27	homogeneity	homogeneity	NOUN
cte-761	41	28	based	base	VERB
cte-761	41	29	on	on	ADP
cte-761	41	30	academic	academic	ADJ
cte-761	41	31	performance	performance	NOUN
cte-761	41	32	criteria	criterion	NOUN
cte-761	41	33	[	[	X
cte-761	41	34	5	5	NUM
cte-761	41	35	]	]	PUNCT
cte-761	41	36	.	.	PUNCT
cte-761	42	1	group	group	NOUN
cte-761	42	2	e	e	PROPN
cte-761	42	3	utilised	utilise	VERB
cte-761	42	4	excel	excel	VERB
cte-761	42	5	spreadsheets	spreadsheet	NOUN
cte-761	42	6	during	during	ADP
cte-761	42	7	mathematical	mathematical	ADJ
cte-761	42	8	modelling	modelling	NOUN
cte-761	42	9	instruction	instruction	NOUN
cte-761	42	10	.	.	PUNCT
cte-761	43	1	group	group	NOUN
cte-761	43	2	g	g	PROPN
cte-761	43	3	employed	employ	VERB
cte-761	43	4	geogebra	geogebra	NOUN
cte-761	43	5	software	software	NOUN
cte-761	43	6	.	.	PUNCT
cte-761	44	1	group	group	NOUN
cte-761	44	2	eg	eg	NOUN
cte-761	44	3	simultaneously	simultaneously	ADV
cte-761	44	4	used	use	VERB
cte-761	44	5	both	both	CCONJ
cte-761	44	6	excel	excel	VERB
cte-761	44	7	and	and	CCONJ
cte-761	44	8	geogebra	geogebra	NOUN
cte-761	44	9	.	.	PUNCT
cte-761	45	1	throughout	throughout	ADP
cte-761	45	2	the	the	DET
cte-761	45	3	study	study	NOUN
cte-761	45	4	,	,	PUNCT
cte-761	45	5	students	student	NOUN
cte-761	45	6	could	could	AUX
cte-761	45	7	migrate	migrate	VERB
cte-761	45	8	between	between	ADP
cte-761	45	9	experimental	experimental	ADJ
cte-761	45	10	groups	group	NOUN
cte-761	45	11	while	while	SCONJ
cte-761	45	12	remaining	remain	VERB
cte-761	45	13	in	in	ADP
cte-761	45	14	their	their	PRON
cte-761	45	15	academic	academic	ADJ
cte-761	45	16	cohorts	cohort	NOUN
cte-761	45	17	,	,	PUNCT
cte-761	45	18	with	with	ADP
cte-761	45	19	final	final	ADJ
cte-761	45	20	group	group	NOUN
cte-761	45	21	composition	composition	NOUN
cte-761	45	22	documented	document	VERB
cte-761	45	23	at	at	ADP
cte-761	45	24	the	the	DET
cte-761	45	25	experiment	experiment	NOUN
cte-761	45	26	’s	’s	PART
cte-761	45	27	conclusion	conclusion	NOUN
cte-761	45	28	.	.	PUNCT
cte-761	46	1	all	all	DET
cte-761	46	2	groups	group	NOUN
cte-761	46	3	received	receive	VERB
cte-761	46	4	instruction	instruction	NOUN
cte-761	46	5	following	follow	VERB
cte-761	46	6	a	a	DET
cte-761	46	7	single	single	ADJ
cte-761	46	8	curriculum	curriculum	NOUN
cte-761	46	9	using	use	VERB
cte-761	46	10	a	a	DET
cte-761	46	11	differentiated	differentiated	ADJ
cte-761	46	12	approach	approach	NOUN
cte-761	46	13	to	to	ADP
cte-761	46	14	teaching	teaching	NOUN
cte-761	46	15	as	as	SCONJ
cte-761	46	16	described	describe	VERB
cte-761	46	17	in	in	ADP
cte-761	46	18	our	our	PRON
cte-761	46	19	previous	previous	ADJ
cte-761	46	20	research	research	NOUN
cte-761	46	21	[	[	X
cte-761	46	22	2	2	NUM
cte-761	46	23	]	]	PUNCT
cte-761	46	24	.	.	PUNCT
cte-761	47	1	the	the	DET
cte-761	47	2	training	training	NOUN
cte-761	47	3	duration	duration	NOUN
cte-761	47	4	was	be	AUX
cte-761	47	5	identical	identical	ADJ
cte-761	47	6	across	across	ADP
cte-761	47	7	all	all	DET
cte-761	47	8	groups	group	NOUN
cte-761	47	9	.	.	PUNCT
cte-761	48	1	each	each	DET
cte-761	48	2	group	group	NOUN
cte-761	48	3	completed	complete	VERB
cte-761	48	4	the	the	DET
cte-761	48	5	same	same	ADJ
cte-761	48	6	learning	learning	NOUN
cte-761	48	7	task	task	NOUN
cte-761	48	8	directly	directly	ADV
cte-761	48	9	relevant	relevant	ADJ
cte-761	48	10	to	to	ADP
cte-761	48	11	agricultural	agricultural	ADJ
cte-761	48	12	engineering	engineering	NOUN
cte-761	48	13	training	training	NOUN
cte-761	48	14	:	:	PUNCT
cte-761	48	15	modelling	model	VERB
cte-761	48	16	the	the	DET
cte-761	48	17	movement	movement	NOUN
cte-761	48	18	of	of	ADP
cte-761	48	19	a	a	DET
cte-761	48	20	spherical	spherical	ADJ
cte-761	48	21	object	object	NOUN
cte-761	48	22	projected	project	VERB
cte-761	48	23	at	at	ADP
cte-761	48	24	an	an	DET
cte-761	48	25	angle	angle	NOUN
cte-761	48	26	to	to	ADP
cte-761	48	27	the	the	DET
cte-761	48	28	horizon	horizon	NOUN
cte-761	48	29	,	,	PUNCT
cte-761	48	30	accounting	account	VERB
cte-761	48	31	for	for	ADP
cte-761	48	32	air	air	NOUN
cte-761	48	33	resistance	resistance	NOUN
cte-761	48	34	,	,	PUNCT
cte-761	48	35	gravitational	gravitational	ADJ
cte-761	48	36	interaction	interaction	NOUN
cte-761	48	37	,	,	PUNCT
cte-761	48	38	electrostatic	electrostatic	ADJ
cte-761	48	39	interaction	interaction	NOUN
cte-761	48	40	,	,	PUNCT
cte-761	48	41	and	and	CCONJ
cte-761	48	42	magnetic	magnetic	ADJ
cte-761	48	43	interaction	interaction	NOUN
cte-761	48	44	.	.	PUNCT
cte-761	49	1	we	we	PRON
cte-761	49	2	implemented	implement	VERB
cte-761	49	3	a	a	DET
cte-761	49	4	differentiated	differentiated	ADJ
cte-761	49	5	approach	approach	NOUN
cte-761	49	6	through	through	ADP
cte-761	49	7	the	the	DET
cte-761	49	8	gradual	gradual	ADJ
cte-761	49	9	construction	construction	NOUN
cte-761	49	10	of	of	ADP
cte-761	49	11	mathematical	mathematical	ADJ
cte-761	49	12	models	model	NOUN
cte-761	49	13	(	(	PUNCT
cte-761	49	14	mm	mm	NOUN
cte-761	49	15	)	)	PUNCT
cte-761	49	16	with	with	ADP
cte-761	49	17	increasing	increase	VERB
cte-761	49	18	complexity	complexity	NOUN
cte-761	49	19	levels	level	NOUN
cte-761	49	20	,	,	PUNCT
cte-761	49	21	their	their	PRON
cte-761	49	22	computer	computer	NOUN
cte-761	49	23	implementation	implementation	NOUN
cte-761	49	24	,	,	PUNCT
cte-761	49	25	and	and	CCONJ
cte-761	49	26	analysis	analysis	NOUN
cte-761	49	27	through	through	ADP
cte-761	49	28	numerical	numerical	ADJ
cte-761	49	29	calculations	calculation	NOUN
cte-761	49	30	.	.	PUNCT
cte-761	50	1	the	the	DET
cte-761	50	2	purpose	purpose	NOUN
cte-761	50	3	was	be	AUX
cte-761	50	4	to	to	PART
cte-761	50	5	familiarise	familiarise	VERB
cte-761	50	6	students	student	NOUN
cte-761	50	7	with	with	ADP
cte-761	50	8	mathematical	mathematical	ADJ
cte-761	50	9	model	model	NOUN
cte-761	50	10	creation	creation	NOUN
cte-761	50	11	practices	practice	NOUN
cte-761	50	12	,	,	PUNCT
cte-761	50	13	computational	computational	ADJ
cte-761	50	14	algorithm	algorithm	NOUN
cte-761	50	15	development	development	NOUN
cte-761	50	16	,	,	PUNCT
cte-761	50	17	computer	computer	NOUN
cte-761	50	18	experiment	experiment	NOUN
cte-761	50	19	execution	execution	NOUN
cte-761	50	20	,	,	PUNCT
cte-761	50	21	and	and	CCONJ
cte-761	50	22	result	result	VERB
cte-761	50	23	interpretation	interpretation	NOUN
cte-761	50	24	.	.	PUNCT
cte-761	51	1	student	student	NOUN
cte-761	51	2	learning	learn	VERB
cte-761	51	3	outcomes	outcome	NOUN
cte-761	51	4	were	be	AUX
cte-761	51	5	assessed	assess	VERB
cte-761	51	6	on	on	ADP
cte-761	51	7	a	a	DET
cte-761	51	8	100	100	NUM
cte-761	51	9	-	-	PUNCT
cte-761	51	10	point	point	NOUN
cte-761	51	11	scale	scale	NOUN
cte-761	51	12	based	base	VERB
cte-761	51	13	on	on	ADP
cte-761	51	14	performance	performance	NOUN
cte-761	51	15	in	in	ADP
cte-761	51	16	individual	individual	ADJ
cte-761	51	17	assignments	assignment	NOUN
cte-761	51	18	[	[	X
cte-761	51	19	2	2	NUM
cte-761	51	20	]	]	PUNCT
cte-761	51	21	.	.	PUNCT
cte-761	52	1	2.2	2.2	NUM
cte-761	52	2	.	.	PUNCT
cte-761	52	3	mathematical	mathematical	ADJ
cte-761	52	4	model	model	NOUN
cte-761	52	5	development	development	NOUN
cte-761	52	6	students	student	NOUN
cte-761	52	7	were	be	AUX
cte-761	52	8	expected	expect	VERB
cte-761	52	9	to	to	PART
cte-761	52	10	employ	employ	VERB
cte-761	52	11	competencies	competency	NOUN
cte-761	52	12	acquired	acquire	VERB
cte-761	52	13	from	from	ADP
cte-761	52	14	previous	previous	ADJ
cte-761	52	15	coursework	coursework	NOUN
cte-761	52	16	in	in	ADP
cte-761	52	17	“	"	PUNCT
cte-761	52	18	higher	high	ADJ
cte-761	52	19	mathematics	mathematic	NOUN
cte-761	52	20	”	"	PUNCT
cte-761	52	21	,	,	PUNCT
cte-761	52	22	“	"	PUNCT
cte-761	52	23	physics	physics	NOUN
cte-761	52	24	”	"	PUNCT
cte-761	52	25	,	,	PUNCT
cte-761	52	26	“	"	PUNCT
cte-761	52	27	theoretical	theoretical	ADJ
cte-761	52	28	mechanics	mechanic	NOUN
cte-761	52	29	”	"	PUNCT
cte-761	52	30	,	,	PUNCT
cte-761	52	31	and	and	CCONJ
cte-761	52	32	“	"	PUNCT
cte-761	52	33	applied	applied	ADJ
cte-761	52	34	mathematics	mathematic	NOUN
cte-761	52	35	”	"	PUNCT
cte-761	52	36	when	when	SCONJ
cte-761	52	37	constructing	construct	VERB
cte-761	52	38	and	and	CCONJ
cte-761	52	39	analysing	analyse	VERB
cte-761	52	40	mathematical	mathematical	ADJ
cte-761	52	41	models	model	NOUN
cte-761	52	42	.	.	PUNCT
cte-761	53	1	236	236	NUM
cte-761	53	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	53	3	cte	cte	NOUN
cte-761	53	4	workshop	workshop	NOUN
cte-761	53	5	proceedings	proceeding	NOUN
cte-761	53	6	,	,	PUNCT
cte-761	53	7	2025	2025	NUM
cte-761	53	8	,	,	PUNCT
cte-761	53	9	vol	vol	NOUN
cte-761	53	10	.	.	PROPN
cte-761	53	11	12	12	NUM
cte-761	53	12	,	,	PUNCT
cte-761	53	13	pp	pp	ADJ
cte-761	53	14	.	.	PUNCT
cte-761	54	1	235–252	235–252	NUM
cte-761	55	1	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	55	2	following	follow	VERB
cte-761	55	3	an	an	DET
cte-761	55	4	algorithm	algorithm	NOUN
cte-761	55	5	detailed	detail	VERB
cte-761	55	6	in	in	ADP
cte-761	55	7	previous	previous	ADJ
cte-761	55	8	publications	publication	NOUN
cte-761	55	9	[	[	X
cte-761	55	10	2	2	NUM
cte-761	55	11	,	,	PUNCT
cte-761	55	12	4	4	NUM
cte-761	55	13	]	]	PUNCT
cte-761	55	14	,	,	PUNCT
cte-761	55	15	students	student	NOUN
cte-761	55	16	developed	develop	VERB
cte-761	55	17	three	three	NUM
cte-761	55	18	mathematical	mathematical	ADJ
cte-761	55	19	models	model	NOUN
cte-761	55	20	of	of	ADP
cte-761	55	21	increasing	increase	VERB
cte-761	55	22	complexity	complexity	NOUN
cte-761	55	23	(	(	PUNCT
cte-761	55	24	mm	mm	INTJ
cte-761	55	25	i	i	PROPN
cte-761	55	26	,	,	PUNCT
cte-761	55	27	mm	mm	PROPN
cte-761	55	28	ii	ii	PROPN
cte-761	55	29	,	,	PUNCT
cte-761	55	30	and	and	CCONJ
cte-761	55	31	mm	mm	PROPN
cte-761	55	32	iii	iii	PROPN
cte-761	55	33	)	)	PUNCT
cte-761	55	34	.	.	PUNCT
cte-761	56	1	these	these	DET
cte-761	56	2	models	model	NOUN
cte-761	56	3	were	be	AUX
cte-761	56	4	constructed	construct	VERB
cte-761	56	5	by	by	ADP
cte-761	56	6	analysing	analyse	VERB
cte-761	56	7	the	the	DET
cte-761	56	8	vector	vector	NOUN
cte-761	56	9	equations	equation	NOUN
cte-761	56	10	of	of	ADP
cte-761	56	11	resultant	resultant	NOUN
cte-761	56	12	forces	force	NOUN
cte-761	56	13	acting	act	VERB
cte-761	56	14	on	on	ADP
cte-761	56	15	a	a	DET
cte-761	56	16	projectile	projectile	NOUN
cte-761	56	17	under	under	ADP
cte-761	56	18	different	different	ADJ
cte-761	56	19	conditions	condition	NOUN
cte-761	56	20	.	.	PUNCT
cte-761	57	1	mm	mm	INTJ
cte-761	57	2	i	i	PRON
cte-761	57	3	describes	describe	VERB
cte-761	57	4	the	the	DET
cte-761	57	5	motion	motion	NOUN
cte-761	57	6	of	of	ADP
cte-761	57	7	an	an	DET
cte-761	57	8	object	object	NOUN
cte-761	57	9	projected	project	VERB
cte-761	57	10	at	at	ADP
cte-761	57	11	an	an	DET
cte-761	57	12	angle	angle	NOUN
cte-761	57	13	to	to	ADP
cte-761	57	14	the	the	DET
cte-761	57	15	horizon	horizon	NOUN
cte-761	57	16	without	without	ADP
cte-761	57	17	air	air	NOUN
cte-761	57	18	resistance	resistance	NOUN
cte-761	57	19	.	.	PUNCT
cte-761	58	1	based	base	VERB
cte-761	58	2	on	on	ADP
cte-761	58	3	the	the	DET
cte-761	58	4	mechanical	mechanical	ADJ
cte-761	58	5	interpretation	interpretation	NOUN
cte-761	58	6	of	of	ADP
cte-761	58	7	derivatives	derivative	NOUN
cte-761	58	8	,	,	PUNCT
cte-761	58	9	all	all	DET
cte-761	58	10	models	model	NOUN
cte-761	58	11	included	include	VERB
cte-761	58	12	two	two	NUM
cte-761	58	13	first	first	ADJ
cte-761	58	14	-	-	PUNCT
cte-761	58	15	order	order	NOUN
cte-761	58	16	ordinary	ordinary	ADJ
cte-761	58	17	differential	differential	ADJ
cte-761	58	18	equations	equation	NOUN
cte-761	58	19	:	:	PUNCT
cte-761	58	20	𝑑𝑥	𝑑𝑥	VERB
cte-761	58	21	𝑑𝑡	𝑑𝑡	ADP
cte-761	58	22	=	=	PUNCT
cte-761	58	23	𝑣𝑥	𝑣𝑥	NOUN
cte-761	58	24	,	,	PUNCT
cte-761	58	25	𝑑𝑦	𝑑𝑦	NOUN
cte-761	58	26	𝑑𝑡	𝑑𝑡	ADP
cte-761	58	27	=	=	SYM
cte-761	58	28	𝑣𝑦.	𝑣𝑦.	ADJ
cte-761	58	29	(	(	PUNCT
cte-761	58	30	1	1	NUM
cte-761	58	31	)	)	PUNCT
cte-761	58	32	when	when	SCONJ
cte-761	58	33	an	an	DET
cte-761	58	34	object	object	NOUN
cte-761	58	35	projected	project	VERB
cte-761	58	36	at	at	ADP
cte-761	58	37	an	an	DET
cte-761	58	38	angle	angle	NOUN
cte-761	58	39	to	to	ADP
cte-761	58	40	the	the	DET
cte-761	58	41	horizon	horizon	NOUN
cte-761	58	42	is	be	AUX
cte-761	58	43	affected	affect	VERB
cte-761	58	44	only	only	ADV
cte-761	58	45	by	by	ADP
cte-761	58	46	gravitational	gravitational	ADJ
cte-761	58	47	force	force	NOUN
cte-761	58	48	𝐹	𝐹	PROPN
cte-761	58	49	𝑇	𝑇	PROPN
cte-761	59	1	=	=	SYM
cte-761	59	2	(	(	PUNCT
cte-761	59	3	𝐹	𝐹	PROPN
cte-761	59	4	𝑥	𝑥	PROPN
cte-761	59	5	1	1	NUM
cte-761	59	6	,	,	PUNCT
cte-761	59	7	𝐹	𝐹	PROPN
cte-761	59	8	𝑦	𝑦	NOUN
cte-761	59	9	1	1	NUM
cte-761	59	10	)	)	PUNCT
cte-761	59	11	where	where	SCONJ
cte-761	59	12	𝐹	𝐹	PROPN
cte-761	59	13	𝑥	𝑥	VERB
cte-761	59	14	1	1	NUM
cte-761	59	15	=	=	SYM
cte-761	59	16	0	0	NUM
cte-761	60	1	and	and	CCONJ
cte-761	60	2	𝐹	𝐹	PROPN
cte-761	60	3	𝑦	𝑦	NOUN
cte-761	60	4	1	1	X
cte-761	60	5	=	=	SYM
cte-761	60	6	−𝑚𝑔	−𝑚𝑔	PROPN
cte-761	60	7	,	,	PUNCT
cte-761	60	8	two	two	NUM
cte-761	60	9	additional	additional	ADJ
cte-761	60	10	equations	equation	NOUN
cte-761	60	11	are	be	AUX
cte-761	60	12	added	add	VERB
cte-761	60	13	to	to	ADP
cte-761	60	14	mm	mm	PROPN
cte-761	60	15	i	i	NOUN
cte-761	60	16	:	:	PUNCT
cte-761	60	17	𝑑𝑣𝑥	𝑑𝑣𝑥	NOUN
cte-761	60	18	𝑑𝑡	𝑑𝑡	ADP
cte-761	60	19	=	=	SYM
cte-761	60	20	0	0	PROPN
cte-761	60	21	,	,	PUNCT
cte-761	60	22	𝑑𝑣𝑦	𝑑𝑣𝑦	VERB
cte-761	60	23	𝑑𝑡	𝑑𝑡	ADP
cte-761	60	24	=	=	SYM
cte-761	60	25	−𝑔.	−𝑔.	X
cte-761	60	26	(	(	PUNCT
cte-761	60	27	2	2	X
cte-761	60	28	)	)	PUNCT
cte-761	60	29	mm	mm	NOUN
cte-761	61	1	i	i	PROPN
cte-761	61	2	,	,	PUNCT
cte-761	61	3	therefore	therefore	ADV
cte-761	61	4	,	,	PUNCT
cte-761	61	5	comprises	comprise	VERB
cte-761	61	6	four	four	NUM
cte-761	61	7	differential	differential	ADJ
cte-761	61	8	equations	equation	NOUN
cte-761	61	9	(	(	PUNCT
cte-761	61	10	1	1	NUM
cte-761	61	11	)	)	PUNCT
cte-761	61	12	and	and	CCONJ
cte-761	61	13	(	(	PUNCT
cte-761	61	14	2	2	NUM
cte-761	61	15	)	)	PUNCT
cte-761	61	16	,	,	PUNCT
cte-761	61	17	representing	represent	VERB
cte-761	61	18	the	the	DET
cte-761	61	19	simplest	simple	ADJ
cte-761	61	20	model	model	NOUN
cte-761	61	21	describing	describe	VERB
cte-761	61	22	projectile	projectile	NOUN
cte-761	61	23	motion	motion	NOUN
cte-761	61	24	in	in	ADP
cte-761	61	25	a	a	DET
cte-761	61	26	gravitational	gravitational	ADJ
cte-761	61	27	field	field	NOUN
cte-761	61	28	without	without	ADP
cte-761	61	29	air	air	NOUN
cte-761	61	30	resistance	resistance	NOUN
cte-761	61	31	.	.	PUNCT
cte-761	62	1	this	this	PRON
cte-761	62	2	corresponds	correspond	VERB
cte-761	62	3	to	to	ADP
cte-761	62	4	the	the	DET
cte-761	62	5	first	first	ADJ
cte-761	62	6	(	(	PUNCT
cte-761	62	7	lowest	low	ADJ
cte-761	62	8	)	)	PUNCT
cte-761	62	9	difficulty	difficulty	NOUN
cte-761	62	10	level	level	NOUN
cte-761	62	11	in	in	ADP
cte-761	62	12	mathematical	mathematical	ADJ
cte-761	62	13	modelling	modelling	NOUN
cte-761	62	14	instruction	instruction	NOUN
cte-761	62	15	.	.	PUNCT
cte-761	63	1	mm	mm	PROPN
cte-761	63	2	ii	ii	PROPN
cte-761	63	3	introduces	introduce	VERB
cte-761	63	4	medium	medium	ADJ
cte-761	63	5	resistance	resistance	NOUN
cte-761	63	6	,	,	PUNCT
cte-761	63	7	representing	represent	VERB
cte-761	63	8	the	the	DET
cte-761	63	9	second	second	ADJ
cte-761	63	10	complexity	complexity	NOUN
cte-761	63	11	level	level	NOUN
cte-761	63	12	.	.	PUNCT
cte-761	64	1	here	here	ADV
cte-761	64	2	,	,	PUNCT
cte-761	64	3	the	the	DET
cte-761	64	4	equivalent	equivalent	ADJ
cte-761	64	5	force	force	NOUN
cte-761	64	6	acting	act	VERB
cte-761	64	7	on	on	ADP
cte-761	64	8	the	the	DET
cte-761	64	9	object	object	NOUN
cte-761	64	10	is	be	AUX
cte-761	64	11	𝐹	𝐹	PROPN
cte-761	64	12	2	2	NUM
cte-761	64	13	=	=	SYM
cte-761	64	14	𝐹	𝐹	PROPN
cte-761	64	15	𝑇	𝑇	PROPN
cte-761	64	16	+	+	CCONJ
cte-761	64	17	𝐹	𝐹	PROPN
cte-761	64	18	𝑟	𝑟	NOUN
cte-761	64	19	,	,	PUNCT
cte-761	64	20	where	where	SCONJ
cte-761	64	21	𝐹	𝐹	PROPN
cte-761	64	22	𝑟	𝑟	X
cte-761	64	23	=	=	SYM
cte-761	64	24	−𝑘2𝑣	−𝑘2𝑣	NUM
cte-761	64	25	2	2	NUM
cte-761	64	26	𝑣	𝑣	PRON
cte-761	64	27	𝑣	𝑣	PROPN
cte-761	64	28	represents	represent	VERB
cte-761	64	29	air	air	NOUN
cte-761	64	30	resistance	resistance	NOUN
cte-761	64	31	force	force	NOUN
cte-761	64	32	,	,	PUNCT
cte-761	64	33	𝑣	𝑣	X
cte-761	64	34	=	=	VERB
cte-761	64	35	√︁	√︁	PROPN
cte-761	64	36	𝑣2𝑥	𝑣2𝑥	PROPN
cte-761	64	37	+	+	CCONJ
cte-761	64	38	𝑣2𝑦	𝑣2𝑦	VERB
cte-761	64	39	is	be	AUX
cte-761	64	40	object	object	NOUN
cte-761	64	41	velocity	velocity	NOUN
cte-761	64	42	,	,	PUNCT
cte-761	64	43	and	and	CCONJ
cte-761	64	44	𝑘2	𝑘2	PROPN
cte-761	64	45	is	be	AUX
cte-761	64	46	the	the	DET
cte-761	64	47	medium	medium	ADJ
cte-761	64	48	resistance	resistance	NOUN
cte-761	64	49	coefficient	coefficient	NOUN
cte-761	64	50	.	.	PUNCT
cte-761	65	1	from	from	ADP
cte-761	65	2	this	this	DET
cte-761	65	3	relationship	relationship	NOUN
cte-761	65	4	:	:	PUNCT
cte-761	65	5	𝐹	𝐹	PROPN
cte-761	65	6	𝑥	𝑥	PROPN
cte-761	65	7	2	2	NUM
cte-761	65	8	=	=	SYM
cte-761	65	9	−𝑘2𝑣𝑥𝑣	−𝑘2𝑣𝑥𝑣	PROPN
cte-761	65	10	,	,	PUNCT
cte-761	65	11	𝐹	𝐹	PROPN
cte-761	65	12	𝑦	𝑦	NOUN
cte-761	65	13	2	2	NUM
cte-761	65	14	=	=	SYM
cte-761	65	15	−𝑔−	−𝑔−	PROPN
cte-761	65	16	𝑘2𝑣𝑦𝑣.	𝑘2𝑣𝑦𝑣.	PROPN
cte-761	65	17	therefore	therefore	ADV
cte-761	65	18	,	,	PUNCT
cte-761	65	19	in	in	ADP
cte-761	65	20	addition	addition	NOUN
cte-761	65	21	to	to	ADP
cte-761	65	22	equation	equation	NOUN
cte-761	65	23	(	(	PUNCT
cte-761	65	24	1	1	NUM
cte-761	65	25	)	)	PUNCT
cte-761	65	26	,	,	PUNCT
cte-761	65	27	mm	mm	PROPN
cte-761	65	28	ii	ii	PROPN
cte-761	65	29	includes	include	VERB
cte-761	65	30	:	:	PUNCT
cte-761	65	31	𝑑𝑣𝑥	𝑑𝑣𝑥	NOUN
cte-761	65	32	𝑑𝑡	𝑑𝑡	ADP
cte-761	65	33	=	=	SYM
cte-761	65	34	−𝑘2	−𝑘2	PROPN
cte-761	65	35	𝑚	𝑚	X
cte-761	65	36	𝑣𝑥	𝑣𝑥	VERB
cte-761	65	37	√︁	√︁	PROPN
cte-761	65	38	𝑣2𝑥	𝑣2𝑥	PROPN
cte-761	65	39	+	+	CCONJ
cte-761	65	40	𝑣2𝑦	𝑣2𝑦	NUM
cte-761	65	41	,	,	PUNCT
cte-761	65	42	𝑑𝑣𝑦	𝑑𝑣𝑦	VERB
cte-761	65	43	𝑑𝑡	𝑑𝑡	ADP
cte-761	65	44	=	=	PUNCT
cte-761	65	45	−𝑔	−𝑔	PROPN
cte-761	65	46	−	−	PROPN
cte-761	65	47	𝑘2	𝑘2	PROPN
cte-761	65	48	𝑚	𝑚	PROPN
cte-761	65	49	𝑣𝑦	𝑣𝑦	PROPN
cte-761	65	50	√︁	√︁	PROPN
cte-761	65	51	𝑣2𝑥	𝑣2𝑥	PROPN
cte-761	66	1	+	+	X
cte-761	66	2	𝑣2𝑦	𝑣2𝑦	NUM
cte-761	66	3	.	.	PUNCT
cte-761	67	1	(	(	PUNCT
cte-761	67	2	3	3	X
cte-761	67	3	)	)	PUNCT
cte-761	67	4	mm	mm	PROPN
cte-761	67	5	iii	iii	NOUN
cte-761	67	6	represents	represent	VERB
cte-761	67	7	the	the	DET
cte-761	67	8	highest	high	ADJ
cte-761	67	9	complexity	complexity	NOUN
cte-761	67	10	level	level	NOUN
cte-761	67	11	,	,	PUNCT
cte-761	67	12	incorporating	incorporate	VERB
cte-761	67	13	the	the	DET
cte-761	67	14	magnus	magnus	PROPN
cte-761	67	15	effect	effect	NOUN
cte-761	67	16	.	.	PUNCT
cte-761	68	1	the	the	DET
cte-761	68	2	equivalent	equivalent	ADJ
cte-761	68	3	force	force	NOUN
cte-761	68	4	becomes	become	VERB
cte-761	68	5	𝐹3	𝐹3	PROPN
cte-761	69	1	=	=	PUNCT
cte-761	69	2	𝐹	𝐹	PROPN
cte-761	69	3	𝑇	𝑇	PROPN
cte-761	69	4	+	+	CCONJ
cte-761	69	5	𝐹	𝐹	PROPN
cte-761	69	6	𝑟	𝑟	NOUN
cte-761	69	7	+	+	CCONJ
cte-761	69	8	𝐹𝑀	𝐹𝑀	PROPN
cte-761	69	9	,	,	PUNCT
cte-761	69	10	where	where	SCONJ
cte-761	69	11	𝐹𝑀	𝐹𝑀	NOUN
cte-761	69	12	=	=	SYM
cte-761	69	13	−𝑘3𝑣	−𝑘3𝑣	NUM
cte-761	69	14	2	2	NUM
cte-761	69	15	(	(	PUNCT
cte-761	69	16	𝜔	𝜔	SYM
cte-761	69	17	×	×	NOUN
cte-761	69	18	𝑣	𝑣	NOUN
cte-761	69	19	)	)	PUNCT
cte-761	69	20	and	and	CCONJ
cte-761	69	21	𝑘3	𝑘3	PROPN
cte-761	69	22	is	be	AUX
cte-761	69	23	the	the	DET
cte-761	69	24	magnus	magnus	PROPN
cte-761	69	25	effect	effect	NOUN
cte-761	69	26	coefficient	coefficient	NOUN
cte-761	69	27	.	.	PUNCT
cte-761	70	1	this	this	DET
cte-761	70	2	yields	yield	VERB
cte-761	70	3	:	:	PUNCT
cte-761	70	4	𝐹	𝐹	PROPN
cte-761	70	5	𝑥	𝑥	PROPN
cte-761	70	6	3	3	NUM
cte-761	70	7	=	=	SYM
cte-761	70	8	−𝑘2𝑣𝑥𝑣	−𝑘2𝑣𝑥𝑣	PROPN
cte-761	70	9	±	±	NUM
cte-761	70	10	𝑘3𝑣𝑦𝑣	𝑘3𝑣𝑦𝑣	PROPN
cte-761	70	11	,	,	PUNCT
cte-761	70	12	𝐹	𝐹	PROPN
cte-761	70	13	𝑦	𝑦	NOUN
cte-761	70	14	3	3	X
cte-761	70	15	=	=	SYM
cte-761	70	16	−𝑔	−𝑔	NOUN
cte-761	70	17	−	−	PROPN
cte-761	71	1	𝑘2𝑣𝑦𝑣	𝑘2𝑣𝑦𝑣	INTJ
cte-761	71	2	∓	∓	PROPN
cte-761	71	3	𝑘3𝑣𝑥𝑣	𝑘3𝑣𝑥𝑣	INTJ
cte-761	72	1	(	(	PUNCT
cte-761	72	2	the	the	DET
cte-761	72	3	positive	positive	ADJ
cte-761	72	4	sign	sign	NOUN
cte-761	72	5	applies	apply	VERB
cte-761	72	6	for	for	ADP
cte-761	72	7	clockwise	clockwise	NOUN
cte-761	72	8	rotation	rotation	NOUN
cte-761	72	9	,	,	PUNCT
cte-761	72	10	the	the	DET
cte-761	72	11	negative	negative	ADJ
cte-761	72	12	sign	sign	NOUN
cte-761	72	13	for	for	ADP
cte-761	72	14	counterclockwise	counterclockwise	NOUN
cte-761	72	15	rotation	rotation	NOUN
cte-761	72	16	)	)	PUNCT
cte-761	72	17	.	.	PUNCT
cte-761	73	1	in	in	ADP
cte-761	73	2	addition	addition	NOUN
cte-761	73	3	to	to	ADP
cte-761	73	4	equation	equation	NOUN
cte-761	73	5	(	(	PUNCT
cte-761	73	6	1	1	NUM
cte-761	73	7	)	)	PUNCT
cte-761	73	8	,	,	PUNCT
cte-761	73	9	mm	mm	PROPN
cte-761	73	10	iii	iii	NOUN
cte-761	73	11	includes	include	VERB
cte-761	73	12	:	:	PUNCT
cte-761	73	13	𝑑𝑣𝑥	𝑑𝑣𝑥	NOUN
cte-761	73	14	𝑑𝑡	𝑑𝑡	ADP
cte-761	73	15	=	=	SYM
cte-761	73	16	(	(	PUNCT
cte-761	73	17	︂	︂	PUNCT
cte-761	73	18	−𝑘2	−𝑘2	NOUN
cte-761	73	19	𝑚	𝑚	ADP
cte-761	73	20	𝑣𝑥	𝑣𝑥	VERB
cte-761	73	21	±	±	NUM
cte-761	73	22	𝑘3	𝑘3	PROPN
cte-761	73	23	𝑚	𝑚	PROPN
cte-761	73	24	𝑣𝑦	𝑣𝑦	PROPN
cte-761	73	25	)	)	PUNCT
cte-761	73	26	︂√︁	︂√︁	NOUN
cte-761	73	27	𝑣2𝑥	𝑣2𝑥	PRON
cte-761	74	1	+	+	CCONJ
cte-761	74	2	𝑣2𝑦	𝑣2𝑦	NUM
cte-761	74	3	,	,	PUNCT
cte-761	74	4	𝑑𝑣𝑦	𝑑𝑣𝑦	VERB
cte-761	74	5	𝑑𝑡	𝑑𝑡	ADP
cte-761	74	6	=	=	PUNCT
cte-761	74	7	−𝑔	−𝑔	PROPN
cte-761	74	8	+	+	CCONJ
cte-761	74	9	(	(	PUNCT
cte-761	74	10	︂	︂	PUNCT
cte-761	74	11	−𝑘2	−𝑘2	NOUN
cte-761	74	12	𝑚	𝑚	ADP
cte-761	74	13	𝑣𝑦	𝑣𝑦	PROPN
cte-761	74	14	∓	∓	PROPN
cte-761	74	15	𝑘3	𝑘3	PROPN
cte-761	74	16	𝑚	𝑚	ADP
cte-761	74	17	𝑣𝑥	𝑣𝑥	PROPN
cte-761	74	18	)	)	PUNCT
cte-761	74	19	︂√︁	︂√︁	NOUN
cte-761	75	1	𝑣2𝑥	𝑣2𝑥	PROPN
cte-761	75	2	+	+	CCONJ
cte-761	75	3	𝑣2𝑦	𝑣2𝑦	NUM
cte-761	75	4	.	.	PUNCT
cte-761	76	1	(	(	PUNCT
cte-761	76	2	4	4	X
cte-761	76	3	)	)	PUNCT
cte-761	76	4	in	in	ADP
cte-761	76	5	these	these	DET
cte-761	76	6	mathematical	mathematical	ADJ
cte-761	76	7	models	model	NOUN
cte-761	76	8	,	,	PUNCT
cte-761	76	9	we	we	PRON
cte-761	76	10	employ	employ	VERB
cte-761	76	11	the	the	DET
cte-761	76	12	following	follow	VERB
cte-761	76	13	notation	notation	NOUN
cte-761	76	14	:	:	PUNCT
cte-761	76	15	•	•	NUM
cte-761	76	16	𝑥	𝑥	NOUN
cte-761	76	17	=	=	SYM
cte-761	76	18	𝑥(𝑡	𝑥(𝑡	NOUN
cte-761	76	19	)	)	PUNCT
cte-761	76	20	,	,	PUNCT
cte-761	76	21	𝑦	𝑦	NOUN
cte-761	76	22	=	=	SYM
cte-761	76	23	𝑦(𝑡	𝑦(𝑡	PROPN
cte-761	76	24	)	)	PUNCT
cte-761	76	25	–	–	PUNCT
cte-761	76	26	coordinates	coordinate	NOUN
cte-761	76	27	of	of	ADP
cte-761	76	28	the	the	DET
cte-761	76	29	object	object	NOUN
cte-761	76	30	’s	’s	PART
cte-761	76	31	centre	centre	NOUN
cte-761	76	32	at	at	ADP
cte-761	76	33	time	time	NOUN
cte-761	76	34	𝑡	𝑡	PROPN
cte-761	76	35	(	(	PUNCT
cte-761	76	36	m	m	NOUN
cte-761	76	37	)	)	PUNCT
cte-761	76	38	•	•	NOUN
cte-761	76	39	𝑣𝑥	𝑣𝑥	X
cte-761	76	40	=	=	SYM
cte-761	76	41	𝑣𝑥(𝑡	𝑣𝑥(𝑡	X
cte-761	76	42	)	)	PUNCT
cte-761	76	43	,	,	PUNCT
cte-761	76	44	𝑣𝑦	𝑣𝑦	NOUN
cte-761	76	45	=	=	SYM
cte-761	76	46	𝑣𝑦(𝑡	𝑣𝑦(𝑡	NUM
cte-761	76	47	)	)	PUNCT
cte-761	76	48	–	–	PUNCT
cte-761	76	49	object	object	VERB
cte-761	76	50	velocity	velocity	NOUN
cte-761	76	51	projections	projection	NOUN
cte-761	76	52	on	on	ADP
cte-761	76	53	coordinate	coordinate	NOUN
cte-761	76	54	axes	axis	NOUN
cte-761	76	55	at	at	ADP
cte-761	76	56	time	time	NOUN
cte-761	76	57	𝑡	𝑡	PROPN
cte-761	76	58	(	(	PUNCT
cte-761	76	59	m	m	PROPN
cte-761	76	60	/	/	SYM
cte-761	76	61	s	s	PART
cte-761	76	62	)	)	PUNCT
cte-761	76	63	•	•	NOUN
cte-761	76	64	𝑔	𝑔	PROPN
cte-761	76	65	–	–	PUNCT
cte-761	76	66	gravitational	gravitational	ADJ
cte-761	76	67	acceleration	acceleration	NOUN
cte-761	76	68	(	(	PUNCT
cte-761	76	69	m	m	NOUN
cte-761	76	70	/	/	SYM
cte-761	76	71	s2	s2	PROPN
cte-761	76	72	)	)	PUNCT
cte-761	76	73	•	•	ADP
cte-761	76	74	𝑚	𝑚	X
cte-761	76	75	–	–	PUNCT
cte-761	76	76	object	object	NOUN
cte-761	76	77	mass	mass	NOUN
cte-761	76	78	(	(	PUNCT
cte-761	76	79	𝑚	𝑚	X
cte-761	76	80	̸=	̸=	PROPN
cte-761	76	81	0	0	NUM
cte-761	76	82	)	)	PUNCT
cte-761	76	83	(	(	PUNCT
cte-761	76	84	kg	kg	NOUN
cte-761	76	85	)	)	PUNCT
cte-761	76	86	•	•	NUM
cte-761	76	87	𝑘2	𝑘2	NOUN
cte-761	76	88	=	=	NOUN
cte-761	76	89	1	1	NUM
cte-761	76	90	2𝐶𝐷𝜌𝑆	2𝐶𝐷𝜌𝑆	NUM
cte-761	76	91	–	–	PUNCT
cte-761	76	92	medium	medium	ADJ
cte-761	76	93	resistance	resistance	NOUN
cte-761	76	94	factor	factor	NOUN
cte-761	76	95	(	(	PUNCT
cte-761	76	96	kg	kg	NOUN
cte-761	76	97	/	/	SYM
cte-761	76	98	m	m	NOUN
cte-761	76	99	)	)	PUNCT
cte-761	76	100	•	•	NUM
cte-761	76	101	𝑘3	𝑘3	PROPN
cte-761	76	102	=	=	NOUN
cte-761	76	103	1	1	NUM
cte-761	76	104	2𝐶𝐿𝜌𝑆	2𝐶𝐿𝜌𝑆	NUM
cte-761	76	105	–	–	PUNCT
cte-761	76	106	magnus	magnus	PROPN
cte-761	76	107	force	force	NOUN
cte-761	76	108	factor	factor	NOUN
cte-761	76	109	(	(	PUNCT
cte-761	76	110	kg	kg	NOUN
cte-761	76	111	/	/	SYM
cte-761	76	112	m	m	NOUN
cte-761	76	113	)	)	PUNCT
cte-761	76	114	•	•	ADP
cte-761	76	115	𝐶𝐷	𝐶𝐷	PROPN
cte-761	76	116	–	–	PUNCT
cte-761	76	117	drag	drag	NOUN
cte-761	76	118	coefficient	coefficient	NOUN
cte-761	76	119	,	,	PUNCT
cte-761	76	120	dependent	dependent	ADJ
cte-761	76	121	on	on	ADP
cte-761	76	122	object	object	NOUN
cte-761	76	123	shape	shape	NOUN
cte-761	76	124	and	and	CCONJ
cte-761	76	125	medium	medium	NOUN
cte-761	76	126	characteristics	characteristic	NOUN
cte-761	76	127	(	(	PUNCT
cte-761	76	128	dimensionless	dimensionless	NOUN
cte-761	76	129	;	;	PUNCT
cte-761	76	130	for	for	ADP
cte-761	76	131	a	a	DET
cte-761	76	132	sphere	sphere	NOUN
cte-761	76	133	in	in	ADP
cte-761	76	134	air	air	NOUN
cte-761	76	135	,	,	PUNCT
cte-761	76	136	𝐶𝐷	𝐶𝐷	PROPN
cte-761	76	137	=	=	SYM
cte-761	76	138	0.47	0.47	NUM
cte-761	76	139	)	)	PUNCT
cte-761	76	140	•	•	NOUN
cte-761	76	141	𝐶𝐿	𝐶𝐿	PROPN
cte-761	76	142	–	–	PUNCT
cte-761	76	143	magnus	magnus	NOUN
cte-761	76	144	effect	effect	NOUN
cte-761	76	145	coefficient	coefficient	NOUN
cte-761	76	146	,	,	PUNCT
cte-761	76	147	dependent	dependent	ADJ
cte-761	76	148	on	on	ADP
cte-761	76	149	object	object	NOUN
cte-761	76	150	shape	shape	NOUN
cte-761	76	151	,	,	PUNCT
cte-761	76	152	surface	surface	NOUN
cte-761	76	153	quality	quality	NOUN
cte-761	76	154	,	,	PUNCT
cte-761	76	155	and	and	CCONJ
cte-761	76	156	medium	medium	NOUN
cte-761	76	157	properties	property	NOUN
cte-761	76	158	(	(	PUNCT
cte-761	76	159	dimensionless	dimensionless	NOUN
cte-761	76	160	;	;	PUNCT
cte-761	76	161	for	for	ADP
cte-761	76	162	a	a	DET
cte-761	76	163	sphere	sphere	NOUN
cte-761	76	164	,	,	PUNCT
cte-761	76	165	0.1	0.1	NUM
cte-761	76	166	≤	≤	NOUN
cte-761	76	167	𝐶𝐿	𝐶𝐿	ADP
cte-761	76	168	≤	≤	NUM
cte-761	76	169	0.6	0.6	NUM
cte-761	76	170	)	)	PUNCT
cte-761	76	171	•	•	ADP
cte-761	76	172	𝜌	𝜌	X
cte-761	76	173	–	–	PUNCT
cte-761	76	174	medium	medium	ADJ
cte-761	76	175	density	density	NOUN
cte-761	76	176	(	(	PUNCT
cte-761	76	177	for	for	ADP
cte-761	76	178	air	air	NOUN
cte-761	76	179	,	,	PUNCT
cte-761	76	180	𝜌	𝜌	X
cte-761	76	181	=	=	SYM
cte-761	76	182	1.213	1.213	NUM
cte-761	76	183	kg	kg	NOUN
cte-761	76	184	/	/	SYM
cte-761	76	185	m3	m3	PROPN
cte-761	76	186	)	)	PUNCT
cte-761	76	187	•	•	ADP
cte-761	76	188	𝑆	𝑆	PROPN
cte-761	76	189	–	–	PUNCT
cte-761	76	190	object	object	VERB
cte-761	76	191	’s	’s	PART
cte-761	76	192	cross	cross	ADJ
cte-761	76	193	-	-	ADJ
cte-761	76	194	sectional	sectional	ADJ
cte-761	76	195	area	area	NOUN
cte-761	76	196	perpendicular	perpendicular	NOUN
cte-761	76	197	to	to	ADP
cte-761	76	198	motion	motion	NOUN
cte-761	76	199	direction	direction	NOUN
cte-761	76	200	(	(	PUNCT
cte-761	76	201	m2	m2	PROPN
cte-761	76	202	)	)	PUNCT
cte-761	76	203	;	;	PUNCT
cte-761	76	204	for	for	ADP
cte-761	76	205	a	a	DET
cte-761	76	206	spherical	spherical	ADJ
cte-761	76	207	object	object	NOUN
cte-761	76	208	,	,	PUNCT
cte-761	76	209	𝑆	𝑆	PROPN
cte-761	76	210	=	=	SYM
cte-761	76	211	𝜋𝑟2	𝜋𝑟2	PROPN
cte-761	76	212	,	,	PUNCT
cte-761	76	213	where	where	SCONJ
cte-761	76	214	𝑟	𝑟	X
cte-761	76	215	is	be	AUX
cte-761	76	216	the	the	DET
cte-761	76	217	sphere	sphere	NOUN
cte-761	76	218	’s	’s	PART
cte-761	76	219	radius	radius	NOUN
cte-761	76	220	(	(	PUNCT
cte-761	76	221	m	m	NOUN
cte-761	76	222	)	)	PUNCT
cte-761	76	223	237	237	NUM
cte-761	76	224	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	76	225	cte	cte	NOUN
cte-761	76	226	workshop	workshop	NOUN
cte-761	76	227	proceedings	proceeding	NOUN
cte-761	76	228	,	,	PUNCT
cte-761	76	229	2025	2025	NUM
cte-761	76	230	,	,	PUNCT
cte-761	76	231	vol	vol	NOUN
cte-761	76	232	.	.	PROPN
cte-761	76	233	12	12	NUM
cte-761	76	234	,	,	PUNCT
cte-761	76	235	pp	pp	ADJ
cte-761	76	236	.	.	PUNCT
cte-761	77	1	235–252	235–252	NUM
cte-761	77	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	77	3	2.3	2.3	NUM
cte-761	77	4	.	.	PUNCT
cte-761	77	5	implementation	implementation	NOUN
cte-761	77	6	in	in	ADP
cte-761	77	7	excel	excel	PROPN
cte-761	77	8	(	(	PUNCT
cte-761	77	9	group	group	NOUN
cte-761	77	10	e	e	NOUN
cte-761	77	11	)	)	PUNCT
cte-761	77	12	we	we	PRON
cte-761	77	13	present	present	VERB
cte-761	77	14	this	this	DET
cte-761	77	15	section	section	NOUN
cte-761	77	16	in	in	ADP
cte-761	77	17	condensed	condense	VERB
cte-761	77	18	form	form	NOUN
cte-761	77	19	,	,	PUNCT
cte-761	77	20	with	with	ADP
cte-761	77	21	detailed	detailed	ADJ
cte-761	77	22	methodological	methodological	ADJ
cte-761	77	23	descriptions	description	NOUN
cte-761	77	24	available	available	ADJ
cte-761	77	25	in	in	ADP
cte-761	77	26	our	our	PRON
cte-761	77	27	previous	previous	ADJ
cte-761	77	28	publication	publication	NOUN
cte-761	77	29	[	[	X
cte-761	77	30	4	4	NUM
cte-761	77	31	]	]	PUNCT
cte-761	77	32	.	.	PUNCT
cte-761	78	1	the	the	DET
cte-761	78	2	computational	computational	ADJ
cte-761	78	3	scheme	scheme	NOUN
cte-761	78	4	for	for	ADP
cte-761	78	5	implementing	implement	VERB
cte-761	78	6	mm	mm	PROPN
cte-761	78	7	i	i	PRON
cte-761	78	8	(	(	PUNCT
cte-761	78	9	equations	equation	NOUN
cte-761	78	10	1	1	NUM
cte-761	78	11	and	and	CCONJ
cte-761	78	12	2	2	NUM
cte-761	78	13	)	)	PUNCT
cte-761	78	14	in	in	ADP
cte-761	78	15	excel	excel	NOUN
cte-761	78	16	is	be	AUX
cte-761	78	17	derived	derive	VERB
cte-761	78	18	directly	directly	ADV
cte-761	78	19	from	from	ADP
cte-761	78	20	its	its	PRON
cte-761	78	21	analytical	analytical	ADJ
cte-761	78	22	solution	solution	NOUN
cte-761	78	23	[	[	X
cte-761	78	24	8	8	NUM
cte-761	78	25	,	,	PUNCT
cte-761	78	26	11	11	NUM
cte-761	78	27	]	]	NUM
cte-761	78	28	:	:	PUNCT
cte-761	78	29	𝑣𝑥𝑖	𝑣𝑥𝑖	NOUN
cte-761	78	30	=	=	SYM
cte-761	78	31	𝑣0𝑥	𝑣0𝑥	PROPN
cte-761	78	32	,	,	PUNCT
cte-761	78	33	𝑣𝑦𝑖	𝑣𝑦𝑖	NOUN
cte-761	78	34	=	=	SYM
cte-761	78	35	𝑣0𝑦	𝑣0𝑦	NUM
cte-761	78	36	−	−	NOUN
cte-761	78	37	𝑔𝑡𝑖	𝑔𝑡𝑖	NOUN
cte-761	78	38	,	,	PUNCT
cte-761	78	39	𝑥𝑖	𝑥𝑖	PRON
cte-761	78	40	=	=	PUNCT
cte-761	78	41	𝑥0	𝑥0	PROPN
cte-761	78	42	+	+	CCONJ
cte-761	78	43	𝑣0𝑥𝑡𝑖	𝑣0𝑥𝑡𝑖	PROPN
cte-761	78	44	,	,	PUNCT
cte-761	78	45	𝑦𝑖	𝑦𝑖	ADP
cte-761	78	46	=	=	NOUN
cte-761	78	47	𝑦0	𝑦0	PROPN
cte-761	79	1	+	+	CCONJ
cte-761	79	2	𝑣0𝑦𝑡𝑖	𝑣0𝑦𝑡𝑖	NUM
cte-761	79	3	−	−	VERB
cte-761	80	1	𝑔𝑡2𝑖	𝑔𝑡2𝑖	ADJ
cte-761	80	2	2	2	NUM
cte-761	80	3	,	,	PUNCT
cte-761	80	4	𝑖	𝑖	NOUN
cte-761	80	5	=	=	SYM
cte-761	80	6	0	0	NUM
cte-761	80	7	,	,	PUNCT
cte-761	80	8	𝑛	𝑛	PRON
cte-761	80	9	(	(	PUNCT
cte-761	80	10	5	5	NUM
cte-761	80	11	)	)	PUNCT
cte-761	81	1	where	where	SCONJ
cte-761	81	2	:	:	PUNCT
cte-761	81	3	•	•	NUM
cte-761	81	4	𝑥0	𝑥0	NOUN
cte-761	81	5	,	,	PUNCT
cte-761	81	6	𝑦0	𝑦0	NOUN
cte-761	81	7	–	–	PUNCT
cte-761	81	8	initial	initial	ADJ
cte-761	81	9	coordinates	coordinate	NOUN
cte-761	81	10	of	of	ADP
cte-761	81	11	the	the	DET
cte-761	81	12	object	object	NOUN
cte-761	81	13	’s	’s	PART
cte-761	81	14	centre	centre	NOUN
cte-761	81	15	(	(	PUNCT
cte-761	81	16	m	m	NOUN
cte-761	81	17	)	)	PUNCT
cte-761	81	18	•	•	ADV
cte-761	81	19	𝑣0	𝑣0	PROPN
cte-761	81	20	–	–	PUNCT
cte-761	81	21	initial	initial	ADJ
cte-761	81	22	object	object	NOUN
cte-761	81	23	velocity	velocity	NOUN
cte-761	81	24	(	(	PUNCT
cte-761	81	25	m	m	NOUN
cte-761	81	26	/	/	SYM
cte-761	81	27	s	s	PART
cte-761	81	28	)	)	PUNCT
cte-761	81	29	•	•	NUM
cte-761	81	30	𝛼0	𝛼0	PROPN
cte-761	81	31	–	–	PUNCT
cte-761	81	32	initial	initial	ADJ
cte-761	81	33	projection	projection	NOUN
cte-761	81	34	angle	angle	NOUN
cte-761	81	35	to	to	ADP
cte-761	81	36	the	the	DET
cte-761	81	37	horizon	horizon	NOUN
cte-761	81	38	(	(	PUNCT
cte-761	81	39	radians	radian	NOUN
cte-761	81	40	)	)	PUNCT
cte-761	81	41	•	•	NUM
cte-761	81	42	𝑣0𝑥	𝑣0𝑥	NUM
cte-761	81	43	=	=	NOUN
cte-761	81	44	𝑣0	𝑣0	NOUN
cte-761	81	45	cos𝛼0	cos𝛼0	PROPN
cte-761	81	46	,	,	PUNCT
cte-761	81	47	𝑣0𝑦	𝑣0𝑦	NUM
cte-761	81	48	=	=	SYM
cte-761	81	49	𝑣0	𝑣0	PROPN
cte-761	81	50	sin𝛼0	sin𝛼0	X
cte-761	81	51	–	–	PUNCT
cte-761	81	52	initial	initial	ADJ
cte-761	81	53	velocity	velocity	NOUN
cte-761	81	54	projections	projection	NOUN
cte-761	81	55	•	•	ADP
cte-761	81	56	𝑡𝑖	𝑡𝑖	NOUN
cte-761	81	57	=	=	SYM
cte-761	81	58	𝑖	𝑖	SYM
cte-761	81	59	·	·	SYM
cte-761	81	60	δ𝑡	δ𝑡	NOUN
cte-761	81	61	,	,	PUNCT
cte-761	81	62	𝑡𝑖	𝑡𝑖	X
cte-761	81	63	∈	∈	NOUN
cte-761	82	1	[	[	X
cte-761	82	2	0	0	NUM
cte-761	82	3	,	,	PUNCT
cte-761	82	4	𝑡𝑀	𝑡𝑀	X
cte-761	82	5	]	]	PUNCT
cte-761	82	6	,	,	PUNCT
cte-761	82	7	𝑖	𝑖	SYM
cte-761	82	8	=	=	SYM
cte-761	82	9	0	0	NUM
cte-761	82	10	,	,	PUNCT
cte-761	82	11	𝑛	𝑛	PRON
cte-761	82	12	–	–	PUNCT
cte-761	82	13	observation	observation	NOUN
cte-761	82	14	time	time	NOUN
cte-761	82	15	points	point	VERB
cte-761	82	16	•	•	NOUN
cte-761	82	17	𝑖	𝑖	NOUN
cte-761	82	18	–	–	PUNCT
cte-761	82	19	position	position	NOUN
cte-761	82	20	observation	observation	NOUN
cte-761	82	21	index	index	NOUN
cte-761	82	22	•	•	NOUN
cte-761	82	23	𝑛	𝑛	PROPN
cte-761	82	24	–	–	PUNCT
cte-761	82	25	number	number	NOUN
cte-761	82	26	of	of	ADP
cte-761	82	27	observation	observation	NOUN
cte-761	82	28	points	point	NOUN
cte-761	82	29	•	•	ADP
cte-761	82	30	δ𝑡	δ𝑡	X
cte-761	82	31	=	=	PUNCT
cte-761	82	32	𝑡𝑀/𝑛	𝑡𝑀/𝑛	NUM
cte-761	82	33	–	–	PUNCT
cte-761	82	34	time	time	NOUN
cte-761	82	35	interval	interval	NOUN
cte-761	82	36	between	between	ADP
cte-761	82	37	observations	observation	NOUN
cte-761	82	38	(	(	PUNCT
cte-761	82	39	s	s	NOUN
cte-761	82	40	)	)	PUNCT
cte-761	82	41	•	•	NUM
cte-761	82	42	𝑡𝑀	𝑡𝑀	NOUN
cte-761	82	43	–	–	PUNCT
cte-761	82	44	simulation	simulation	NOUN
cte-761	82	45	duration	duration	NOUN
cte-761	82	46	(	(	PUNCT
cte-761	82	47	s	s	NOUN
cte-761	82	48	)	)	PUNCT
cte-761	82	49	•	•	NOUN
cte-761	82	50	𝑥𝑖	𝑥𝑖	PROPN
cte-761	82	51	=	=	PUNCT
cte-761	82	52	𝑥(𝑡𝑖	𝑥(𝑡𝑖	X
cte-761	82	53	)	)	PUNCT
cte-761	82	54	,	,	PUNCT
cte-761	82	55	𝑦𝑖	𝑦𝑖	PROPN
cte-761	82	56	=	=	PUNCT
cte-761	82	57	𝑦(𝑡𝑖	𝑦(𝑡𝑖	PROPN
cte-761	82	58	)	)	PUNCT
cte-761	82	59	–	–	PUNCT
cte-761	82	60	object	object	NOUN
cte-761	82	61	centre	centre	NOUN
cte-761	82	62	coordinates	coordinate	NOUN
cte-761	82	63	at	at	ADP
cte-761	82	64	time	time	NOUN
cte-761	82	65	𝑡𝑖	𝑡𝑖	NOUN
cte-761	82	66	•	•	ADJ
cte-761	82	67	𝑣𝑥𝑖	𝑣𝑥𝑖	NOUN
cte-761	82	68	=	=	PUNCT
cte-761	82	69	𝑣𝑥(𝑡𝑖	𝑣𝑥(𝑡𝑖	PROPN
cte-761	82	70	)	)	PUNCT
cte-761	82	71	,	,	PUNCT
cte-761	82	72	𝑣𝑦𝑖	𝑣𝑦𝑖	NOUN
cte-761	82	73	=	=	SYM
cte-761	82	74	𝑣𝑦(𝑡𝑖	𝑣𝑦(𝑡𝑖	ADJ
cte-761	82	75	)	)	PUNCT
cte-761	82	76	–	–	PUNCT
cte-761	82	77	velocity	velocity	NOUN
cte-761	82	78	projections	projection	NOUN
cte-761	82	79	at	at	ADP
cte-761	82	80	time	time	NOUN
cte-761	82	81	𝑡𝑖	𝑡𝑖	NOUN
cte-761	82	82	•	•	NOUN
cte-761	82	83	𝑣𝑖	𝑣𝑖	ADP
cte-761	82	84	=	=	PUNCT
cte-761	82	85	√︁	√︁	PROPN
cte-761	82	86	𝑣2𝑥𝑖	𝑣2𝑥𝑖	X
cte-761	82	87	+	+	CCONJ
cte-761	82	88	𝑣2𝑦𝑖	𝑣2𝑦𝑖	NUM
cte-761	82	89	–	–	PUNCT
cte-761	82	90	object	object	VERB
cte-761	82	91	velocity	velocity	NOUN
cte-761	82	92	magnitude	magnitude	NOUN
cte-761	82	93	at	at	ADP
cte-761	82	94	time	time	NOUN
cte-761	82	95	𝑡𝑖	𝑡𝑖	NOUN
cte-761	82	96	models	model	VERB
cte-761	82	97	mm	mm	PROPN
cte-761	82	98	ii	ii	PROPN
cte-761	83	1	(	(	PUNCT
cte-761	83	2	equations	equation	NOUN
cte-761	83	3	1	1	NUM
cte-761	83	4	and	and	CCONJ
cte-761	83	5	3	3	NUM
cte-761	83	6	)	)	PUNCT
cte-761	83	7	and	and	CCONJ
cte-761	83	8	mm	mm	PROPN
cte-761	83	9	iii	iii	PROPN
cte-761	83	10	(	(	PUNCT
cte-761	83	11	equations	equation	NOUN
cte-761	83	12	1	1	NUM
cte-761	83	13	and	and	CCONJ
cte-761	83	14	4	4	NUM
cte-761	83	15	)	)	PUNCT
cte-761	83	16	,	,	PUNCT
cte-761	83	17	unlike	unlike	ADP
cte-761	83	18	mm	mm	PROPN
cte-761	83	19	i	i	NOUN
cte-761	83	20	,	,	PUNCT
cte-761	83	21	lack	lack	VERB
cte-761	83	22	analytical	analytical	ADJ
cte-761	83	23	solutions	solution	NOUN
cte-761	83	24	.	.	PUNCT
cte-761	84	1	therefore	therefore	ADV
cte-761	84	2	,	,	PUNCT
cte-761	84	3	we	we	PRON
cte-761	84	4	implemented	implement	VERB
cte-761	84	5	computational	computational	ADJ
cte-761	84	6	schemes	scheme	NOUN
cte-761	84	7	based	base	VERB
cte-761	84	8	on	on	ADP
cte-761	84	9	numerical	numerical	ADJ
cte-761	84	10	methods	method	NOUN
cte-761	84	11	.	.	PUNCT
cte-761	85	1	the	the	DET
cte-761	85	2	computational	computational	ADJ
cte-761	85	3	scheme	scheme	NOUN
cte-761	85	4	for	for	ADP
cte-761	85	5	mm	mm	PROPN
cte-761	85	6	ii	ii	PROPN
cte-761	85	7	using	use	VERB
cte-761	85	8	the	the	DET
cte-761	85	9	euler	euler	NOUN
cte-761	85	10	method	method	NOUN
cte-761	85	11	for	for	ADP
cte-761	85	12	first	first	ADJ
cte-761	85	13	-	-	PUNCT
cte-761	85	14	order	order	NOUN
cte-761	85	15	ode	ode	NOUN
cte-761	85	16	systems	system	NOUN
cte-761	85	17	is	be	AUX
cte-761	85	18	[	[	X
cte-761	85	19	10	10	NUM
cte-761	85	20	]	]	PUNCT
cte-761	85	21	:	:	PUNCT
cte-761	85	22	𝑣𝑥	𝑣𝑥	X
cte-761	85	23	𝑖+1	𝑖+1	PUNCT
cte-761	85	24	=	=	PUNCT
cte-761	85	25	𝑣𝑥𝑖	𝑣𝑥𝑖	NOUN
cte-761	85	26	−	−	PROPN
cte-761	85	27	𝑘2	𝑘2	PROPN
cte-761	85	28	𝑚	𝑚	PROPN
cte-761	85	29	𝑣𝑥𝑖𝑣𝑖δ𝑡	𝑣𝑥𝑖𝑣𝑖δ𝑡	NOUN
cte-761	85	30	,	,	PUNCT
cte-761	85	31	𝑣𝑦	𝑣𝑦	X
cte-761	85	32	𝑖+1	𝑖+1	X
cte-761	85	33	=	=	PUNCT
cte-761	85	34	𝑣𝑦𝑖	𝑣𝑦𝑖	NOUN
cte-761	86	1	−	−	PROPN
cte-761	87	1	(	(	PUNCT
cte-761	87	2	︂	︂	PUNCT
cte-761	87	3	𝑔	𝑔	PROPN
cte-761	87	4	+	+	CCONJ
cte-761	87	5	𝑘2	𝑘2	PROPN
cte-761	87	6	𝑚	𝑚	X
cte-761	87	7	𝑣𝑦𝑖𝑣𝑖	𝑣𝑦𝑖𝑣𝑖	PROPN
cte-761	87	8	)	)	PUNCT
cte-761	87	9	︂	︂	PUNCT
cte-761	87	10	δ𝑡	δ𝑡	PROPN
cte-761	87	11	,	,	PUNCT
cte-761	87	12	𝑥𝑖+1	𝑥𝑖+1	PROPN
cte-761	87	13	=	=	SYM
cte-761	87	14	𝑥𝑖	𝑥𝑖	PROPN
cte-761	87	15	+	+	CCONJ
cte-761	87	16	𝑣𝑥𝑖δ𝑡	𝑣𝑥𝑖δ𝑡	NOUN
cte-761	87	17	,	,	PUNCT
cte-761	87	18	𝑦𝑖+1	𝑦𝑖+1	PROPN
cte-761	87	19	=	=	SYM
cte-761	87	20	𝑦𝑖	𝑦𝑖	PROPN
cte-761	87	21	+	+	NUM
cte-761	87	22	𝑣𝑦𝑖δ𝑡.	𝑣𝑦𝑖δ𝑡.	PROPN
cte-761	87	23	(	(	PUNCT
cte-761	87	24	6	6	NUM
cte-761	87	25	)	)	PUNCT
cte-761	87	26	for	for	ADP
cte-761	87	27	mm	mm	PROPN
cte-761	87	28	iii	iii	PROPN
cte-761	87	29	,	,	PUNCT
cte-761	87	30	the	the	DET
cte-761	87	31	computational	computational	ADJ
cte-761	87	32	scheme	scheme	NOUN
cte-761	87	33	is	be	AUX
cte-761	87	34	similar	similar	ADJ
cte-761	87	35	,	,	PUNCT
cte-761	87	36	but	but	CCONJ
cte-761	87	37	the	the	DET
cte-761	87	38	first	first	ADJ
cte-761	87	39	two	two	NUM
cte-761	87	40	equations	equation	NOUN
cte-761	87	41	in	in	ADP
cte-761	87	42	(	(	PUNCT
cte-761	87	43	6	6	NUM
cte-761	87	44	)	)	PUNCT
cte-761	87	45	are	be	AUX
cte-761	87	46	replaced	replace	VERB
cte-761	87	47	by	by	ADP
cte-761	87	48	:	:	PUNCT
cte-761	87	49	𝑣𝑥	𝑣𝑥	NOUN
cte-761	87	50	𝑖+1	𝑖+1	PUNCT
cte-761	87	51	=	=	PUNCT
cte-761	87	52	𝑣𝑥𝑖	𝑣𝑥𝑖	NOUN
cte-761	87	53	−	−	PROPN
cte-761	88	1	(	(	PUNCT
cte-761	88	2	︂	︂	PUNCT
cte-761	88	3	𝑘2	𝑘2	PROPN
cte-761	88	4	𝑚	𝑚	PROPN
cte-761	88	5	𝑣𝑥𝑖𝑣𝑖	𝑣𝑥𝑖𝑣𝑖	PROPN
cte-761	88	6	−	−	PROPN
cte-761	88	7	rot	rot	VERB
cte-761	88	8	𝑘3	𝑘3	PROPN
cte-761	88	9	𝑚	𝑚	ADP
cte-761	88	10	𝑣𝑦𝑖𝑣𝑖	𝑣𝑦𝑖𝑣𝑖	PROPN
cte-761	88	11	)	)	PUNCT
cte-761	88	12	︂	︂	PUNCT
cte-761	88	13	δ𝑡	δ𝑡	ADP
cte-761	88	14	,	,	PUNCT
cte-761	88	15	𝑣𝑦	𝑣𝑦	PRON
cte-761	88	16	𝑖+1	𝑖+1	PUNCT
cte-761	88	17	=	=	PUNCT
cte-761	88	18	𝑣𝑦𝑖	𝑣𝑦𝑖	NOUN
cte-761	88	19	−	−	PROPN
cte-761	89	1	(	(	PUNCT
cte-761	89	2	︂	︂	PUNCT
cte-761	89	3	𝑔	𝑔	PROPN
cte-761	89	4	+	+	CCONJ
cte-761	89	5	𝑘2	𝑘2	PROPN
cte-761	89	6	𝑚	𝑚	X
cte-761	89	7	𝑣𝑦𝑖𝑣𝑖	𝑣𝑦𝑖𝑣𝑖	NOUN
cte-761	89	8	+	+	CCONJ
cte-761	89	9	rot	rot	VERB
cte-761	89	10	𝑘3	𝑘3	PROPN
cte-761	89	11	𝑚	𝑚	PROPN
cte-761	89	12	𝑣𝑥𝑖𝑣𝑖	𝑣𝑥𝑖𝑣𝑖	PROPN
cte-761	89	13	)	)	PUNCT
cte-761	89	14	︂	︂	PUNCT
cte-761	89	15	δ𝑡.	δ𝑡.	PROPN
cte-761	90	1	(	(	PUNCT
cte-761	90	2	7	7	X
cte-761	90	3	)	)	PUNCT
cte-761	90	4	the	the	DET
cte-761	90	5	initial	initial	ADJ
cte-761	90	6	parameters	parameter	NOUN
cte-761	90	7	for	for	ADP
cte-761	90	8	calculations	calculation	NOUN
cte-761	90	9	using	use	VERB
cte-761	90	10	equations	equation	NOUN
cte-761	90	11	(	(	PUNCT
cte-761	90	12	5	5	NUM
cte-761	90	13	)	)	PUNCT
cte-761	90	14	,	,	PUNCT
cte-761	90	15	(	(	PUNCT
cte-761	90	16	6	6	NUM
cte-761	90	17	)	)	PUNCT
cte-761	90	18	,	,	PUNCT
cte-761	90	19	and	and	CCONJ
cte-761	90	20	(	(	PUNCT
cte-761	90	21	7	7	X
cte-761	90	22	)	)	PUNCT
cte-761	90	23	include	include	VERB
cte-761	90	24	𝑔	𝑔	NOUN
cte-761	90	25	,	,	PUNCT
cte-761	90	26	𝑥0	𝑥0	NOUN
cte-761	90	27	,	,	PUNCT
cte-761	90	28	𝑦0	𝑦0	NOUN
cte-761	90	29	,	,	PUNCT
cte-761	90	30	𝑣0	𝑣0	PROPN
cte-761	90	31	,	,	PUNCT
cte-761	90	32	𝛼0	𝛼0	PROPN
cte-761	90	33	,	,	PUNCT
cte-761	90	34	𝑟	𝑟	X
cte-761	90	35	,	,	PUNCT
cte-761	90	36	𝑚	𝑚	PROPN
cte-761	90	37	,	,	PUNCT
cte-761	90	38	𝜌	𝜌	X
cte-761	90	39	,	,	PUNCT
cte-761	90	40	𝐶𝐷	𝐶𝐷	PROPN
cte-761	90	41	,	,	PUNCT
cte-761	90	42	𝐶𝐿	𝐶𝐿	PROPN
cte-761	90	43	,	,	PUNCT
cte-761	90	44	𝑡0	𝑡0	NOUN
cte-761	90	45	,	,	PUNCT
cte-761	90	46	𝑡𝑀	𝑡𝑀	NOUN
cte-761	90	47	,	,	PUNCT
cte-761	90	48	𝑛	𝑛	NOUN
cte-761	90	49	,	,	PUNCT
cte-761	90	50	and	and	CCONJ
cte-761	90	51	rot	rot	VERB
cte-761	90	52	.	.	PUNCT
cte-761	91	1	the	the	DET
cte-761	91	2	parameter	parameter	NOUN
cte-761	91	3	rot	rot	NOUN
cte-761	91	4	can	can	AUX
cte-761	91	5	take	take	VERB
cte-761	91	6	three	three	NUM
cte-761	91	7	fixed	fixed	ADJ
cte-761	91	8	values	value	NOUN
cte-761	91	9	:	:	PUNCT
cte-761	91	10	−1	−1	NOUN
cte-761	91	11	for	for	ADP
cte-761	91	12	counterclockwise	counterclockwise	NOUN
cte-761	91	13	rotation	rotation	NOUN
cte-761	91	14	,	,	PUNCT
cte-761	91	15	+1	+1	NOUN
cte-761	91	16	for	for	ADP
cte-761	91	17	clockwise	clockwise	NOUN
cte-761	91	18	rotation	rotation	NOUN
cte-761	91	19	,	,	PUNCT
cte-761	91	20	and	and	CCONJ
cte-761	91	21	0	0	NUM
cte-761	91	22	for	for	ADP
cte-761	91	23	no	no	DET
cte-761	91	24	rotation	rotation	NOUN
cte-761	91	25	.	.	PUNCT
cte-761	92	1	mm	mm	PROPN
cte-761	92	2	iii	iii	PROPN
cte-761	92	3	with	with	ADP
cte-761	92	4	rot	rot	NOUN
cte-761	92	5	=	=	SYM
cte-761	92	6	−1	−1	NOUN
cte-761	92	7	is	be	AUX
cte-761	92	8	denoted	denote	VERB
cte-761	92	9	as	as	ADP
cte-761	92	10	mm	mm	PROPN
cte-761	92	11	iii−	iii−	PROPN
cte-761	92	12	,	,	PUNCT
cte-761	92	13	and	and	CCONJ
cte-761	92	14	with	with	ADP
cte-761	92	15	rot	rot	NOUN
cte-761	92	16	=	=	SYM
cte-761	92	17	+1	+1	PROPN
cte-761	92	18	as	as	ADP
cte-761	92	19	mm	mm	PROPN
cte-761	92	20	iii+	iii+	PROPN
cte-761	92	21	.	.	PROPN
cte-761	92	22	with	with	ADP
cte-761	92	23	rot	rot	NOUN
cte-761	92	24	=	=	SYM
cte-761	92	25	0	0	NUM
cte-761	92	26	,	,	PUNCT
cte-761	92	27	mm	mm	PROPN
cte-761	92	28	iii	iii	NOUN
cte-761	92	29	becomes	become	VERB
cte-761	92	30	equivalent	equivalent	ADJ
cte-761	92	31	to	to	ADP
cte-761	92	32	mm	mm	PROPN
cte-761	92	33	ii	ii	PROPN
cte-761	92	34	.	.	PUNCT
cte-761	93	1	numerical	numerical	PROPN
cte-761	93	2	calculations	calculation	NOUN
cte-761	93	3	using	use	VERB
cte-761	93	4	equations	equation	NOUN
cte-761	93	5	(	(	PUNCT
cte-761	93	6	5	5	NUM
cte-761	93	7	)	)	PUNCT
cte-761	93	8	,	,	PUNCT
cte-761	93	9	(	(	PUNCT
cte-761	93	10	6	6	NUM
cte-761	93	11	)	)	PUNCT
cte-761	93	12	,	,	PUNCT
cte-761	93	13	and	and	CCONJ
cte-761	93	14	(	(	PUNCT
cte-761	93	15	7	7	X
cte-761	93	16	)	)	PUNCT
cte-761	93	17	were	be	AUX
cte-761	93	18	performed	perform	VERB
cte-761	93	19	using	use	VERB
cte-761	93	20	standard	standard	ADJ
cte-761	93	21	excel	excel	NOUN
cte-761	93	22	functions	function	NOUN
cte-761	93	23	.	.	PUNCT
cte-761	94	1	the	the	DET
cte-761	94	2	results	result	NOUN
cte-761	94	3	provided	provide	VERB
cte-761	94	4	tables	table	NOUN
cte-761	94	5	of	of	ADP
cte-761	94	6	values	value	NOUN
cte-761	94	7	including	include	VERB
cte-761	94	8	𝑖	𝑖	NUM
cte-761	94	9	,	,	PUNCT
cte-761	94	10	𝑡𝑖	𝑡𝑖	NOUN
cte-761	94	11	,	,	PUNCT
cte-761	94	12	𝑥𝑖	𝑥𝑖	PROPN
cte-761	94	13	,	,	PUNCT
cte-761	94	14	𝑦𝑖	𝑦𝑖	PROPN
cte-761	94	15	,	,	PUNCT
cte-761	94	16	𝑣𝑥𝑖	𝑣𝑥𝑖	NOUN
cte-761	94	17	,	,	PUNCT
cte-761	94	18	𝑣𝑦𝑖	𝑣𝑦𝑖	NOUN
cte-761	94	19	,	,	PUNCT
cte-761	94	20	𝑣𝑖	𝑣𝑖	ADP
cte-761	94	21	,	,	PUNCT
cte-761	94	22	𝛼𝑖	𝛼𝑖	PROPN
cte-761	94	23	,	,	PUNCT
cte-761	94	24	𝐸𝑘𝑖	𝐸𝑘𝑖	PROPN
cte-761	94	25	,	,	PUNCT
cte-761	94	26	𝐸𝑝𝑖	𝐸𝑝𝑖	PROPN
cte-761	94	27	,	,	PUNCT
cte-761	94	28	where	where	SCONJ
cte-761	94	29	𝛼𝑖	𝛼𝑖	PROPN
cte-761	94	30	=	=	PUNCT
cte-761	94	31	arctan	arctan	PROPN
cte-761	94	32	𝑣𝑦𝑖	𝑣𝑦𝑖	PROPN
cte-761	94	33	𝑣𝑥𝑖	𝑣𝑥𝑖	NOUN
cte-761	94	34	represents	represent	VERB
cte-761	94	35	the	the	DET
cte-761	94	36	trajectory	trajectory	NOUN
cte-761	94	37	angle	angle	NOUN
cte-761	94	38	to	to	ADP
cte-761	94	39	the	the	DET
cte-761	94	40	238	238	NUM
cte-761	94	41	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	94	42	cte	cte	NOUN
cte-761	94	43	workshop	workshop	NOUN
cte-761	94	44	proceedings	proceeding	NOUN
cte-761	94	45	,	,	PUNCT
cte-761	94	46	2025	2025	NUM
cte-761	94	47	,	,	PUNCT
cte-761	94	48	vol	vol	NOUN
cte-761	94	49	.	.	PROPN
cte-761	94	50	12	12	NUM
cte-761	94	51	,	,	PUNCT
cte-761	94	52	pp	pp	ADJ
cte-761	94	53	.	.	PUNCT
cte-761	95	1	235–252	235–252	NUM
cte-761	95	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	95	3	horizon	horizon	NOUN
cte-761	95	4	,	,	PUNCT
cte-761	95	5	𝐸𝑘𝑖	𝐸𝑘𝑖	PROPN
cte-761	95	6	=	=	SYM
cte-761	95	7	𝑚𝑣2𝑖	𝑚𝑣2𝑖	NOUN
cte-761	95	8	2	2	NUM
cte-761	95	9	is	be	AUX
cte-761	95	10	the	the	DET
cte-761	95	11	translational	translational	ADJ
cte-761	95	12	kinetic	kinetic	ADJ
cte-761	95	13	energy	energy	NOUN
cte-761	95	14	,	,	PUNCT
cte-761	95	15	and	and	CCONJ
cte-761	95	16	𝐸𝑝𝑖	𝐸𝑝𝑖	PROPN
cte-761	95	17	=	=	NOUN
cte-761	95	18	𝑚𝑔𝑦𝑖	𝑚𝑔𝑦𝑖	NOUN
cte-761	95	19	is	be	AUX
cte-761	95	20	the	the	DET
cte-761	95	21	potential	potential	ADJ
cte-761	95	22	energy	energy	NOUN
cte-761	95	23	.	.	PUNCT
cte-761	96	1	figure	figure	NOUN
cte-761	96	2	1	1	NUM
cte-761	96	3	demonstrates	demonstrate	VERB
cte-761	96	4	the	the	DET
cte-761	96	5	data	datum	NOUN
cte-761	96	6	input	input	NOUN
cte-761	96	7	interface	interface	NOUN
cte-761	96	8	for	for	ADP
cte-761	96	9	models	model	NOUN
cte-761	97	1	mm	mm	INTJ
cte-761	98	1	i	i	PROPN
cte-761	98	2	,	,	PUNCT
cte-761	98	3	mm	mm	PROPN
cte-761	98	4	ii	ii	PROPN
cte-761	98	5	,	,	PUNCT
cte-761	98	6	and	and	CCONJ
cte-761	98	7	mm	mm	PROPN
cte-761	98	8	iii	iii	PROPN
cte-761	98	9	in	in	ADP
cte-761	98	10	excel	excel	NOUN
cte-761	98	11	.	.	PUNCT
cte-761	99	1	figure	figure	NOUN
cte-761	99	2	2	2	NUM
cte-761	99	3	displays	display	VERB
cte-761	99	4	a	a	DET
cte-761	99	5	portion	portion	NOUN
cte-761	99	6	of	of	ADP
cte-761	99	7	the	the	DET
cte-761	99	8	calculation	calculation	NOUN
cte-761	99	9	table	table	NOUN
cte-761	99	10	for	for	ADP
cte-761	99	11	mm	mm	PROPN
cte-761	99	12	iii	iii	PROPN
cte-761	99	13	with	with	ADP
cte-761	99	14	the	the	DET
cte-761	99	15	input	input	NOUN
cte-761	99	16	parameters	parameter	NOUN
cte-761	99	17	shown	show	VERB
cte-761	99	18	in	in	ADP
cte-761	99	19	figure	figure	NOUN
cte-761	99	20	1	1	NUM
cte-761	99	21	.	.	PUNCT
cte-761	99	22	figure	figure	NOUN
cte-761	99	23	1	1	NUM
cte-761	99	24	:	:	PUNCT
cte-761	99	25	interface	interface	NOUN
cte-761	99	26	for	for	ADP
cte-761	99	27	parameter	parameter	NOUN
cte-761	99	28	input	input	NOUN
cte-761	99	29	for	for	ADP
cte-761	99	30	mathematical	mathematical	ADJ
cte-761	99	31	models	model	NOUN
cte-761	100	1	mm	mm	INTJ
cte-761	101	1	i	i	PROPN
cte-761	101	2	-	-	PUNCT
cte-761	101	3	mm	mm	PROPN
cte-761	101	4	iii	iii	PROPN
cte-761	101	5	in	in	ADP
cte-761	101	6	excel	excel	NOUN
cte-761	101	7	.	.	PUNCT
cte-761	102	1	figure	figure	NOUN
cte-761	102	2	2	2	NUM
cte-761	102	3	:	:	PUNCT
cte-761	102	4	fragment	fragment	NOUN
cte-761	102	5	of	of	ADP
cte-761	102	6	the	the	DET
cte-761	102	7	calculation	calculation	NOUN
cte-761	102	8	table	table	NOUN
cte-761	102	9	for	for	ADP
cte-761	102	10	mm	mm	PROPN
cte-761	102	11	i	i	PROPN
cte-761	102	12	in	in	ADP
cte-761	102	13	excel	excel	NOUN
cte-761	102	14	.	.	PUNCT
cte-761	103	1	figures	figure	NOUN
cte-761	103	2	3	3	NUM
cte-761	103	3	and	and	CCONJ
cte-761	103	4	4	4	NUM
cte-761	103	5	display	display	VERB
cte-761	103	6	the	the	DET
cte-761	103	7	trajectories	trajectory	NOUN
cte-761	103	8	and	and	CCONJ
cte-761	103	9	velocities	velocity	NOUN
cte-761	103	10	calculated	calculate	VERB
cte-761	103	11	for	for	ADP
cte-761	103	12	all	all	DET
cte-761	103	13	models	model	NOUN
cte-761	103	14	.	.	PUNCT
cte-761	104	1	visual	visual	ADJ
cte-761	104	2	comparison	comparison	NOUN
cte-761	104	3	and	and	CCONJ
cte-761	104	4	analysis	analysis	NOUN
cte-761	104	5	of	of	ADP
cte-761	104	6	these	these	DET
cte-761	104	7	graphs	graph	NOUN
cte-761	104	8	allowed	allow	VERB
cte-761	104	9	students	student	NOUN
cte-761	104	10	to	to	PART
cte-761	104	11	formulate	formulate	VERB
cte-761	104	12	substantive	substantive	ADJ
cte-761	104	13	conclusions	conclusion	NOUN
cte-761	104	14	about	about	ADP
cte-761	104	15	movement	movement	NOUN
cte-761	104	16	characteristics	characteristic	NOUN
cte-761	104	17	under	under	ADP
cte-761	104	18	various	various	ADJ
cte-761	104	19	conditions	condition	NOUN
cte-761	104	20	.	.	PUNCT
cte-761	105	1	239	239	NUM
cte-761	105	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	105	3	cte	cte	NOUN
cte-761	105	4	workshop	workshop	NOUN
cte-761	105	5	proceedings	proceeding	NOUN
cte-761	105	6	,	,	PUNCT
cte-761	105	7	2025	2025	NUM
cte-761	105	8	,	,	PUNCT
cte-761	105	9	vol	vol	NOUN
cte-761	105	10	.	.	PROPN
cte-761	105	11	12	12	NUM
cte-761	105	12	,	,	PUNCT
cte-761	105	13	pp	pp	ADJ
cte-761	105	14	.	.	PUNCT
cte-761	106	1	235–252	235–252	NUM
cte-761	106	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	106	3	figure	figure	NOUN
cte-761	106	4	3	3	NUM
cte-761	106	5	:	:	PUNCT
cte-761	106	6	trajectories	trajectory	NOUN
cte-761	106	7	of	of	ADP
cte-761	106	8	projectiles	projectile	NOUN
cte-761	106	9	for	for	ADP
cte-761	106	10	models	model	NOUN
cte-761	106	11	mm	mm	INTJ
cte-761	107	1	i	i	PROPN
cte-761	107	2	-	-	PUNCT
cte-761	107	3	mm	mm	PROPN
cte-761	107	4	iii	iii	PROPN
cte-761	107	5	in	in	ADP
cte-761	107	6	excel	excel	NOUN
cte-761	107	7	.	.	PUNCT
cte-761	108	1	figure	figure	VERB
cte-761	108	2	4	4	NUM
cte-761	108	3	:	:	PUNCT
cte-761	108	4	velocity	velocity	NOUN
cte-761	108	5	graphs	graph	NOUN
cte-761	108	6	for	for	ADP
cte-761	108	7	models	model	NOUN
cte-761	108	8	mm	mm	INTJ
cte-761	109	1	i	i	PROPN
cte-761	109	2	-	-	PUNCT
cte-761	109	3	mm	mm	PROPN
cte-761	109	4	iii	iii	PROPN
cte-761	109	5	in	in	ADP
cte-761	109	6	excel	excel	NOUN
cte-761	109	7	.	.	PUNCT
cte-761	110	1	students	student	NOUN
cte-761	110	2	using	use	VERB
cte-761	110	3	excel	excel	NOUN
cte-761	110	4	analysed	analyse	VERB
cte-761	110	5	the	the	DET
cte-761	110	6	numerical	numerical	ADJ
cte-761	110	7	results	result	NOUN
cte-761	110	8	and	and	CCONJ
cte-761	110	9	graphs	graph	NOUN
cte-761	110	10	by	by	ADP
cte-761	110	11	varying	vary	VERB
cte-761	110	12	model	model	NOUN
cte-761	110	13	input	input	NOUN
cte-761	110	14	parameters	parameter	NOUN
cte-761	110	15	,	,	PUNCT
cte-761	110	16	enabling	enable	VERB
cte-761	110	17	them	they	PRON
cte-761	110	18	to	to	PART
cte-761	110	19	study	study	VERB
cte-761	110	20	projectile	projectile	NOUN
cte-761	110	21	motion	motion	NOUN
cte-761	110	22	under	under	ADP
cte-761	110	23	different	different	ADJ
cte-761	110	24	conditions	condition	NOUN
cte-761	110	25	,	,	PUNCT
cte-761	110	26	answer	answer	VERB
cte-761	110	27	self	self	NOUN
cte-761	110	28	-	-	PUNCT
cte-761	110	29	assessment	assessment	NOUN
cte-761	110	30	questions	question	NOUN
cte-761	110	31	,	,	PUNCT
cte-761	110	32	and	and	CCONJ
cte-761	110	33	prepare	prepare	VERB
cte-761	110	34	for	for	ADP
cte-761	110	35	examinations	examination	NOUN
cte-761	110	36	.	.	PUNCT
cte-761	111	1	2.4	2.4	NUM
cte-761	111	2	.	.	PUNCT
cte-761	111	3	implementation	implementation	NOUN
cte-761	111	4	in	in	ADP
cte-761	111	5	geogebra	geogebra	NOUN
cte-761	111	6	(	(	PUNCT
cte-761	111	7	group	group	NOUN
cte-761	111	8	g	g	NOUN
cte-761	111	9	)	)	PUNCT
cte-761	111	10	implementing	implement	VERB
cte-761	111	11	mathematical	mathematical	ADJ
cte-761	111	12	models	model	NOUN
cte-761	111	13	of	of	ADP
cte-761	111	14	mechanical	mechanical	ADJ
cte-761	111	15	motion	motion	NOUN
cte-761	111	16	in	in	ADP
cte-761	111	17	geogebra	geogebra	NOUN
cte-761	111	18	involves	involve	VERB
cte-761	111	19	distinct	distinct	ADJ
cte-761	111	20	approaches	approach	NOUN
cte-761	111	21	as	as	SCONJ
cte-761	111	22	previously	previously	ADV
cte-761	111	23	noted	note	VERB
cte-761	111	24	in	in	ADP
cte-761	111	25	[	[	X
cte-761	111	26	2	2	NUM
cte-761	111	27	]	]	PUNCT
cte-761	111	28	.	.	PUNCT
cte-761	112	1	the	the	DET
cte-761	112	2	primary	primary	ADJ
cte-761	112	3	characteristic	characteristic	NOUN
cte-761	112	4	of	of	ADP
cte-761	112	5	geogebra	geogebra	NOUN
cte-761	112	6	is	be	AUX
cte-761	112	7	its	its	PRON
cte-761	112	8	algebraic	algebraic	ADJ
cte-761	112	9	-	-	PUNCT
cte-761	112	10	geometric	geometric	ADJ
cte-761	112	11	approach	approach	NOUN
cte-761	112	12	to	to	ADP
cte-761	112	13	mathematical	mathematical	ADJ
cte-761	112	14	object	object	NOUN
cte-761	112	15	description	description	NOUN
cte-761	112	16	.	.	PUNCT
cte-761	113	1	solving	solve	VERB
cte-761	113	2	mm	mm	PROPN
cte-761	113	3	i	i	PROPN
cte-761	113	4	,	,	PUNCT
cte-761	113	5	mm	mm	PROPN
cte-761	113	6	ii	ii	PROPN
cte-761	113	7	,	,	PUNCT
cte-761	113	8	and	and	CCONJ
cte-761	113	9	mm	mm	NOUN
cte-761	113	10	iii	iii	NOUN
cte-761	113	11	in	in	ADP
cte-761	113	12	geogebra	geogebra	NOUN
cte-761	113	13	is	be	AUX
cte-761	113	14	considerably	considerably	ADV
cte-761	113	15	more	more	ADV
cte-761	113	16	straightforward	straightforward	ADJ
cte-761	113	17	than	than	ADP
cte-761	113	18	in	in	ADP
cte-761	113	19	excel	excel	NOUN
cte-761	113	20	.	.	PUNCT
cte-761	114	1	geogebra	geogebra	NOUN
cte-761	114	2	eliminates	eliminate	VERB
cte-761	114	3	the	the	DET
cte-761	114	4	need	need	NOUN
cte-761	114	5	240	240	NUM
cte-761	114	6	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	114	7	cte	cte	NOUN
cte-761	114	8	workshop	workshop	NOUN
cte-761	114	9	proceedings	proceeding	NOUN
cte-761	114	10	,	,	PUNCT
cte-761	114	11	2025	2025	NUM
cte-761	114	12	,	,	PUNCT
cte-761	114	13	vol	vol	NOUN
cte-761	114	14	.	.	PROPN
cte-761	114	15	12	12	NUM
cte-761	114	16	,	,	PUNCT
cte-761	114	17	pp	pp	ADJ
cte-761	114	18	.	.	PUNCT
cte-761	115	1	235–252	235–252	NUM
cte-761	115	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	115	3	to	to	PART
cte-761	115	4	construct	construct	VERB
cte-761	115	5	detailed	detailed	ADJ
cte-761	115	6	calculation	calculation	NOUN
cte-761	115	7	schemes	scheme	NOUN
cte-761	115	8	for	for	ADP
cte-761	115	9	solving	solve	VERB
cte-761	115	10	differential	differential	ADJ
cte-761	115	11	equation	equation	NOUN
cte-761	115	12	systems	system	NOUN
cte-761	115	13	.	.	PUNCT
cte-761	116	1	instead	instead	ADV
cte-761	116	2	,	,	PUNCT
cte-761	116	3	models	model	NOUN
cte-761	116	4	can	can	AUX
cte-761	116	5	be	be	AUX
cte-761	116	6	entered	enter	VERB
cte-761	116	7	using	use	VERB
cte-761	116	8	geogebra	geogebra	NOUN
cte-761	116	9	syntax	syntax	NOUN
cte-761	116	10	and	and	CCONJ
cte-761	116	11	solved	solve	VERB
cte-761	116	12	using	use	VERB
cte-761	116	13	the	the	DET
cte-761	116	14	built	build	VERB
cte-761	116	15	-	-	PUNCT
cte-761	116	16	in	in	ADP
cte-761	116	17	nsolveode	nsolveode	NOUN
cte-761	116	18	command	command	NOUN
cte-761	116	19	for	for	ADP
cte-761	116	20	numerical	numerical	ADJ
cte-761	116	21	solutions	solution	NOUN
cte-761	116	22	of	of	ADP
cte-761	116	23	first	first	ADJ
cte-761	116	24	-	-	PUNCT
cte-761	116	25	order	order	NOUN
cte-761	116	26	differential	differential	ADJ
cte-761	116	27	equations	equation	NOUN
cte-761	116	28	[	[	X
cte-761	116	29	6	6	NUM
cte-761	116	30	]	]	PUNCT
cte-761	116	31	.	.	PUNCT
cte-761	117	1	mm	mm	INTJ
cte-761	117	2	i	i	PRON
cte-761	117	3	is	be	AUX
cte-761	117	4	represented	represent	VERB
cte-761	117	5	in	in	ADP
cte-761	117	6	geogebra	geogebra	NOUN
cte-761	117	7	based	base	VERB
cte-761	117	8	on	on	ADP
cte-761	117	9	equations	equation	NOUN
cte-761	117	10	(	(	PUNCT
cte-761	117	11	1	1	NUM
cte-761	117	12	)	)	PUNCT
cte-761	117	13	and	and	CCONJ
cte-761	117	14	(	(	PUNCT
cte-761	117	15	2	2	X
cte-761	117	16	)	)	PUNCT
cte-761	117	17	as	as	SCONJ
cte-761	117	18	follows	follow	VERB
cte-761	117	19	(	(	PUNCT
cte-761	117	20	all	all	DET
cte-761	117	21	commands	command	NOUN
cte-761	117	22	are	be	AUX
cte-761	117	23	entered	enter	VERB
cte-761	117	24	through	through	ADP
cte-761	117	25	the	the	DET
cte-761	117	26	command	command	NOUN
cte-761	117	27	line	line	NOUN
cte-761	117	28	):	):	PUNCT
cte-761	118	1	x1’(t	x1’(t	NUM
cte-761	118	2	,	,	PUNCT
cte-761	118	3	x1,y1,vx1,vy1	x1,y1,vx1,vy1	ADJ
cte-761	118	4	)	)	PUNCT
cte-761	118	5	=	=	SYM
cte-761	118	6	vx1	vx1	NOUN
cte-761	118	7	y1’(t	y1’(t	NUM
cte-761	118	8	,	,	PUNCT
cte-761	118	9	x1,y1,vx1,vy1	x1,y1,vx1,vy1	PROPN
cte-761	118	10	)	)	PUNCT
cte-761	118	11	=	=	PUNCT
cte-761	118	12	vy1	vy1	NOUN
cte-761	118	13	vx1’(t	vx1’(t	NOUN
cte-761	118	14	,	,	PUNCT
cte-761	118	15	x1,y1,vx1,vy1	x1,y1,vx1,vy1	PROPN
cte-761	118	16	)	)	PUNCT
cte-761	118	17	=	=	SYM
cte-761	118	18	0	0	NUM
cte-761	118	19	vy1’(t	vy1’(t	NOUN
cte-761	118	20	,	,	PUNCT
cte-761	118	21	x1,y1,vx1,vy1	x1,y1,vx1,vy1	PROPN
cte-761	118	22	)	)	PUNCT
cte-761	118	23	=	=	SYM
cte-761	119	1	-g	-g	PUNCT
cte-761	119	2	the	the	DET
cte-761	119	3	following	follow	VERB
cte-761	119	4	command	command	NOUN
cte-761	119	5	solves	solve	VERB
cte-761	119	6	this	this	DET
cte-761	119	7	differential	differential	ADJ
cte-761	119	8	equation	equation	NOUN
cte-761	119	9	system	system	NOUN
cte-761	119	10	:	:	PUNCT
cte-761	119	11	nsolveode({x1’,y1’,vx1’,vy1’},0,{x0,y0,vx0,vy0},tm	nsolveode({x1’,y1’,vx1’,vy1’},0,{x0,y0,vx0,vy0},tm	NUM
cte-761	119	12	)	)	PUNCT
cte-761	119	13	the	the	DET
cte-761	119	14	nsolveode	nsolveode	NOUN
cte-761	119	15	command	command	NOUN
cte-761	119	16	implements	implement	VERB
cte-761	119	17	the	the	DET
cte-761	119	18	4th	4th	ADJ
cte-761	119	19	-	-	PUNCT
cte-761	119	20	order	order	NOUN
cte-761	119	21	runge	runge	NOUN
cte-761	119	22	-	-	PUNCT
cte-761	119	23	kutta	kutta	NOUN
cte-761	119	24	method	method	NOUN
cte-761	119	25	[	[	X
cte-761	119	26	6	6	NUM
cte-761	119	27	,	,	PUNCT
cte-761	119	28	7	7	NUM
cte-761	119	29	,	,	PUNCT
cte-761	119	30	9	9	NUM
cte-761	119	31	]	]	PUNCT
cte-761	119	32	,	,	PUNCT
cte-761	119	33	generating	generate	VERB
cte-761	119	34	a	a	DET
cte-761	119	35	numerical	numerical	ADJ
cte-761	119	36	solution	solution	NOUN
cte-761	119	37	to	to	ADP
cte-761	119	38	the	the	DET
cte-761	119	39	cauchy	cauchy	ADJ
cte-761	119	40	problem	problem	NOUN
cte-761	119	41	over	over	ADP
cte-761	119	42	the	the	DET
cte-761	119	43	interval	interval	NOUN
cte-761	119	44	𝑡	𝑡	PROPN
cte-761	119	45	∈	∈	PROPN
cte-761	120	1	[	[	X
cte-761	120	2	0	0	NUM
cte-761	120	3	,	,	PUNCT
cte-761	120	4	𝑡𝑀	𝑡𝑀	X
cte-761	120	5	]	]	PUNCT
cte-761	120	6	with	with	ADP
cte-761	120	7	initial	initial	ADJ
cte-761	120	8	conditions	condition	NOUN
cte-761	120	9	{	{	PUNCT
cte-761	120	10	x0	x0	PROPN
cte-761	120	11	,	,	PUNCT
cte-761	120	12	y0	y0	PROPN
cte-761	120	13	,	,	PUNCT
cte-761	120	14	vx0	vx0	NOUN
cte-761	120	15	,	,	PUNCT
cte-761	120	16	vy0	vy0	NOUN
cte-761	120	17	}	}	PUNCT
cte-761	120	18	and	and	CCONJ
cte-761	120	19	other	other	ADJ
cte-761	120	20	input	input	NOUN
cte-761	120	21	parameters	parameter	NOUN
cte-761	120	22	.	.	PUNCT
cte-761	121	1	this	this	PRON
cte-761	121	2	tabulates	tabulate	VERB
cte-761	121	3	four	four	NUM
cte-761	121	4	functions	function	NOUN
cte-761	121	5	:	:	PUNCT
cte-761	121	6	𝑥1	𝑥1	NOUN
cte-761	121	7	=	=	NOUN
cte-761	121	8	𝑥1(𝑡	𝑥1(𝑡	NUM
cte-761	121	9	)	)	PUNCT
cte-761	121	10	,	,	PUNCT
cte-761	121	11	𝑦1	𝑦1	PROPN
cte-761	121	12	=	=	SYM
cte-761	121	13	𝑦1(𝑡	𝑦1(𝑡	PROPN
cte-761	121	14	)	)	PUNCT
cte-761	121	15	,	,	PUNCT
cte-761	121	16	𝑣𝑥1	𝑣𝑥1	NOUN
cte-761	121	17	=	=	SYM
cte-761	121	18	𝑣𝑥1(𝑡	𝑣𝑥1(𝑡	NOUN
cte-761	121	19	)	)	PUNCT
cte-761	121	20	,	,	PUNCT
cte-761	121	21	𝑣𝑦1	𝑣𝑦1	NOUN
cte-761	121	22	=	=	SYM
cte-761	121	23	𝑣𝑦1(𝑡	𝑣𝑦1(𝑡	PROPN
cte-761	121	24	)	)	PUNCT
cte-761	121	25	,	,	PUNCT
cte-761	121	26	which	which	PRON
cte-761	121	27	are	be	AUX
cte-761	121	28	assigned	assign	VERB
cte-761	121	29	identifiers	identifier	NOUN
cte-761	121	30	:	:	PUNCT
cte-761	121	31	numericalintegral1	numericalintegral1	X
cte-761	122	1	=	=	SYM
cte-761	122	2	x1(t	x1(t	PROPN
cte-761	122	3	)	)	PUNCT
cte-761	122	4	numericalintegral2	numericalintegral2	NOUN
cte-761	123	1	=	=	PUNCT
cte-761	123	2	y1(t	y1(t	ADJ
cte-761	123	3	)	)	PUNCT
cte-761	123	4	numericalintegral3	numericalintegral3	NOUN
cte-761	124	1	=	=	SYM
cte-761	124	2	vx1(t	vx1(t	PROPN
cte-761	124	3	)	)	PUNCT
cte-761	124	4	numericalintegral4	numericalintegral4	NOUN
cte-761	125	1	=	=	SYM
cte-761	125	2	vy1(t	vy1(t	PROPN
cte-761	125	3	)	)	PUNCT
cte-761	125	4	similarly	similarly	ADV
cte-761	125	5	,	,	PUNCT
cte-761	125	6	mm	mm	PROPN
cte-761	125	7	ii	ii	NOUN
cte-761	125	8	based	base	VERB
cte-761	125	9	on	on	ADP
cte-761	125	10	equations	equation	NOUN
cte-761	125	11	(	(	PUNCT
cte-761	125	12	1	1	NUM
cte-761	125	13	)	)	PUNCT
cte-761	125	14	and	and	CCONJ
cte-761	125	15	(	(	PUNCT
cte-761	125	16	3	3	X
cte-761	125	17	)	)	PUNCT
cte-761	125	18	is	be	AUX
cte-761	125	19	represented	represent	VERB
cte-761	125	20	as	as	ADP
cte-761	125	21	:	:	PUNCT
cte-761	125	22	x2’(t	x2’(t	NUM
cte-761	125	23	,	,	PUNCT
cte-761	125	24	x2,y2,vx2,vy2)=vx2	x2,y2,vx2,vy2)=vx2	X
cte-761	126	1	y2’(t	y2’(t	NOUN
cte-761	126	2	,	,	PUNCT
cte-761	126	3	x2,y2,vx2,vy2)=vy2	x2,y2,vx2,vy2)=vy2	PROPN
cte-761	126	4	vx2’(t	vx2’(t	ADJ
cte-761	126	5	,	,	PUNCT
cte-761	126	6	x2,y2,vx2,vy2)=-k_2*vx2*sqrt(vx2	x2,y2,vx2,vy2)=-k_2*vx2*sqrt(vx2	PROPN
cte-761	126	7	^	^	SYM
cte-761	126	8	2	2	NUM
cte-761	126	9	+	+	NUM
cte-761	126	10	vy2	vy2	NOUN
cte-761	126	11	^	^	NUM
cte-761	126	12	2)/m	2)/m	NUM
cte-761	126	13	vy2’(t	vy2’(t	NOUN
cte-761	126	14	,	,	PUNCT
cte-761	126	15	x2,y2,vx2,vy2)=-g	x2,y2,vx2,vy2)=-g	PROPN
cte-761	126	16	k_2*vy2*sqrt(vx2	k_2*vy2*sqrt(vx2	PROPN
cte-761	126	17	^	^	SYM
cte-761	126	18	2	2	NUM
cte-761	126	19	+	+	NUM
cte-761	126	20	vy2	vy2	PROPN
cte-761	126	21	^	^	SYM
cte-761	126	22	2)/m	2)/m	NUM
cte-761	126	23	mm	mm	PROPN
cte-761	126	24	ii	ii	NOUN
cte-761	126	25	is	be	AUX
cte-761	126	26	solved	solve	VERB
cte-761	126	27	using	use	VERB
cte-761	126	28	:	:	PUNCT
cte-761	126	29	nsolveode({x2’,y2’,vx2’,vy2’},0,{x0,y0,vx0,vy0},tm	nsolveode({x2’,y2’,vx2’,vy2’},0,{x0,y0,vx0,vy0},tm	X
cte-761	126	30	)	)	PUNCT
cte-761	126	31	this	this	PRON
cte-761	126	32	yields	yield	VERB
cte-761	126	33	four	four	NUM
cte-761	126	34	functions	function	NOUN
cte-761	126	35	as	as	ADP
cte-761	126	36	the	the	DET
cte-761	126	37	mm	mm	PROPN
cte-761	126	38	ii	ii	PROPN
cte-761	126	39	solution	solution	NOUN
cte-761	126	40	:	:	PUNCT
cte-761	126	41	numericalintegral5	numericalintegral5	PUNCT
cte-761	127	1	=	=	SYM
cte-761	127	2	x2(t	x2(t	PROPN
cte-761	127	3	)	)	PUNCT
cte-761	127	4	numericalintegral6	numericalintegral6	NOUN
cte-761	128	1	=	=	SYM
cte-761	128	2	y2(t	y2(t	PROPN
cte-761	128	3	)	)	PUNCT
cte-761	128	4	numericalintegral7	numericalintegral7	NOUN
cte-761	129	1	=	=	SYM
cte-761	129	2	vx2(t	vx2(t	PROPN
cte-761	129	3	)	)	PUNCT
cte-761	129	4	numericalintegral8	numericalintegral8	NOUN
cte-761	130	1	=	=	SYM
cte-761	130	2	vy2(t	vy2(t	PROPN
cte-761	130	3	)	)	PUNCT
cte-761	131	1	likewise	likewise	ADV
cte-761	131	2	,	,	PUNCT
cte-761	131	3	mm	mm	PROPN
cte-761	131	4	iii	iii	NUM
cte-761	131	5	based	base	VERB
cte-761	131	6	on	on	ADP
cte-761	131	7	equations	equation	NOUN
cte-761	131	8	(	(	PUNCT
cte-761	131	9	1	1	NUM
cte-761	131	10	)	)	PUNCT
cte-761	131	11	and	and	CCONJ
cte-761	131	12	(	(	PUNCT
cte-761	131	13	4	4	X
cte-761	131	14	)	)	PUNCT
cte-761	131	15	is	be	AUX
cte-761	131	16	represented	represent	VERB
cte-761	131	17	as	as	ADP
cte-761	131	18	:	:	PUNCT
cte-761	131	19	x3’(t	x3’(t	PROPN
cte-761	131	20	,	,	PUNCT
cte-761	131	21	x3,y3,vx3,vy3)=vx3	x3,y3,vx3,vy3)=vx3	PROPN
cte-761	131	22	y3’(t	y3’(t	PROPN
cte-761	131	23	,	,	PUNCT
cte-761	131	24	x3,y3,vx3,vy3)=vy3	x3,y3,vx3,vy3)=vy3	PROPN
cte-761	131	25	vx3’(t	vx3’(t	PROPN
cte-761	131	26	,	,	PUNCT
cte-761	131	27	x3,y3,vx3,vy3)=-k_2*vx3*sqrt(vx3	x3,y3,vx3,vy3)=-k_2*vx3*sqrt(vx3	PROPN
cte-761	131	28	^	^	SYM
cte-761	131	29	2	2	NUM
cte-761	131	30	+	+	CCONJ
cte-761	131	31	vy3	vy3	ADJ
cte-761	131	32	^	^	NUM
cte-761	131	33	2)/m	2)/m	NUM
cte-761	131	34	+	+	SYM
cte-761	131	35	+	+	NUM
cte-761	131	36	rot*k_3*vy3*sqrt(vx3	rot*k_3*vy3*sqrt(vx3	ADJ
cte-761	131	37	^	^	NUM
cte-761	131	38	2	2	NUM
cte-761	131	39	+	+	CCONJ
cte-761	131	40	vy3	vy3	ADJ
cte-761	131	41	^	^	NOUN
cte-761	131	42	2)/m	2)/m	NUM
cte-761	131	43	vy3’(t	vy3’(t	NUM
cte-761	131	44	,	,	PUNCT
cte-761	131	45	x3,y3,vx3,vy3)=-g	x3,y3,vx3,vy3)=-g	PROPN
cte-761	131	46	k_2*vy3*sqrt(vx3	k_2*vy3*sqrt(vx3	NOUN
cte-761	131	47	^	^	SYM
cte-761	131	48	2	2	NUM
cte-761	131	49	+	+	CCONJ
cte-761	131	50	vy3	vy3	ADJ
cte-761	131	51	^	^	NUM
cte-761	131	52	2)/m	2)/m	NUM
cte-761	131	53	rot*k_3*vx3*sqrt(vx3	rot*k_3*vx3*sqrt(vx3	NOUN
cte-761	131	54	^	^	SYM
cte-761	131	55	2	2	NUM
cte-761	131	56	+	+	CCONJ
cte-761	131	57	vy3	vy3	ADJ
cte-761	131	58	^	^	SYM
cte-761	131	59	2)/m	2)/m	NUM
cte-761	131	60	mm	mm	PROPN
cte-761	131	61	iii	iii	NOUN
cte-761	131	62	is	be	AUX
cte-761	131	63	solved	solve	VERB
cte-761	131	64	using	use	VERB
cte-761	131	65	:	:	PUNCT
cte-761	131	66	nsolveode({x3’,y3’,vx3’,vy3’},0,{x0,y0,vx0,vy0},tm	nsolveode({x3’,y3’,vx3’,vy3’},0,{x0,y0,vx0,vy0},tm	ADJ
cte-761	131	67	)	)	PUNCT
cte-761	131	68	241	241	NUM
cte-761	131	69	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	131	70	cte	cte	NOUN
cte-761	131	71	workshop	workshop	NOUN
cte-761	131	72	proceedings	proceeding	NOUN
cte-761	131	73	,	,	PUNCT
cte-761	131	74	2025	2025	NUM
cte-761	131	75	,	,	PUNCT
cte-761	131	76	vol	vol	NOUN
cte-761	131	77	.	.	PROPN
cte-761	131	78	12	12	NUM
cte-761	131	79	,	,	PUNCT
cte-761	131	80	pp	pp	ADJ
cte-761	131	81	.	.	PUNCT
cte-761	132	1	235–252	235–252	NUM
cte-761	132	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	132	3	this	this	PRON
cte-761	132	4	produces	produce	VERB
cte-761	132	5	the	the	DET
cte-761	132	6	mm	mm	PROPN
cte-761	132	7	iii	iii	NOUN
cte-761	132	8	solution	solution	NOUN
cte-761	132	9	:	:	PUNCT
cte-761	132	10	numericalintegral9	numericalintegral9	PROPN
cte-761	132	11	=	=	SYM
cte-761	132	12	x3(t	x3(t	PROPN
cte-761	132	13	)	)	PUNCT
cte-761	132	14	numericalintegral10	numericalintegral10	NOUN
cte-761	132	15	=	=	SYM
cte-761	132	16	y3(t	y3(t	PROPN
cte-761	132	17	)	)	PUNCT
cte-761	132	18	numericalintegral11	numericalintegral11	NOUN
cte-761	132	19	=	=	SYM
cte-761	132	20	vx3(t	vx3(t	ADJ
cte-761	132	21	)	)	PUNCT
cte-761	132	22	numericalintegral12	numericalintegral12	NOUN
cte-761	133	1	=	=	SYM
cte-761	133	2	vy3(t	vy3(t	PROPN
cte-761	133	3	)	)	PUNCT
cte-761	133	4	due	due	ADP
cte-761	133	5	to	to	ADP
cte-761	133	6	the	the	DET
cte-761	133	7	uniformity	uniformity	NOUN
cte-761	133	8	of	of	ADP
cte-761	133	9	operations	operation	NOUN
cte-761	133	10	,	,	PUNCT
cte-761	133	11	solutions	solution	NOUN
cte-761	133	12	for	for	ADP
cte-761	133	13	all	all	DET
cte-761	133	14	three	three	NUM
cte-761	133	15	models	model	NOUN
cte-761	133	16	were	be	AUX
cte-761	133	17	obtained	obtain	VERB
cte-761	133	18	in	in	ADP
cte-761	133	19	geogebra	geogebra	NOUN
cte-761	133	20	significantly	significantly	ADV
cte-761	133	21	faster	fast	ADV
cte-761	133	22	than	than	ADP
cte-761	133	23	in	in	ADP
cte-761	133	24	excel	excel	NOUN
cte-761	133	25	.	.	PUNCT
cte-761	134	1	once	once	SCONJ
cte-761	134	2	solutions	solution	NOUN
cte-761	134	3	were	be	AUX
cte-761	134	4	available	available	ADJ
cte-761	134	5	,	,	PUNCT
cte-761	134	6	students	student	NOUN
cte-761	134	7	could	could	AUX
cte-761	134	8	readily	readily	ADV
cte-761	134	9	visualise	visualise	VERB
cte-761	134	10	the	the	DET
cte-761	134	11	dynamic	dynamic	ADJ
cte-761	134	12	characteristics	characteristic	NOUN
cte-761	134	13	of	of	ADP
cte-761	134	14	the	the	DET
cte-761	134	15	mathematical	mathematical	ADJ
cte-761	134	16	models	model	NOUN
cte-761	134	17	as	as	SCONJ
cte-761	134	18	shown	show	VERB
cte-761	134	19	in	in	ADP
cte-761	134	20	figure	figure	NOUN
cte-761	134	21	5	5	NUM
cte-761	134	22	.	.	PUNCT
cte-761	134	23	figure	figure	VERB
cte-761	134	24	5	5	NUM
cte-761	134	25	:	:	PUNCT
cte-761	134	26	trajectories	trajectory	NOUN
cte-761	134	27	of	of	ADP
cte-761	134	28	projectiles	projectile	NOUN
cte-761	134	29	for	for	ADP
cte-761	134	30	models	model	NOUN
cte-761	135	1	mm	mm	INTJ
cte-761	136	1	i	i	PROPN
cte-761	136	2	-	-	PUNCT
cte-761	136	3	mm	mm	PROPN
cte-761	136	4	iii	iii	PROPN
cte-761	136	5	in	in	ADP
cte-761	136	6	geogebra	geogebra	NOUN
cte-761	136	7	for	for	ADP
cte-761	136	8	different	different	ADJ
cte-761	136	9	initial	initial	ADJ
cte-761	136	10	angles	angle	NOUN
cte-761	136	11	𝛼0	𝛼0	NUM
cte-761	136	12	:	:	PUNCT
cte-761	137	1	30∘	30∘	NUM
cte-761	137	2	,	,	PUNCT
cte-761	137	3	45∘	45∘	NOUN
cte-761	137	4	,	,	PUNCT
cte-761	137	5	and	and	CCONJ
cte-761	137	6	60∘.	60∘.	NUM
cte-761	137	7	the	the	DET
cte-761	137	8	trajectories	trajectory	NOUN
cte-761	137	9	visualised	visualise	VERB
cte-761	137	10	in	in	ADP
cte-761	137	11	geogebra	geogebra	NOUN
cte-761	137	12	for	for	ADP
cte-761	137	13	models	model	NOUN
cte-761	138	1	mm	mm	INTJ
cte-761	138	2	i	i	PROPN
cte-761	138	3	,	,	PUNCT
cte-761	138	4	mm	mm	PROPN
cte-761	138	5	ii	ii	PROPN
cte-761	138	6	,	,	PUNCT
cte-761	138	7	and	and	CCONJ
cte-761	138	8	mm	mm	PROPN
cte-761	138	9	iii	iii	NUM
cte-761	138	10	used	use	VERB
cte-761	138	11	the	the	DET
cte-761	138	12	function	function	NOUN
cte-761	138	13	values	value	NOUN
cte-761	138	14	x1(t	x1(t	PUNCT
cte-761	138	15	)	)	PUNCT
cte-761	138	16	,	,	PUNCT
cte-761	138	17	y1(t	y1(t	PROPN
cte-761	138	18	)	)	PUNCT
cte-761	138	19	,	,	PUNCT
cte-761	138	20	x2(t	x2(t	PROPN
cte-761	138	21	)	)	PUNCT
cte-761	138	22	,	,	PUNCT
cte-761	138	23	y2(t	y2(t	PROPN
cte-761	138	24	)	)	PUNCT
cte-761	138	25	,	,	PUNCT
cte-761	138	26	x3(t	x3(t	PROPN
cte-761	138	27	)	)	PUNCT
cte-761	138	28	,	,	PUNCT
cte-761	138	29	y3(t	y3(t	PROPN
cte-761	138	30	)	)	PUNCT
cte-761	138	31	as	as	ADP
cte-761	138	32	coordinates	coordinate	NOUN
cte-761	138	33	for	for	ADP
cte-761	138	34	three	three	NUM
cte-761	138	35	moving	move	VERB
cte-761	138	36	points	point	NOUN
cte-761	138	37	a=(x1	a=(x1	ADJ
cte-761	138	38	,	,	PUNCT
cte-761	138	39	y1	y1	NOUN
cte-761	138	40	)	)	PUNCT
cte-761	138	41	,	,	PUNCT
cte-761	138	42	b=(x2	b=(x2	NOUN
cte-761	138	43	,	,	PUNCT
cte-761	138	44	y2	y2	PROPN
cte-761	138	45	)	)	PUNCT
cte-761	138	46	,	,	PUNCT
cte-761	138	47	and	and	CCONJ
cte-761	138	48	c=(x3	c=(x3	NOUN
cte-761	138	49	,	,	PUNCT
cte-761	138	50	y3	y3	PROPN
cte-761	138	51	)	)	PUNCT
cte-761	138	52	respectively	respectively	ADV
cte-761	138	53	.	.	PUNCT
cte-761	139	1	figure	figure	NOUN
cte-761	139	2	5	5	NUM
cte-761	139	3	shows	show	VERB
cte-761	139	4	projectile	projectile	NOUN
cte-761	139	5	trajectories	trajectory	NOUN
cte-761	139	6	(	(	PUNCT
cte-761	139	7	for	for	ADP
cte-761	139	8	three	three	NUM
cte-761	139	9	different	different	ADJ
cte-761	139	10	initial	initial	ADJ
cte-761	139	11	angles	angle	NOUN
cte-761	139	12	)	)	PUNCT
cte-761	139	13	for	for	ADP
cte-761	139	14	models	model	NOUN
cte-761	139	15	:	:	PUNCT
cte-761	140	1	mm	mm	INTJ
cte-761	140	2	i	i	PRON
cte-761	140	3	(	(	PUNCT
cte-761	140	4	red	red	ADJ
cte-761	140	5	line	line	NOUN
cte-761	140	6	)	)	PUNCT
cte-761	140	7	,	,	PUNCT
cte-761	140	8	mm	mm	PROPN
cte-761	140	9	ii	ii	PROPN
cte-761	140	10	(	(	PUNCT
cte-761	140	11	green	green	ADJ
cte-761	140	12	line	line	NOUN
cte-761	140	13	)	)	PUNCT
cte-761	140	14	,	,	PUNCT
cte-761	140	15	and	and	CCONJ
cte-761	140	16	mm	mm	PROPN
cte-761	140	17	iii	iii	PROPN
cte-761	140	18	(	(	PUNCT
cte-761	140	19	blue	blue	ADJ
cte-761	140	20	line	line	NOUN
cte-761	140	21	)	)	PUNCT
cte-761	140	22	.	.	PUNCT
cte-761	141	1	the	the	DET
cte-761	141	2	arrows	arrow	NOUN
cte-761	141	3	in	in	ADP
cte-761	141	4	figure	figure	NOUN
cte-761	141	5	5	5	NUM
cte-761	141	6	represent	represent	VERB
cte-761	141	7	velocity	velocity	NOUN
cte-761	141	8	vectors	vector	NOUN
cte-761	141	9	and	and	CCONJ
cte-761	141	10	force	force	NOUN
cte-761	141	11	vectors	vector	NOUN
cte-761	141	12	acting	act	VERB
cte-761	141	13	on	on	ADP
cte-761	141	14	the	the	DET
cte-761	141	15	projectiles	projectile	NOUN
cte-761	141	16	during	during	ADP
cte-761	141	17	motion	motion	NOUN
cte-761	141	18	.	.	PUNCT
cte-761	142	1	geogebra	geogebra	NOUN
cte-761	142	2	provides	provide	VERB
cte-761	142	3	superior	superior	ADJ
cte-761	142	4	graphics	graphic	NOUN
cte-761	142	5	compared	compare	VERB
cte-761	142	6	to	to	PART
cte-761	142	7	excel	excel	VERB
cte-761	142	8	and	and	CCONJ
cte-761	142	9	offers	offer	VERB
cte-761	142	10	additional	additional	ADJ
cte-761	142	11	visualisation	visualisation	NOUN
cte-761	142	12	capabilities	capability	NOUN
cte-761	142	13	.	.	PUNCT
cte-761	143	1	using	use	VERB
cte-761	143	2	the	the	DET
cte-761	143	3	animation	animation	NOUN
cte-761	143	4	feature	feature	NOUN
cte-761	143	5	,	,	PUNCT
cte-761	143	6	students	student	NOUN
cte-761	143	7	could	could	AUX
cte-761	143	8	observe	observe	VERB
cte-761	143	9	dynamic	dynamic	ADJ
cte-761	143	10	function	function	NOUN
cte-761	143	11	graphing	graphing	NOUN
cte-761	143	12	—	—	PUNCT
cte-761	143	13	effectively	effectively	ADV
cte-761	143	14	simulating	simulate	VERB
cte-761	143	15	the	the	DET
cte-761	143	16	mechanical	mechanical	ADJ
cte-761	143	17	motion	motion	NOUN
cte-761	143	18	process	process	NOUN
cte-761	143	19	.	.	PUNCT
cte-761	144	1	figures	figure	NOUN
cte-761	144	2	6	6	NUM
cte-761	144	3	and	and	CCONJ
cte-761	144	4	7	7	NUM
cte-761	144	5	show	show	NOUN
cte-761	144	6	trajectory	trajectory	NOUN
cte-761	144	7	fragments	fragment	NOUN
cte-761	144	8	for	for	ADP
cte-761	144	9	opposite	opposite	ADJ
cte-761	144	10	rotation	rotation	NOUN
cte-761	144	11	directions	direction	NOUN
cte-761	144	12	controlled	control	VERB
cte-761	144	13	in	in	ADP
cte-761	144	14	mm	mm	PROPN
cte-761	144	15	iii	iii	NOUN
cte-761	144	16	by	by	ADP
cte-761	144	17	the	the	DET
cte-761	144	18	parameter	parameter	NOUN
cte-761	144	19	𝑟𝑜𝑡.	𝑟𝑜𝑡.	PUNCT
cte-761	144	20	velocity	velocity	NOUN
cte-761	144	21	vectors	vector	NOUN
cte-761	144	22	in	in	ADP
cte-761	144	23	figures	figure	NOUN
cte-761	144	24	6	6	NUM
cte-761	144	25	and	and	CCONJ
cte-761	144	26	7	7	NUM
cte-761	144	27	are	be	AUX
cte-761	144	28	defined	define	VERB
cte-761	144	29	in	in	ADP
cte-761	144	30	geogebra	geogebra	NOUN
cte-761	144	31	as	as	SCONJ
cte-761	144	32	follows	follow	VERB
cte-761	144	33	:	:	PUNCT
cte-761	144	34	v1	v1	NOUN
cte-761	144	35	=	=	SYM
cte-761	144	36	vector(a	vector(a	PROPN
cte-761	144	37	,	,	PUNCT
cte-761	144	38	a3	a3	NOUN
cte-761	144	39	)	)	PUNCT
cte-761	144	40	,	,	PUNCT
cte-761	144	41	where	where	SCONJ
cte-761	144	42	a3=(x(a1	a3=(x(a1	NOUN
cte-761	144	43	)	)	PUNCT
cte-761	144	44	,	,	PUNCT
cte-761	144	45	y(a2	y(a2	NOUN
cte-761	144	46	)	)	PUNCT
cte-761	144	47	)	)	PUNCT
cte-761	144	48	,	,	PUNCT
cte-761	144	49	a1=(x(a	a1=(x(a	NOUN
cte-761	144	50	)	)	PUNCT
cte-761	144	51	+	+	SYM
cte-761	144	52	mst*y(point(numericalintegral3	mst*y(point(numericalintegral3	NOUN
cte-761	144	53	,	,	PUNCT
cte-761	144	54	c	c	NOUN
cte-761	144	55	)	)	PUNCT
cte-761	144	56	)	)	PUNCT
cte-761	144	57	,	,	PUNCT
cte-761	144	58	y(a	y(a	PROPN
cte-761	144	59	)	)	PUNCT
cte-761	144	60	)	)	PUNCT
cte-761	144	61	or	or	CCONJ
cte-761	144	62	a1=(x(a	a1=(x(a	NOUN
cte-761	144	63	)	)	PUNCT
cte-761	144	64	+	+	CCONJ
cte-761	144	65	mst*vx1	mst*vx1	PROPN
cte-761	144	66	,	,	PUNCT
cte-761	144	67	y(a	y(a	X
cte-761	144	68	)	)	PUNCT
cte-761	144	69	)	)	PUNCT
cte-761	144	70	,	,	PUNCT
cte-761	144	71	a2=(x(a	a2=(x(a	NOUN
cte-761	144	72	)	)	PUNCT
cte-761	144	73	,	,	PUNCT
cte-761	144	74	y(a	y(a	X
cte-761	144	75	)	)	PUNCT
cte-761	145	1	+	+	CCONJ
cte-761	145	2	mst*y(point(numericalintegral4	mst*y(point(numericalintegral4	NOUN
cte-761	145	3	,	,	PUNCT
cte-761	145	4	c	c	NOUN
cte-761	145	5	)	)	PUNCT
cte-761	145	6	)	)	PUNCT
cte-761	145	7	)	)	PUNCT
cte-761	145	8	or	or	CCONJ
cte-761	145	9	a2=(x(a	a2=(x(a	NOUN
cte-761	145	10	)	)	PUNCT
cte-761	145	11	,	,	PUNCT
cte-761	145	12	y(a	y(a	X
cte-761	145	13	)	)	PUNCT
cte-761	146	1	+	+	CCONJ
cte-761	146	2	mst	mst	PROPN
cte-761	146	3	*	*	PROPN
cte-761	146	4	vy1	vy1	PROPN
cte-761	146	5	)	)	PUNCT
cte-761	146	6	;	;	PUNCT
cte-761	146	7	242	242	NUM
cte-761	146	8	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	146	9	cte	cte	NOUN
cte-761	146	10	workshop	workshop	NOUN
cte-761	146	11	proceedings	proceeding	NOUN
cte-761	146	12	,	,	PUNCT
cte-761	146	13	2025	2025	NUM
cte-761	146	14	,	,	PUNCT
cte-761	146	15	vol	vol	NOUN
cte-761	146	16	.	.	PROPN
cte-761	146	17	12	12	NUM
cte-761	146	18	,	,	PUNCT
cte-761	146	19	pp	pp	ADJ
cte-761	146	20	.	.	PUNCT
cte-761	147	1	235–252	235–252	NUM
cte-761	147	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	147	3	figure	figure	NOUN
cte-761	147	4	6	6	NUM
cte-761	147	5	:	:	PUNCT
cte-761	147	6	trajectory	trajectory	NOUN
cte-761	147	7	fragments	fragment	NOUN
cte-761	147	8	plotted	plot	VERB
cte-761	147	9	in	in	ADP
cte-761	147	10	geogebra	geogebra	NOUN
cte-761	147	11	for	for	ADP
cte-761	147	12	models	model	NOUN
cte-761	147	13	mm	mm	INTJ
cte-761	148	1	i	i	PROPN
cte-761	148	2	-	-	PUNCT
cte-761	148	3	mm	mm	PROPN
cte-761	148	4	iii	iii	NOUN
cte-761	148	5	with	with	ADP
cte-761	148	6	rot=+1	rot=+1	NOUN
cte-761	148	7	(	(	PUNCT
cte-761	148	8	clockwise	clockwise	NOUN
cte-761	148	9	rotation	rotation	NOUN
cte-761	148	10	)	)	PUNCT
cte-761	148	11	.	.	PUNCT
cte-761	149	1	v2	v2	NOUN
cte-761	149	2	=	=	SYM
cte-761	149	3	vector(b	vector(b	PROPN
cte-761	149	4	,	,	PUNCT
cte-761	149	5	b3	b3	PROPN
cte-761	149	6	)	)	PUNCT
cte-761	149	7	,	,	PUNCT
cte-761	149	8	where	where	SCONJ
cte-761	149	9	b3=(x(b1	b3=(x(b1	NOUN
cte-761	149	10	)	)	PUNCT
cte-761	149	11	,	,	PUNCT
cte-761	149	12	y(b2	y(b2	NOUN
cte-761	149	13	)	)	PUNCT
cte-761	149	14	)	)	PUNCT
cte-761	149	15	,	,	PUNCT
cte-761	149	16	b1=(x(b	b1=(x(b	NUM
cte-761	149	17	)	)	PUNCT
cte-761	149	18	+	+	CCONJ
cte-761	149	19	mst*y(point(numericalintegral7	mst*y(point(numericalintegral7	ADJ
cte-761	149	20	,	,	PUNCT
cte-761	149	21	c	c	NOUN
cte-761	149	22	)	)	PUNCT
cte-761	149	23	)	)	PUNCT
cte-761	149	24	,	,	PUNCT
cte-761	149	25	y(b	y(b	PROPN
cte-761	149	26	)	)	PUNCT
cte-761	149	27	)	)	PUNCT
cte-761	149	28	or	or	CCONJ
cte-761	149	29	b1=(x(b	b1=(x(b	NUM
cte-761	149	30	)	)	PUNCT
cte-761	149	31	+	+	X
cte-761	149	32	mst*vx2	mst*vx2	PROPN
cte-761	149	33	,	,	PUNCT
cte-761	149	34	y(b	y(b	PROPN
cte-761	149	35	)	)	PUNCT
cte-761	149	36	)	)	PUNCT
cte-761	149	37	,	,	PUNCT
cte-761	149	38	b2=(x(b	b2=(x(b	PROPN
cte-761	149	39	)	)	PUNCT
cte-761	149	40	,	,	PUNCT
cte-761	149	41	y(b	y(b	PROPN
cte-761	149	42	)	)	PUNCT
cte-761	150	1	+	+	X
cte-761	150	2	mst*y(point(numericalintegral8	mst*y(point(numericalintegral8	ADJ
cte-761	150	3	,	,	PUNCT
cte-761	150	4	c	c	NOUN
cte-761	150	5	)	)	PUNCT
cte-761	150	6	)	)	PUNCT
cte-761	150	7	)	)	PUNCT
cte-761	151	1	or	or	CCONJ
cte-761	151	2	b2=(x(b	b2=(x(b	VERB
cte-761	151	3	)	)	PUNCT
cte-761	151	4	,	,	PUNCT
cte-761	151	5	y(b	y(b	PROPN
cte-761	151	6	)	)	PUNCT
cte-761	151	7	+	+	CCONJ
cte-761	151	8	mst	mst	PROPN
cte-761	151	9	*	*	PROPN
cte-761	151	10	vy2	vy2	PROPN
cte-761	151	11	)	)	PUNCT
cte-761	151	12	;	;	PUNCT
cte-761	152	1	v3	v3	PROPN
cte-761	152	2	=	=	SYM
cte-761	152	3	vector(c	vector(c	PROPN
cte-761	152	4	,	,	PUNCT
cte-761	152	5	c3	c3	PROPN
cte-761	152	6	)	)	PUNCT
cte-761	152	7	,	,	PUNCT
cte-761	152	8	where	where	SCONJ
cte-761	152	9	c3=(x(c1	c3=(x(c1	NOUN
cte-761	152	10	)	)	PUNCT
cte-761	152	11	,	,	PUNCT
cte-761	152	12	y(c2	y(c2	NOUN
cte-761	152	13	)	)	PUNCT
cte-761	152	14	)	)	PUNCT
cte-761	152	15	,	,	PUNCT
cte-761	152	16	c1=(x(c	c1=(x(c	PRON
cte-761	152	17	)	)	PUNCT
cte-761	152	18	+	+	X
cte-761	152	19	mst*y(point(numericalintegral11	mst*y(point(numericalintegral11	ADJ
cte-761	152	20	,	,	PUNCT
cte-761	152	21	c	c	NOUN
cte-761	152	22	)	)	PUNCT
cte-761	152	23	)	)	PUNCT
cte-761	152	24	,	,	PUNCT
cte-761	152	25	y(c	y(c	NOUN
cte-761	152	26	)	)	PUNCT
cte-761	152	27	)	)	PUNCT
cte-761	152	28	or	or	CCONJ
cte-761	152	29	c1=(x(c	c1=(x(c	NOUN
cte-761	152	30	)	)	PUNCT
cte-761	152	31	+	+	CCONJ
cte-761	152	32	mst*vx3	mst*vx3	NOUN
cte-761	152	33	,	,	PUNCT
cte-761	152	34	y(c	y(c	PROPN
cte-761	152	35	)	)	PUNCT
cte-761	152	36	)	)	PUNCT
cte-761	152	37	,	,	PUNCT
cte-761	152	38	c2=(x(c	c2=(x(c	PROPN
cte-761	152	39	)	)	PUNCT
cte-761	152	40	,	,	PUNCT
cte-761	152	41	y(c	y(c	PROPN
cte-761	152	42	)	)	PUNCT
cte-761	153	1	+	+	X
cte-761	153	2	mst*y(point(numericalintegral12	mst*y(point(numericalintegral12	ADJ
cte-761	153	3	,	,	PUNCT
cte-761	153	4	c	c	NOUN
cte-761	153	5	)	)	PUNCT
cte-761	153	6	)	)	PUNCT
cte-761	153	7	)	)	PUNCT
cte-761	153	8	or	or	CCONJ
cte-761	153	9	c2=(x(c	c2=(x(c	PROPN
cte-761	153	10	)	)	PUNCT
cte-761	153	11	,	,	PUNCT
cte-761	153	12	y(c	y(c	PROPN
cte-761	153	13	)	)	PUNCT
cte-761	154	1	+	+	CCONJ
cte-761	154	2	mst	mst	PROPN
cte-761	154	3	*	*	PROPN
cte-761	154	4	vy3	vy3	PROPN
cte-761	154	5	)	)	PUNCT
cte-761	154	6	;	;	PUNCT
cte-761	154	7	where	where	SCONJ
cte-761	154	8	mst	mst	PROPN
cte-761	154	9	is	be	AUX
cte-761	154	10	a	a	DET
cte-761	154	11	scale	scale	NOUN
cte-761	154	12	factor	factor	NOUN
cte-761	154	13	for	for	ADP
cte-761	154	14	vector	vector	NOUN
cte-761	154	15	display	display	NOUN
cte-761	154	16	(	(	PUNCT
cte-761	154	17	in	in	ADP
cte-761	154	18	figures	figure	NOUN
cte-761	154	19	5	5	NUM
cte-761	154	20	-	-	SYM
cte-761	154	21	7	7	NUM
cte-761	154	22	,	,	PUNCT
cte-761	154	23	mst	mst	PROPN
cte-761	154	24	=	=	PROPN
cte-761	154	25	0.1	0.1	NUM
cte-761	154	26	)	)	PUNCT
cte-761	154	27	.	.	PUNCT
cte-761	155	1	force	force	NOUN
cte-761	155	2	vectors	vector	NOUN
cte-761	155	3	acting	act	VERB
cte-761	155	4	on	on	ADP
cte-761	155	5	the	the	DET
cte-761	155	6	projectiles	projectile	NOUN
cte-761	155	7	and	and	CCONJ
cte-761	155	8	their	their	PRON
cte-761	155	9	resultants	resultant	NOUN
cte-761	155	10	for	for	ADP
cte-761	155	11	the	the	DET
cte-761	155	12	three	three	NUM
cte-761	155	13	models	model	NOUN
cte-761	155	14	are	be	AUX
cte-761	155	15	represented	represent	VERB
cte-761	155	16	in	in	ADP
cte-761	155	17	figures	figure	NOUN
cte-761	155	18	6	6	NUM
cte-761	155	19	and	and	CCONJ
cte-761	155	20	7	7	NUM
cte-761	155	21	.	.	NOUN
cte-761	155	22	indices	index	NOUN
cte-761	155	23	1	1	NUM
cte-761	155	24	,	,	PUNCT
cte-761	155	25	2	2	NUM
cte-761	155	26	,	,	PUNCT
cte-761	155	27	and	and	CCONJ
cte-761	155	28	3	3	NUM
cte-761	155	29	refer	refer	VERB
cte-761	155	30	to	to	ADP
cte-761	155	31	models	model	NOUN
cte-761	155	32	mm	mm	INTJ
cte-761	156	1	i	i	PROPN
cte-761	156	2	,	,	PUNCT
cte-761	156	3	mm	mm	PROPN
cte-761	156	4	ii	ii	PROPN
cte-761	156	5	,	,	PUNCT
cte-761	156	6	and	and	CCONJ
cte-761	156	7	mm	mm	PROPN
cte-761	156	8	iii	iii	NOUN
cte-761	156	9	respectively	respectively	ADV
cte-761	156	10	:	:	PUNCT
cte-761	156	11	f1	f1	PROPN
cte-761	156	12	=	=	SYM
cte-761	156	13	vector(a	vector(a	PROPN
cte-761	156	14	,	,	PUNCT
cte-761	156	15	f1_1	f1_1	PRON
cte-761	156	16	)	)	PUNCT
cte-761	156	17	,	,	PUNCT
cte-761	156	18	where	where	SCONJ
cte-761	156	19	f1_1=(x(a),y(a	f1_1=(x(a),y(a	X
cte-761	156	20	)	)	PUNCT
cte-761	156	21	mst*m*g	mst*m*g	NOUN
cte-761	156	22	)	)	PUNCT
cte-761	156	23	;	;	PUNCT
cte-761	156	24	243	243	NUM
cte-761	156	25	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	156	26	cte	cte	NOUN
cte-761	156	27	workshop	workshop	NOUN
cte-761	156	28	proceedings	proceeding	NOUN
cte-761	156	29	,	,	PUNCT
cte-761	156	30	2025	2025	NUM
cte-761	156	31	,	,	PUNCT
cte-761	156	32	vol	vol	NOUN
cte-761	156	33	.	.	PROPN
cte-761	156	34	12	12	NUM
cte-761	156	35	,	,	PUNCT
cte-761	156	36	pp	pp	ADJ
cte-761	156	37	.	.	PUNCT
cte-761	157	1	235–252	235–252	NUM
cte-761	157	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	157	3	figure	figure	NOUN
cte-761	157	4	7	7	NUM
cte-761	157	5	:	:	PUNCT
cte-761	157	6	trajectory	trajectory	NOUN
cte-761	157	7	fragments	fragment	NOUN
cte-761	157	8	plotted	plot	VERB
cte-761	157	9	in	in	ADP
cte-761	157	10	geogebra	geogebra	NOUN
cte-761	157	11	for	for	ADP
cte-761	157	12	models	model	NOUN
cte-761	157	13	mm	mm	INTJ
cte-761	158	1	i	i	PROPN
cte-761	158	2	-	-	PUNCT
cte-761	158	3	mm	mm	PROPN
cte-761	158	4	iii	iii	NOUN
cte-761	158	5	with	with	ADP
cte-761	158	6	rot=-1	rot=-1	NOUN
cte-761	158	7	(	(	PUNCT
cte-761	158	8	counterclockwise	counterclockwise	NOUN
cte-761	158	9	rotation	rotation	NOUN
cte-761	158	10	)	)	PUNCT
cte-761	158	11	.	.	PUNCT
cte-761	159	1	f2	f2	PROPN
cte-761	159	2	=	=	SYM
cte-761	159	3	vector(b	vector(b	PROPN
cte-761	159	4	,	,	PUNCT
cte-761	159	5	f2_3	f2_3	ADJ
cte-761	159	6	)	)	PUNCT
cte-761	159	7	,	,	PUNCT
cte-761	159	8	where	where	SCONJ
cte-761	159	9	f2_3	f2_3	ADV
cte-761	159	10	=	=	PRON
cte-761	159	11	f2_1	f2_1	ADJ
cte-761	159	12	+	+	CCONJ
cte-761	159	13	f2_2	f2_2	ADJ
cte-761	159	14	b	b	NOUN
cte-761	159	15	,	,	PUNCT
cte-761	159	16	f2_1=(x(b),y(b	f2_1=(x(b),y(b	NOUN
cte-761	159	17	)	)	PUNCT
cte-761	159	18	mst*m*g	mst*m*g	NOUN
cte-761	159	19	)	)	PUNCT
cte-761	159	20	,	,	PUNCT
cte-761	159	21	f2_2=(x(b	f2_2=(x(b	PROPN
cte-761	159	22	)	)	PUNCT
cte-761	160	1	+	+	CCONJ
cte-761	160	2	mst*f2x_r	mst*f2x_r	PROPN
cte-761	160	3	,	,	PUNCT
cte-761	160	4	y(b	y(b	PROPN
cte-761	160	5	)	)	PUNCT
cte-761	161	1	+	+	NUM
cte-761	161	2	mst*f2y_r	mst*f2y_r	NOUN
cte-761	161	3	)	)	PUNCT
cte-761	161	4	,	,	PUNCT
cte-761	161	5	f2x_r=	f2x_r=	X
cte-761	161	6	-k_2*vx2*sqrt(vx2	-k_2*vx2*sqrt(vx2	SYM
cte-761	161	7	^	^	SYM
cte-761	161	8	2	2	NUM
cte-761	161	9	+	+	NUM
cte-761	161	10	vy2	vy2	NOUN
cte-761	161	11	^	^	SYM
cte-761	161	12	2	2	NUM
cte-761	161	13	)	)	PUNCT
cte-761	161	14	,	,	PUNCT
cte-761	161	15	f2y_r=	f2y_r=	X
cte-761	161	16	-k_2*vy2*sqrt(vx2	-k_2*vy2*sqrt(vx2	SYM
cte-761	161	17	^	^	SYM
cte-761	161	18	2	2	NUM
cte-761	161	19	+	+	CCONJ
cte-761	161	20	vy2	vy2	NOUN
cte-761	161	21	^	^	SYM
cte-761	161	22	2	2	NUM
cte-761	161	23	)	)	PUNCT
cte-761	161	24	;	;	PUNCT
cte-761	161	25	f2_t=	f2_t=	NOUN
cte-761	161	26	vector(b	vector(b	PROPN
cte-761	161	27	,	,	PUNCT
cte-761	161	28	f2_1	f2_1	NUM
cte-761	161	29	)	)	PUNCT
cte-761	161	30	;	;	PUNCT
cte-761	161	31	f2_r=	f2_r=	PROPN
cte-761	161	32	vector(b	vector(b	PROPN
cte-761	161	33	,	,	PUNCT
cte-761	161	34	f2_2	f2_2	ADJ
cte-761	161	35	)	)	PUNCT
cte-761	161	36	;	;	PUNCT
cte-761	161	37	f3=	f3=	PROPN
cte-761	161	38	vector(c	vector(c	PROPN
cte-761	161	39	,	,	PUNCT
cte-761	161	40	f3_5	f3_5	ADJ
cte-761	161	41	)	)	PUNCT
cte-761	161	42	,	,	PUNCT
cte-761	161	43	where	where	SCONJ
cte-761	161	44	f3_5=	f3_5=	NOUN
cte-761	161	45	f3_3	f3_3	X
cte-761	161	46	+	+	CCONJ
cte-761	161	47	f3_4	f3_4	ADJ
cte-761	161	48	c	c	PROPN
cte-761	161	49	,	,	PUNCT
cte-761	161	50	f3_3=	f3_3=	VERB
cte-761	161	51	f3_1	f3_1	NOUN
cte-761	161	52	+	+	CCONJ
cte-761	161	53	f3_2	f3_2	ADJ
cte-761	161	54	c	c	NOUN
cte-761	161	55	,	,	PUNCT
cte-761	161	56	f3_4=	f3_4=	ADJ
cte-761	161	57	(	(	PUNCT
cte-761	161	58	x(c	x(c	PROPN
cte-761	161	59	)	)	PUNCT
cte-761	162	1	+	+	X
cte-761	162	2	mst*f3x_m	mst*f3x_m	NOUN
cte-761	162	3	,	,	PUNCT
cte-761	162	4	y(c	y(c	PROPN
cte-761	162	5	)	)	PUNCT
cte-761	163	1	+	+	CCONJ
cte-761	163	2	mst*f3y_m	mst*f3y_m	NOUN
cte-761	163	3	)	)	PUNCT
cte-761	163	4	,	,	PUNCT
cte-761	163	5	f3_1=	f3_1=	X
cte-761	163	6	(	(	PUNCT
cte-761	163	7	x(c	x(c	PROPN
cte-761	163	8	)	)	PUNCT
cte-761	163	9	,	,	PUNCT
cte-761	163	10	y(c	y(c	PROPN
cte-761	163	11	)	)	PUNCT
cte-761	163	12	mst*m*g	mst*m*g	PROPN
cte-761	163	13	)	)	PUNCT
cte-761	163	14	,	,	PUNCT
cte-761	163	15	f3_2=	f3_2=	NOUN
cte-761	163	16	(	(	PUNCT
cte-761	163	17	x(c	x(c	PROPN
cte-761	163	18	)	)	PUNCT
cte-761	164	1	+	+	X
cte-761	164	2	mst*f3x_r	mst*f3x_r	ADJ
cte-761	164	3	,	,	PUNCT
cte-761	164	4	y(c	y(c	ADJ
cte-761	164	5	)	)	PUNCT
cte-761	165	1	+	+	NUM
cte-761	165	2	mst*f3y_r	mst*f3y_r	NOUN
cte-761	165	3	)	)	PUNCT
cte-761	165	4	,	,	PUNCT
cte-761	165	5	f3x_r=-k_2*vx3*sqrt(vx3	f3x_r=-k_2*vx3*sqrt(vx3	NOUN
cte-761	165	6	^	^	SYM
cte-761	165	7	2	2	NUM
cte-761	165	8	+	+	CCONJ
cte-761	165	9	vy3	vy3	ADJ
cte-761	165	10	^	^	SYM
cte-761	165	11	2	2	NUM
cte-761	165	12	)	)	PUNCT
cte-761	165	13	,	,	PUNCT
cte-761	165	14	f3y_r=-k_2*vy3*sqrt(vx3	f3y_r=-k_2*vy3*sqrt(vx3	NOUN
cte-761	165	15	^	^	SYM
cte-761	165	16	2	2	NUM
cte-761	165	17	+	+	CCONJ
cte-761	165	18	vy3	vy3	ADJ
cte-761	165	19	^	^	SYM
cte-761	165	20	2	2	NUM
cte-761	165	21	)	)	PUNCT
cte-761	165	22	,	,	PUNCT
cte-761	165	23	f3x_m	f3x_m	NOUN
cte-761	165	24	=	=	SYM
cte-761	165	25	rot*(-k_3)*vy3*sqrt(vx3	rot*(-k_3)*vy3*sqrt(vx3	NOUN
cte-761	165	26	^	^	SYM
cte-761	165	27	2	2	NUM
cte-761	165	28	+	+	CCONJ
cte-761	165	29	vy3	vy3	ADJ
cte-761	165	30	^	^	SYM
cte-761	165	31	2	2	NUM
cte-761	165	32	)	)	PUNCT
cte-761	165	33	,	,	PUNCT
cte-761	165	34	f3y_m	f3y_m	NOUN
cte-761	165	35	=	=	PRON
cte-761	165	36	rot*k_3*vx3*sqrt(vx3	rot*k_3*vx3*sqrt(vx3	NOUN
cte-761	165	37	^	^	SYM
cte-761	165	38	2	2	NUM
cte-761	165	39	+	+	CCONJ
cte-761	165	40	vy3	vy3	ADJ
cte-761	165	41	^	^	NOUN
cte-761	165	42	2	2	NUM
cte-761	165	43	)	)	PUNCT
cte-761	165	44	.	.	PUNCT
cte-761	166	1	the	the	DET
cte-761	166	2	vectors	vector	NOUN
cte-761	166	3	of	of	ADP
cte-761	166	4	gravitational	gravitational	ADJ
cte-761	166	5	force	force	NOUN
cte-761	166	6	𝐹3_𝑇	𝐹3_𝑇	PROPN
cte-761	166	7	,	,	PUNCT
cte-761	166	8	air	air	NOUN
cte-761	166	9	resistance	resistance	PROPN
cte-761	166	10	force	force	PROPN
cte-761	166	11	𝐹3_𝑟	𝐹3_𝑟	NUM
cte-761	166	12	,	,	PUNCT
cte-761	166	13	and	and	CCONJ
cte-761	166	14	magnus	magnus	PROPN
cte-761	166	15	force	force	PROPN
cte-761	166	16	𝐹3_𝑀	𝐹3_𝑀	PROPN
cte-761	166	17	for	for	ADP
cte-761	166	18	model	model	NOUN
cte-761	166	19	mm	mm	PROPN
cte-761	166	20	iii	iii	NOUN
cte-761	166	21	are	be	AUX
cte-761	166	22	represented	represent	VERB
cte-761	166	23	as	as	ADP
cte-761	166	24	:	:	PUNCT
cte-761	166	25	f3_t	f3_t	ADJ
cte-761	166	26	=	=	SYM
cte-761	166	27	vector(c	vector(c	NOUN
cte-761	166	28	,	,	PUNCT
cte-761	166	29	f3_1	f3_1	NOUN
cte-761	166	30	)	)	PUNCT
cte-761	166	31	,	,	PUNCT
cte-761	166	32	f3_r	f3_r	ADJ
cte-761	166	33	=	=	SYM
cte-761	166	34	vector(c	vector(c	NOUN
cte-761	166	35	,	,	PUNCT
cte-761	166	36	f3_2	f3_2	ADJ
cte-761	166	37	)	)	PUNCT
cte-761	166	38	,	,	PUNCT
cte-761	166	39	f3_m	f3_m	NOUN
cte-761	166	40	=	=	SYM
cte-761	166	41	vector(c	vector(c	NOUN
cte-761	166	42	,	,	PUNCT
cte-761	166	43	f3_4	f3_4	VERB
cte-761	166	44	)	)	PUNCT
cte-761	166	45	through	through	ADP
cte-761	166	46	this	this	DET
cte-761	166	47	implementation	implementation	NOUN
cte-761	166	48	,	,	PUNCT
cte-761	166	49	students	student	NOUN
cte-761	166	50	created	create	VERB
cte-761	166	51	mathematical	mathematical	ADJ
cte-761	166	52	models	model	NOUN
cte-761	167	1	mm	mm	INTJ
cte-761	168	1	i	i	PROPN
cte-761	168	2	,	,	PUNCT
cte-761	168	3	mm	mm	PROPN
cte-761	168	4	ii	ii	PROPN
cte-761	168	5	,	,	PUNCT
cte-761	168	6	and	and	CCONJ
cte-761	168	7	mm	mm	NOUN
cte-761	168	8	iii	iii	NOUN
cte-761	168	9	in	in	ADP
cte-761	168	10	geogebra	geogebra	NOUN
cte-761	168	11	using	use	VERB
cte-761	168	12	coordinate	coordinate	NOUN
cte-761	168	13	methods	method	NOUN
cte-761	168	14	and	and	CCONJ
cte-761	168	15	elementary	elementary	ADJ
cte-761	168	16	vector	vector	NOUN
cte-761	168	17	algebra	algebra	NOUN
cte-761	168	18	.	.	PUNCT
cte-761	169	1	244	244	NUM
cte-761	169	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	169	3	cte	cte	NOUN
cte-761	169	4	workshop	workshop	NOUN
cte-761	169	5	proceedings	proceeding	NOUN
cte-761	169	6	,	,	PUNCT
cte-761	169	7	2025	2025	NUM
cte-761	169	8	,	,	PUNCT
cte-761	169	9	vol	vol	NOUN
cte-761	169	10	.	.	PROPN
cte-761	169	11	12	12	NUM
cte-761	169	12	,	,	PUNCT
cte-761	169	13	pp	pp	ADJ
cte-761	169	14	.	.	PUNCT
cte-761	170	1	235–252	235–252	NUM
cte-761	171	1	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	171	2	figures	figure	NOUN
cte-761	171	3	8	8	NUM
cte-761	171	4	,	,	PUNCT
cte-761	171	5	9	9	NUM
cte-761	171	6	,	,	PUNCT
cte-761	171	7	and	and	CCONJ
cte-761	171	8	10	10	NUM
cte-761	171	9	show	show	VERB
cte-761	171	10	the	the	DET
cte-761	171	11	dynamic	dynamic	ADJ
cte-761	171	12	vector	vector	NOUN
cte-761	171	13	diagrams	diagram	NOUN
cte-761	171	14	that	that	PRON
cte-761	171	15	students	student	NOUN
cte-761	171	16	plotted	plot	VERB
cte-761	171	17	and	and	CCONJ
cte-761	171	18	studied	study	VERB
cte-761	171	19	in	in	ADP
cte-761	171	20	geogebra	geogebra	NOUN
cte-761	171	21	.	.	PUNCT
cte-761	172	1	these	these	DET
cte-761	172	2	diagrams	diagram	NOUN
cte-761	172	3	display	display	VERB
cte-761	172	4	velocity	velocity	NOUN
cte-761	172	5	vectors	vector	NOUN
cte-761	172	6	and	and	CCONJ
cte-761	172	7	their	their	PRON
cte-761	172	8	projections	projection	NOUN
cte-761	172	9	in	in	ADP
cte-761	172	10	different	different	ADJ
cte-761	172	11	motion	motion	NOUN
cte-761	172	12	phases	phase	NOUN
cte-761	172	13	according	accord	VERB
cte-761	172	14	to	to	ADP
cte-761	172	15	models	model	NOUN
cte-761	172	16	mm	mm	INTJ
cte-761	173	1	i	i	PROPN
cte-761	173	2	,	,	PUNCT
cte-761	173	3	mm	mm	PROPN
cte-761	173	4	ii	ii	PROPN
cte-761	173	5	,	,	PUNCT
cte-761	173	6	and	and	CCONJ
cte-761	173	7	mm	mm	PROPN
cte-761	173	8	iii	iii	NOUN
cte-761	173	9	.	.	PUNCT
cte-761	174	1	these	these	DET
cte-761	174	2	diagrams	diagram	NOUN
cte-761	174	3	are	be	AUX
cte-761	174	4	dynamic	dynamic	ADJ
cte-761	174	5	and	and	CCONJ
cte-761	174	6	interactive	interactive	ADJ
cte-761	174	7	,	,	PUNCT
cte-761	174	8	automatically	automatically	ADV
cte-761	174	9	updating	update	VERB
cte-761	174	10	when	when	SCONJ
cte-761	174	11	model	model	NOUN
cte-761	174	12	parameters	parameter	NOUN
cte-761	174	13	are	be	AUX
cte-761	174	14	adjusted	adjust	VERB
cte-761	174	15	using	use	VERB
cte-761	174	16	the	the	DET
cte-761	174	17	sliders	slider	NOUN
cte-761	174	18	shown	show	VERB
cte-761	174	19	in	in	ADP
cte-761	174	20	the	the	DET
cte-761	174	21	figures	figure	NOUN
cte-761	174	22	.	.	PUNCT
cte-761	175	1	the	the	DET
cte-761	175	2	figures	figure	NOUN
cte-761	175	3	represent	represent	VERB
cte-761	175	4	results	result	NOUN
cte-761	175	5	at	at	ADP
cte-761	175	6	time	time	NOUN
cte-761	175	7	ti	ti	NOUN
cte-761	175	8	=	=	SYM
cte-761	175	9	1.04	1.04	NUM
cte-761	175	10	s	s	NOUN
cte-761	175	11	with	with	ADP
cte-761	175	12	the	the	DET
cte-761	175	13	following	following	ADJ
cte-761	175	14	initial	initial	ADJ
cte-761	175	15	parameters	parameter	NOUN
cte-761	175	16	:	:	PUNCT
cte-761	175	17	g=9.81	g=9.81	ADJ
cte-761	175	18	,	,	PUNCT
cte-761	175	19	ro=1.213	ro=1.213	NOUN
cte-761	175	20	,	,	PUNCT
cte-761	175	21	m=0.68	m=0.68	NOUN
cte-761	175	22	,	,	PUNCT
cte-761	175	23	r=0.12	r=0.12	NOUN
cte-761	175	24	,	,	PUNCT
cte-761	175	25	x0=0	x0=0	PROPN
cte-761	175	26	,	,	PUNCT
cte-761	175	27	y0=0	y0=0	PROPN
cte-761	175	28	,	,	PUNCT
cte-761	175	29	v0=9.3	v0=9.3	PROPN
cte-761	175	30	,	,	PUNCT
cte-761	175	31	alpha0=45	alpha0=45	PROPN
cte-761	175	32	,	,	PUNCT
cte-761	175	33	rot=1	rot=1	PROPN
cte-761	175	34	,	,	PUNCT
cte-761	175	35	mst=0.1	mst=0.1	PROPN
cte-761	175	36	.	.	PUNCT
cte-761	176	1	figure	figure	NOUN
cte-761	176	2	8	8	NUM
cte-761	176	3	:	:	PUNCT
cte-761	176	4	dynamic	dynamic	ADJ
cte-761	176	5	vector	vector	NOUN
cte-761	176	6	diagram	diagram	NOUN
cte-761	176	7	of	of	ADP
cte-761	176	8	velocities	velocity	NOUN
cte-761	176	9	plotted	plot	VERB
cte-761	176	10	in	in	ADP
cte-761	176	11	geogebra	geogebra	NOUN
cte-761	176	12	for	for	ADP
cte-761	176	13	models	model	NOUN
cte-761	176	14	mm	mm	INTJ
cte-761	177	1	i	i	PROPN
cte-761	177	2	-	-	PUNCT
cte-761	177	3	mm	mm	PROPN
cte-761	177	4	iii+	iii+	PROPN
cte-761	177	5	(	(	PUNCT
cte-761	177	6	clockwise	clockwise	NOUN
cte-761	177	7	rotation	rotation	NOUN
cte-761	177	8	)	)	PUNCT
cte-761	177	9	at	at	ADP
cte-761	177	10	movement	movement	NOUN
cte-761	177	11	initiation	initiation	NOUN
cte-761	177	12	:	:	PUNCT
cte-761	177	13	ti=0	ti=0	ADJ
cte-761	177	14	,	,	PUNCT
cte-761	177	15	rot=+1	rot=+1	NOUN
cte-761	177	16	.	.	PUNCT
cte-761	178	1	the	the	DET
cte-761	178	2	use	use	NOUN
cte-761	178	3	of	of	ADP
cte-761	178	4	dynamic	dynamic	ADJ
cte-761	178	5	vector	vector	NOUN
cte-761	178	6	diagrams	diagram	NOUN
cte-761	178	7	for	for	ADP
cte-761	178	8	visualising	visualise	VERB
cte-761	178	9	mechanical	mechanical	ADJ
cte-761	178	10	movement	movement	NOUN
cte-761	178	11	characteristics	characteristic	NOUN
cte-761	178	12	represents	represent	VERB
cte-761	178	13	the	the	DET
cte-761	178	14	primary	primary	ADJ
cte-761	178	15	innovation	innovation	NOUN
cte-761	178	16	in	in	ADP
cte-761	178	17	this	this	DET
cte-761	178	18	study	study	NOUN
cte-761	178	19	compared	compare	VERB
cte-761	178	20	to	to	ADP
cte-761	178	21	our	our	PRON
cte-761	178	22	previous	previous	ADJ
cte-761	178	23	research	research	NOUN
cte-761	178	24	.	.	PUNCT
cte-761	179	1	computer	computer	NOUN
cte-761	179	2	simulation	simulation	NOUN
cte-761	179	3	and	and	CCONJ
cte-761	179	4	analysis	analysis	NOUN
cte-761	179	5	of	of	ADP
cte-761	179	6	dynamic	dynamic	ADJ
cte-761	179	7	vector	vector	NOUN
cte-761	179	8	diagrams	diagram	NOUN
cte-761	179	9	allowed	allow	VERB
cte-761	179	10	students	student	NOUN
cte-761	179	11	to	to	PART
cte-761	179	12	interactively	interactively	ADV
cte-761	179	13	observe	observe	VERB
cte-761	179	14	velocity	velocity	NOUN
cte-761	179	15	vector	vector	NOUN
cte-761	179	16	changes	change	NOUN
cte-761	179	17	in	in	ADP
cte-761	179	18	response	response	NOUN
cte-761	179	19	to	to	ADP
cte-761	179	20	parameter	parameter	NOUN
cte-761	179	21	adjustments	adjustment	NOUN
cte-761	179	22	,	,	PUNCT
cte-761	179	23	and	and	CCONJ
cte-761	179	24	to	to	PART
cte-761	179	25	compare	compare	VERB
cte-761	179	26	dynamic	dynamic	ADJ
cte-761	179	27	motion	motion	NOUN
cte-761	179	28	characteristics	characteristic	NOUN
cte-761	179	29	at	at	ADP
cte-761	179	30	different	different	ADJ
cte-761	179	31	phases	phase	NOUN
cte-761	179	32	.	.	PUNCT
cte-761	180	1	students	student	NOUN
cte-761	180	2	simultaneously	simultaneously	ADV
cte-761	180	3	analysed	analyse	VERB
cte-761	180	4	velocity	velocity	NOUN
cte-761	180	5	plots	plot	NOUN
cte-761	180	6	and	and	CCONJ
cte-761	180	7	their	their	PRON
cte-761	180	8	projections	projection	NOUN
cte-761	180	9	on	on	ADP
cte-761	180	10	coordinate	coordinate	NOUN
cte-761	180	11	axes	axis	NOUN
cte-761	180	12	,	,	PUNCT
cte-761	180	13	similar	similar	ADJ
cte-761	180	14	to	to	ADP
cte-761	180	15	those	those	PRON
cte-761	180	16	shown	show	VERB
cte-761	180	17	in	in	ADP
cte-761	180	18	figures	figure	NOUN
cte-761	180	19	11	11	NUM
cte-761	180	20	and	and	CCONJ
cte-761	180	21	12	12	NUM
cte-761	180	22	.	.	PUNCT
cte-761	181	1	while	while	SCONJ
cte-761	181	2	these	these	DET
cte-761	181	3	plots	plot	NOUN
cte-761	181	4	effectively	effectively	ADV
cte-761	181	5	illustrate	illustrate	VERB
cte-761	181	6	velocity	velocity	NOUN
cte-761	181	7	magnitude	magnitude	NOUN
cte-761	181	8	changes	change	NOUN
cte-761	181	9	,	,	PUNCT
cte-761	181	10	dynamic	dynamic	ADJ
cte-761	181	11	vector	vector	NOUN
cte-761	181	12	diagrams	diagram	NOUN
cte-761	181	13	provide	provide	VERB
cte-761	181	14	clearer	clear	ADJ
cte-761	181	15	visual	visual	ADJ
cte-761	181	16	representations	representation	NOUN
cte-761	181	17	of	of	ADP
cte-761	181	18	both	both	DET
cte-761	181	19	velocity	velocity	NOUN
cte-761	181	20	magnitude	magnitude	NOUN
cte-761	181	21	(	(	PUNCT
cte-761	181	22	indicated	indicate	VERB
cte-761	181	23	by	by	ADP
cte-761	181	24	vector	vector	NOUN
cte-761	181	25	length	length	NOUN
cte-761	181	26	)	)	PUNCT
cte-761	181	27	and	and	CCONJ
cte-761	181	28	motion	motion	NOUN
cte-761	181	29	direction	direction	NOUN
cte-761	181	30	(	(	PUNCT
cte-761	181	31	indicated	indicate	VERB
cte-761	181	32	by	by	ADP
cte-761	181	33	vector	vector	NOUN
cte-761	181	34	orientation	orientation	NOUN
cte-761	181	35	)	)	PUNCT
cte-761	181	36	.	.	PUNCT
cte-761	182	1	figure	figure	VERB
cte-761	182	2	11	11	NUM
cte-761	182	3	compares	compare	VERB
cte-761	182	4	horizontal	horizontal	ADJ
cte-761	182	5	velocity	velocity	NOUN
cte-761	182	6	projection	projection	NOUN
cte-761	182	7	plots	plot	NOUN
cte-761	182	8	vx1(t	vx1(t	NOUN
cte-761	182	9	)	)	PUNCT
cte-761	182	10	,	,	PUNCT
cte-761	182	11	vx2(t	vx2(t	PROPN
cte-761	182	12	)	)	PUNCT
cte-761	182	13	,	,	PUNCT
cte-761	182	14	vx3(t	vx3(t	VERB
cte-761	182	15	)	)	PUNCT
cte-761	182	16	for	for	ADP
cte-761	182	17	the	the	DET
cte-761	182	18	three	three	NUM
cte-761	182	19	models	model	NOUN
cte-761	182	20	with	with	ADP
cte-761	182	21	clockwise	clockwise	NOUN
cte-761	182	22	rotation	rotation	NOUN
cte-761	182	23	(	(	PUNCT
cte-761	182	24	rot=+1	rot=+1	NOUN
cte-761	182	25	)	)	PUNCT
cte-761	182	26	and	and	CCONJ
cte-761	182	27	counterclockwise	counterclockwise	NOUN
cte-761	182	28	rotation	rotation	NOUN
cte-761	182	29	(	(	PUNCT
cte-761	182	30	rot=-1	rot=-1	NOUN
cte-761	182	31	)	)	PUNCT
cte-761	182	32	.	.	PUNCT
cte-761	183	1	these	these	DET
cte-761	183	2	plots	plot	NOUN
cte-761	183	3	clearly	clearly	ADV
cte-761	183	4	demonstrate	demonstrate	VERB
cte-761	183	5	that	that	SCONJ
cte-761	183	6	with	with	ADP
cte-761	183	7	clockwise	clockwise	NOUN
cte-761	183	8	rotation	rotation	NOUN
cte-761	183	9	,	,	PUNCT
cte-761	183	10	the	the	DET
cte-761	183	11	projectile	projectile	NOUN
cte-761	183	12	initially	initially	ADV
cte-761	183	13	accelerates	accelerate	VERB
cte-761	183	14	horizontally	horizontally	ADV
cte-761	183	15	before	before	ADP
cte-761	183	16	decelerating	decelerate	VERB
cte-761	183	17	,	,	PUNCT
cte-761	183	18	whereas	whereas	SCONJ
cte-761	183	19	with	with	ADP
cte-761	183	20	counterclockwise	counterclockwise	NOUN
cte-761	183	21	rotation	rotation	NOUN
cte-761	183	22	(	(	PUNCT
cte-761	183	23	rot=-1	rot=-1	NOUN
cte-761	183	24	)	)	PUNCT
cte-761	183	25	,	,	PUNCT
cte-761	183	26	it	it	PRON
cte-761	183	27	initially	initially	ADV
cte-761	183	28	decelerates	decelerate	VERB
cte-761	183	29	but	but	CCONJ
cte-761	183	30	slightly	slightly	ADV
cte-761	183	31	accelerates	accelerate	VERB
cte-761	183	32	towards	towards	ADP
cte-761	183	33	the	the	DET
cte-761	183	34	end	end	NOUN
cte-761	183	35	of	of	ADP
cte-761	183	36	its	its	PRON
cte-761	183	37	trajectory	trajectory	NOUN
cte-761	183	38	.	.	PUNCT
cte-761	184	1	geogebra	geogebra	NOUN
cte-761	184	2	facilitates	facilitate	VERB
cte-761	184	3	the	the	DET
cte-761	184	4	construction	construction	NOUN
cte-761	184	5	of	of	ADP
cte-761	184	6	similar	similar	ADJ
cte-761	184	7	plots	plot	NOUN
cte-761	184	8	and	and	CCONJ
cte-761	184	9	diagrams	diagram	NOUN
cte-761	184	10	for	for	ADP
cte-761	184	11	all	all	DET
cte-761	184	12	model	model	NOUN
cte-761	184	13	solution	solution	NOUN
cte-761	184	14	functions	function	NOUN
cte-761	184	15	.	.	PUNCT
cte-761	185	1	interactive	interactive	ADJ
cte-761	185	2	comparison	comparison	NOUN
cte-761	185	3	with	with	ADP
cte-761	185	4	various	various	ADJ
cte-761	185	5	parameter	parameter	NOUN
cte-761	185	6	values	value	NOUN
cte-761	185	7	enables	enable	VERB
cte-761	185	8	rapid	rapid	ADJ
cte-761	185	9	investigation	investigation	NOUN
cte-761	185	10	to	to	PART
cte-761	185	11	establish	establish	VERB
cte-761	185	12	facts	fact	NOUN
cte-761	185	13	or	or	CCONJ
cte-761	185	14	identify	identify	VERB
cte-761	185	15	patterns	pattern	NOUN
cte-761	185	16	.	.	PUNCT
cte-761	186	1	for	for	ADP
cte-761	186	2	example	example	NOUN
cte-761	186	3	,	,	PUNCT
cte-761	186	4	figure	figure	VERB
cte-761	186	5	12	12	NUM
cte-761	186	6	displays	display	NOUN
cte-761	186	7	plots	plot	NOUN
cte-761	186	8	of	of	ADP
cte-761	186	9	functions	function	NOUN
cte-761	186	10	y1(t	y1(t	PRON
cte-761	186	11	)	)	PUNCT
cte-761	186	12	,	,	PUNCT
cte-761	186	13	y2(t	y2(t	PROPN
cte-761	186	14	)	)	PUNCT
cte-761	186	15	,	,	PUNCT
cte-761	186	16	y3(t	y3(t	PROPN
cte-761	186	17	)	)	PUNCT
cte-761	186	18	for	for	ADP
cte-761	186	19	models	model	NOUN
cte-761	187	1	mm	mm	INTJ
cte-761	187	2	i	i	PROPN
cte-761	187	3	,	,	PUNCT
cte-761	187	4	mm	mm	PROPN
cte-761	187	5	ii	ii	PROPN
cte-761	187	6	,	,	PUNCT
cte-761	187	7	and	and	CCONJ
cte-761	187	8	mm	mm	NOUN
cte-761	187	9	iii	iii	NOUN
cte-761	187	10	with	with	ADP
cte-761	187	11	both	both	DET
cte-761	187	12	rotation	rotation	NOUN
cte-761	187	13	directions	direction	NOUN
cte-761	187	14	.	.	PUNCT
cte-761	188	1	by	by	ADP
cte-761	188	2	adjusting	adjust	VERB
cte-761	188	3	the	the	DET
cte-761	188	4	simulation	simulation	NOUN
cte-761	188	5	time	time	NOUN
cte-761	188	6	tm	tm	PROPN
cte-761	188	7	,	,	PUNCT
cte-761	188	8	students	student	NOUN
cte-761	188	9	could	could	AUX
cte-761	188	10	quickly	quickly	ADV
cte-761	188	11	determine	determine	VERB
cte-761	188	12	when	when	SCONJ
cte-761	188	13	the	the	DET
cte-761	188	14	projectile	projectile	NOUN
cte-761	188	15	contacts	contact	VERB
cte-761	188	16	the	the	DET
cte-761	188	17	ground	ground	NOUN
cte-761	188	18	(	(	PUNCT
cte-761	188	19	flight	flight	NOUN
cte-761	188	20	time	time	NOUN
cte-761	188	21	)	)	PUNCT
cte-761	188	22	.	.	PUNCT
cte-761	189	1	as	as	ADP
cte-761	189	2	245	245	NUM
cte-761	189	3	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	189	4	cte	cte	NOUN
cte-761	189	5	workshop	workshop	NOUN
cte-761	189	6	proceedings	proceeding	NOUN
cte-761	189	7	,	,	PUNCT
cte-761	189	8	2025	2025	NUM
cte-761	189	9	,	,	PUNCT
cte-761	189	10	vol	vol	NOUN
cte-761	189	11	.	.	PROPN
cte-761	189	12	12	12	NUM
cte-761	189	13	,	,	PUNCT
cte-761	189	14	pp	pp	ADJ
cte-761	189	15	.	.	PUNCT
cte-761	190	1	235–252	235–252	NUM
cte-761	190	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	190	3	figure	figure	NOUN
cte-761	190	4	9	9	NUM
cte-761	190	5	:	:	PUNCT
cte-761	190	6	dynamic	dynamic	ADJ
cte-761	190	7	vector	vector	NOUN
cte-761	190	8	diagram	diagram	NOUN
cte-761	190	9	of	of	ADP
cte-761	190	10	velocities	velocity	NOUN
cte-761	190	11	plotted	plot	VERB
cte-761	190	12	in	in	ADP
cte-761	190	13	geogebra	geogebra	NOUN
cte-761	190	14	for	for	ADP
cte-761	190	15	models	model	NOUN
cte-761	190	16	mm	mm	INTJ
cte-761	190	17	i	i	PROPN
cte-761	190	18	-	-	PUNCT
cte-761	190	19	mm	mm	PROPN
cte-761	190	20	iii+	iii+	PROPN
cte-761	190	21	(	(	PUNCT
cte-761	190	22	clockwise	clockwise	NOUN
cte-761	190	23	rotation	rotation	NOUN
cte-761	190	24	)	)	PUNCT
cte-761	190	25	at	at	ADP
cte-761	190	26	motion	motion	NOUN
cte-761	190	27	termination	termination	NOUN
cte-761	190	28	:	:	PUNCT
cte-761	190	29	ti=1.34	ti=1.34	ADJ
cte-761	190	30	,	,	PUNCT
cte-761	190	31	rot=+1	rot=+1	PROPN
cte-761	190	32	.	.	PUNCT
cte-761	190	33	demonstrated	demonstrate	VERB
cte-761	190	34	,	,	PUNCT
cte-761	190	35	with	with	ADP
cte-761	190	36	identical	identical	ADJ
cte-761	190	37	input	input	NOUN
cte-761	190	38	parameters	parameter	NOUN
cte-761	190	39	,	,	PUNCT
cte-761	190	40	a	a	DET
cte-761	190	41	clockwise	clockwise	NOUN
cte-761	190	42	-	-	PUNCT
cte-761	190	43	rotating	rotate	VERB
cte-761	190	44	projectile	projectile	NOUN
cte-761	190	45	falls	fall	VERB
cte-761	190	46	to	to	ADP
cte-761	190	47	the	the	DET
cte-761	190	48	ground	ground	NOUN
cte-761	190	49	in	in	ADP
cte-761	190	50	1.14	1.14	NUM
cte-761	190	51	s	s	NOUN
cte-761	190	52	,	,	PUNCT
cte-761	190	53	while	while	SCONJ
cte-761	190	54	a	a	DET
cte-761	190	55	counterclockwise	counterclockwise	NOUN
cte-761	190	56	-	-	PUNCT
cte-761	190	57	rotating	rotate	VERB
cte-761	190	58	projectile	projectile	NOUN
cte-761	190	59	remains	remain	VERB
cte-761	190	60	airborne	airborne	ADJ
cte-761	190	61	for	for	ADP
cte-761	190	62	1.46	1.46	NUM
cte-761	190	63	s.	s.	PROPN
cte-761	190	64	this	this	DET
cte-761	190	65	approach	approach	NOUN
cte-761	190	66	effectively	effectively	ADV
cte-761	190	67	develops	develop	VERB
cte-761	190	68	students	student	NOUN
cte-761	190	69	’	'	PUNCT
cte-761	190	70	research	research	NOUN
cte-761	190	71	competencies	competency	NOUN
cte-761	190	72	.	.	PUNCT
cte-761	191	1	similarly	similarly	ADV
cte-761	191	2	,	,	PUNCT
cte-761	191	3	students	student	NOUN
cte-761	191	4	studied	study	VERB
cte-761	191	5	and	and	CCONJ
cte-761	191	6	analysed	analyse	VERB
cte-761	191	7	forces	force	NOUN
cte-761	191	8	acting	act	VERB
cte-761	191	9	on	on	ADP
cte-761	191	10	the	the	DET
cte-761	191	11	projectile	projectile	NOUN
cte-761	191	12	during	during	ADP
cte-761	191	13	motion	motion	NOUN
cte-761	191	14	,	,	PUNCT
cte-761	191	15	including	include	VERB
cte-761	191	16	resultant	resultant	NOUN
cte-761	191	17	forces	force	NOUN
cte-761	191	18	and	and	CCONJ
cte-761	191	19	their	their	PRON
cte-761	191	20	projections	projection	NOUN
cte-761	191	21	.	.	PUNCT
cte-761	192	1	through	through	ADP
cte-761	192	2	computer	computer	NOUN
cte-761	192	3	simulations	simulation	NOUN
cte-761	192	4	in	in	ADP
cte-761	192	5	excel	excel	NOUN
cte-761	192	6	and	and	CCONJ
cte-761	192	7	geogebra	geogebra	NOUN
cte-761	192	8	,	,	PUNCT
cte-761	192	9	they	they	PRON
cte-761	192	10	observed	observe	VERB
cte-761	192	11	and	and	CCONJ
cte-761	192	12	analysed	analyse	VERB
cte-761	192	13	how	how	SCONJ
cte-761	192	14	these	these	DET
cte-761	192	15	vector	vector	NOUN
cte-761	192	16	quantities	quantity	NOUN
cte-761	192	17	changed	change	VERB
cte-761	192	18	over	over	ADP
cte-761	192	19	time	time	NOUN
cte-761	192	20	,	,	PUNCT
cte-761	192	21	identifying	identify	VERB
cte-761	192	22	trends	trend	NOUN
cte-761	192	23	and	and	CCONJ
cte-761	192	24	their	their	PRON
cte-761	192	25	significance	significance	NOUN
cte-761	192	26	.	.	PUNCT
cte-761	193	1	this	this	DET
cte-761	193	2	approach	approach	NOUN
cte-761	193	3	contributed	contribute	VERB
cte-761	193	4	to	to	ADP
cte-761	193	5	the	the	DET
cte-761	193	6	development	development	NOUN
cte-761	193	7	of	of	ADP
cte-761	193	8	analytical	analytical	ADJ
cte-761	193	9	abilities	ability	NOUN
cte-761	193	10	and	and	CCONJ
cte-761	193	11	graphic	graphic	ADJ
cte-761	193	12	analysis	analysis	NOUN
cte-761	193	13	skills	skill	NOUN
cte-761	193	14	.	.	PUNCT
cte-761	194	1	2.5	2.5	NUM
cte-761	194	2	.	.	PUNCT
cte-761	195	1	combined	combine	VERB
cte-761	195	2	implementation	implementation	NOUN
cte-761	195	3	with	with	ADP
cte-761	195	4	excel	excel	NOUN
cte-761	195	5	and	and	CCONJ
cte-761	195	6	geogebra	geogebra	NOUN
cte-761	195	7	(	(	PUNCT
cte-761	195	8	group	group	NOUN
cte-761	195	9	eg	eg	NOUN
cte-761	195	10	)	)	PUNCT
cte-761	195	11	the	the	DET
cte-761	195	12	experimental	experimental	ADJ
cte-761	195	13	group	group	NOUN
cte-761	195	14	eg	eg	PROPN
cte-761	195	15	completed	complete	VERB
cte-761	195	16	identical	identical	ADJ
cte-761	195	17	learning	learning	NOUN
cte-761	195	18	exercises	exercise	NOUN
cte-761	195	19	using	use	VERB
cte-761	195	20	both	both	PRON
cte-761	195	21	excel	excel	VERB
cte-761	195	22	and	and	CCONJ
cte-761	195	23	geogebra	geogebra	NOUN
cte-761	195	24	simultaneously	simultaneously	ADV
cte-761	195	25	,	,	PUNCT
cte-761	195	26	following	follow	VERB
cte-761	195	27	the	the	DET
cte-761	195	28	methodology	methodology	NOUN
cte-761	195	29	described	describe	VERB
cte-761	195	30	above	above	ADV
cte-761	195	31	.	.	PUNCT
cte-761	196	1	participants	participant	NOUN
cte-761	196	2	in	in	ADP
cte-761	196	3	this	this	DET
cte-761	196	4	group	group	NOUN
cte-761	196	5	used	use	VERB
cte-761	196	6	excel	excel	VERB
cte-761	196	7	when	when	SCONJ
cte-761	196	8	necessary	necessary	ADJ
cte-761	196	9	for	for	ADP
cte-761	196	10	numerical	numerical	ADJ
cte-761	196	11	calculations	calculation	NOUN
cte-761	196	12	or	or	CCONJ
cte-761	196	13	tabular	tabular	NOUN
cte-761	196	14	presentation	presentation	NOUN
cte-761	196	15	of	of	ADP
cte-761	196	16	results	result	NOUN
cte-761	196	17	,	,	PUNCT
cte-761	196	18	and	and	CCONJ
cte-761	196	19	geogebra	geogebra	NOUN
cte-761	196	20	for	for	ADP
cte-761	196	21	visual	visual	ADJ
cte-761	196	22	representation	representation	NOUN
cte-761	196	23	and	and	CCONJ
cte-761	196	24	analysis	analysis	NOUN
cte-761	196	25	of	of	ADP
cte-761	196	26	dynamic	dynamic	ADJ
cte-761	196	27	characteristics	characteristic	NOUN
cte-761	196	28	through	through	ADP
cte-761	196	29	plots	plot	NOUN
cte-761	196	30	and	and	CCONJ
cte-761	196	31	dynamic	dynamic	ADJ
cte-761	196	32	vector	vector	NOUN
cte-761	196	33	diagrams	diagram	NOUN
cte-761	196	34	.	.	PUNCT
cte-761	197	1	figures	figure	NOUN
cte-761	197	2	13	13	NUM
cte-761	197	3	and	and	CCONJ
cte-761	197	4	14	14	NUM
cte-761	197	5	show	show	VERB
cte-761	197	6	examples	example	NOUN
cte-761	197	7	of	of	ADP
cte-761	197	8	vector	vector	NOUN
cte-761	197	9	diagrams	diagram	NOUN
cte-761	197	10	of	of	ADP
cte-761	197	11	forces	force	NOUN
cte-761	197	12	acting	act	VERB
cte-761	197	13	on	on	ADP
cte-761	197	14	the	the	DET
cte-761	197	15	projectile	projectile	NOUN
cte-761	197	16	according	accord	VERB
cte-761	197	17	to	to	ADP
cte-761	197	18	models	model	NOUN
cte-761	197	19	mm	mm	INTJ
cte-761	197	20	i	i	PROPN
cte-761	197	21	,	,	PUNCT
cte-761	197	22	mm	mm	PROPN
cte-761	197	23	ii	ii	PROPN
cte-761	197	24	,	,	PUNCT
cte-761	197	25	and	and	CCONJ
cte-761	197	26	mm	mm	PROPN
cte-761	197	27	iii	iii	PROPN
cte-761	197	28	.	.	PROPN
cte-761	198	1	3	3	X
cte-761	198	2	.	.	NOUN
cte-761	198	3	results	result	NOUN
cte-761	198	4	table	table	NOUN
cte-761	198	5	1	1	NUM
cte-761	198	6	presents	present	VERB
cte-761	198	7	the	the	DET
cte-761	198	8	final	final	ADJ
cte-761	198	9	learning	learn	VERB
cte-761	198	10	outcomes	outcome	NOUN
cte-761	198	11	of	of	ADP
cte-761	198	12	engineering	engineer	VERB
cte-761	198	13	students	student	NOUN
cte-761	198	14	after	after	ADP
cte-761	198	15	completing	complete	VERB
cte-761	198	16	the	the	DET
cte-761	198	17	experiment	experiment	NOUN
cte-761	198	18	on	on	ADP
cte-761	198	19	teaching	teach	VERB
cte-761	198	20	mathematical	mathematical	ADJ
cte-761	198	21	modelling	modelling	NOUN
cte-761	198	22	fundamentals	fundamental	NOUN
cte-761	198	23	using	use	VERB
cte-761	198	24	dynamic	dynamic	ADJ
cte-761	198	25	vector	vector	NOUN
cte-761	198	26	diagrams	diagram	NOUN
cte-761	198	27	to	to	PART
cte-761	198	28	visualise	visualise	VERB
cte-761	198	29	mechanical	mechanical	ADJ
cte-761	198	30	movement	movement	NOUN
cte-761	198	31	characteristics	characteristic	NOUN
cte-761	198	32	.	.	PUNCT
cte-761	199	1	these	these	DET
cte-761	199	2	results	result	NOUN
cte-761	199	3	are	be	AUX
cte-761	199	4	presented	present	VERB
cte-761	199	5	on	on	ADP
cte-761	199	6	a	a	DET
cte-761	199	7	100	100	NUM
cte-761	199	8	-	-	PUNCT
cte-761	199	9	point	point	NOUN
cte-761	199	10	scale	scale	NOUN
cte-761	199	11	[	[	X
cte-761	199	12	2	2	NUM
cte-761	199	13	]	]	PUNCT
cte-761	199	14	.	.	PUNCT
cte-761	200	1	primary	primary	ADJ
cte-761	200	2	statistical	statistical	ADJ
cte-761	200	3	analysis	analysis	NOUN
cte-761	200	4	of	of	ADP
cte-761	200	5	experimental	experimental	ADJ
cte-761	200	6	data	datum	NOUN
cte-761	200	7	(	(	PUNCT
cte-761	200	8	table	table	NOUN
cte-761	200	9	2	2	NUM
cte-761	200	10	)	)	PUNCT
cte-761	200	11	revealed	reveal	VERB
cte-761	200	12	differing	differ	VERB
cte-761	200	13	average	average	ADJ
cte-761	200	14	scores	score	NOUN
cte-761	200	15	across	across	ADP
cte-761	200	16	all	all	DET
cte-761	200	17	groups	group	NOUN
cte-761	200	18	.	.	PUNCT
cte-761	201	1	the	the	DET
cte-761	201	2	means	mean	NOUN
cte-761	201	3	for	for	ADP
cte-761	201	4	groups	group	NOUN
cte-761	201	5	e	e	NOUN
cte-761	201	6	and	and	CCONJ
cte-761	201	7	g	g	PROPN
cte-761	201	8	were	be	AUX
cte-761	201	9	similar	similar	ADJ
cte-761	201	10	(	(	PUNCT
cte-761	201	11	73.36	73.36	NUM
cte-761	201	12	and	and	CCONJ
cte-761	201	13	73.85	73.85	NUM
cte-761	201	14	,	,	PUNCT
cte-761	201	15	respectively	respectively	ADV
cte-761	201	16	)	)	PUNCT
cte-761	201	17	but	but	CCONJ
cte-761	201	18	both	both	CCONJ
cte-761	201	19	lower	low	ADJ
cte-761	201	20	than	than	ADP
cte-761	201	21	the	the	DET
cte-761	201	22	mean	mean	NOUN
cte-761	201	23	for	for	ADP
cte-761	201	24	group	group	NOUN
cte-761	201	25	eg	eg	NOUN
cte-761	201	26	(	(	PUNCT
cte-761	201	27	78.25	78.25	NUM
cte-761	201	28	)	)	PUNCT
cte-761	201	29	.	.	PUNCT
cte-761	202	1	the	the	DET
cte-761	202	2	mean	mean	ADJ
cte-761	202	3	values	value	NOUN
cte-761	202	4	for	for	ADP
cte-761	202	5	groups	group	NOUN
cte-761	202	6	e	e	NOUN
cte-761	202	7	and	and	CCONJ
cte-761	202	8	g	g	PROPN
cte-761	202	9	closely	closely	ADV
cte-761	202	10	corresponded	correspond	VERB
cte-761	202	11	to	to	ADP
cte-761	202	12	similar	similar	ADJ
cte-761	202	13	indicators	indicator	NOUN
cte-761	202	14	from	from	ADP
cte-761	202	15	previous	previous	ADJ
cte-761	202	16	years	year	NOUN
cte-761	202	17	:	:	PUNCT
cte-761	202	18	meane=73.36	meane=73.36	NUM
cte-761	202	19	(	(	PUNCT
cte-761	202	20	2019	2019	NUM
cte-761	202	21	)	)	PUNCT
cte-761	202	22	vs.	vs.	ADP
cte-761	202	23	73.4	73.4	NUM
cte-761	202	24	(	(	PUNCT
cte-761	202	25	2018	2018	NUM
cte-761	202	26	)	)	PUNCT
cte-761	202	27	;	;	PUNCT
cte-761	202	28	meang=73.85	meang=73.85	NUM
cte-761	202	29	(	(	PUNCT
cte-761	202	30	2019	2019	NUM
cte-761	202	31	)	)	PUNCT
cte-761	202	32	vs.	vs.	ADP
cte-761	202	33	73.7	73.7	NUM
cte-761	202	34	(	(	PUNCT
cte-761	202	35	2018	2018	NUM
cte-761	202	36	)	)	PUNCT
cte-761	202	37	.	.	PUNCT
cte-761	203	1	conversely	conversely	ADV
cte-761	203	2	,	,	PUNCT
cte-761	203	3	246	246	NUM
cte-761	203	4	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	203	5	cte	cte	NOUN
cte-761	203	6	workshop	workshop	NOUN
cte-761	203	7	proceedings	proceeding	NOUN
cte-761	203	8	,	,	PUNCT
cte-761	203	9	2025	2025	NUM
cte-761	203	10	,	,	PUNCT
cte-761	203	11	vol	vol	NOUN
cte-761	203	12	.	.	PROPN
cte-761	203	13	12	12	NUM
cte-761	203	14	,	,	PUNCT
cte-761	203	15	pp	pp	ADJ
cte-761	203	16	.	.	PUNCT
cte-761	204	1	235–252	235–252	NUM
cte-761	204	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	204	3	figure	figure	NOUN
cte-761	204	4	10	10	NUM
cte-761	204	5	:	:	PUNCT
cte-761	204	6	dynamic	dynamic	ADJ
cte-761	204	7	vector	vector	NOUN
cte-761	204	8	diagram	diagram	NOUN
cte-761	204	9	of	of	ADP
cte-761	204	10	velocities	velocity	NOUN
cte-761	204	11	plotted	plot	VERB
cte-761	204	12	in	in	ADP
cte-761	204	13	geogebra	geogebra	NOUN
cte-761	204	14	for	for	ADP
cte-761	204	15	models	model	NOUN
cte-761	204	16	mm	mm	INTJ
cte-761	205	1	i	i	PROPN
cte-761	205	2	-	-	PUNCT
cte-761	205	3	mm	mm	PROPN
cte-761	205	4	iii(counterclockwise	iii(counterclockwise	NOUN
cte-761	205	5	rotation	rotation	NOUN
cte-761	205	6	)	)	PUNCT
cte-761	205	7	at	at	ADP
cte-761	205	8	motion	motion	NOUN
cte-761	205	9	termination	termination	NOUN
cte-761	205	10	:	:	PUNCT
cte-761	205	11	ti=1.34	ti=1.34	ADJ
cte-761	205	12	,	,	PUNCT
cte-761	205	13	rot=-1	rot=-1	NOUN
cte-761	205	14	.	.	PUNCT
cte-761	206	1	figure	figure	VERB
cte-761	206	2	11	11	NUM
cte-761	206	3	:	:	PUNCT
cte-761	206	4	visual	visual	ADJ
cte-761	206	5	analysis	analysis	NOUN
cte-761	206	6	in	in	ADP
cte-761	206	7	geogebra	geogebra	NOUN
cte-761	206	8	:	:	PUNCT
cte-761	206	9	𝑣𝑥	𝑣𝑥	NOUN
cte-761	206	10	projection	projection	NOUN
cte-761	206	11	changes	change	NOUN
cte-761	206	12	for	for	ADP
cte-761	206	13	models	model	NOUN
cte-761	207	1	mm	mm	INTJ
cte-761	208	1	i	i	PROPN
cte-761	208	2	,	,	PUNCT
cte-761	208	3	mm	mm	PROPN
cte-761	208	4	ii	ii	PROPN
cte-761	208	5	,	,	PUNCT
cte-761	208	6	mm	mm	PROPN
cte-761	208	7	iii+	iii+	ADP
cte-761	208	8	(	(	PUNCT
cte-761	208	9	rot=+1	rot=+1	NOUN
cte-761	208	10	,	,	PUNCT
cte-761	208	11	clockwise	clockwise	NOUN
cte-761	208	12	rotation	rotation	NOUN
cte-761	208	13	)	)	PUNCT
cte-761	208	14	and	and	CCONJ
cte-761	208	15	mm	mm	PROPN
cte-761	208	16	iii(rot=-1	iii(rot=-1	ADJ
cte-761	208	17	,	,	PUNCT
cte-761	208	18	counterclockwise	counterclockwise	NOUN
cte-761	208	19	rotation	rotation	NOUN
cte-761	208	20	)	)	PUNCT
cte-761	208	21	.	.	PUNCT
cte-761	209	1	the	the	DET
cte-761	209	2	mean	mean	ADJ
cte-761	209	3	value	value	NOUN
cte-761	209	4	for	for	ADP
cte-761	209	5	group	group	NOUN
cte-761	209	6	eg-2019	eg-2019	AUX
cte-761	209	7	exceeded	exceed	VERB
cte-761	209	8	that	that	PRON
cte-761	209	9	of	of	ADP
cte-761	209	10	group	group	NOUN
cte-761	209	11	eg-2018	eg-2018	NOUN
cte-761	209	12	:	:	PUNCT
cte-761	209	13	meaneg=78.25	meaneg=78.25	PROPN
cte-761	209	14	(	(	PUNCT
cte-761	209	15	2019	2019	NUM
cte-761	209	16	)	)	PUNCT
cte-761	209	17	vs.	vs.	ADP
cte-761	209	18	77.4	77.4	NUM
cte-761	209	19	(	(	PUNCT
cte-761	209	20	2018	2018	NUM
cte-761	209	21	)	)	PUNCT
cte-761	209	22	.	.	PUNCT
cte-761	210	1	the	the	DET
cte-761	210	2	shapiro	shapiro	PROPN
cte-761	210	3	-	-	PUNCT
cte-761	210	4	wilk	wilk	NOUN
cte-761	210	5	test	test	NOUN
cte-761	210	6	results	result	NOUN
cte-761	210	7	supported	support	VERB
cte-761	210	8	the	the	DET
cte-761	210	9	hypothesis	hypothesis	NOUN
cte-761	210	10	of	of	ADP
cte-761	210	11	normal	normal	ADJ
cte-761	210	12	data	datum	NOUN
cte-761	210	13	distribution	distribution	NOUN
cte-761	210	14	across	across	ADP
cte-761	210	15	all	all	DET
cte-761	210	16	groups	group	NOUN
cte-761	210	17	.	.	PUNCT
cte-761	211	1	analysis	analysis	NOUN
cte-761	211	2	of	of	ADP
cte-761	211	3	variance	variance	NOUN
cte-761	211	4	(	(	PUNCT
cte-761	211	5	anova	anova	PROPN
cte-761	211	6	)	)	PUNCT
cte-761	211	7	demonstrated	demonstrate	VERB
cte-761	211	8	a	a	DET
cte-761	211	9	statistically	statistically	ADV
cte-761	211	10	significant	significant	ADJ
cte-761	211	11	difference	difference	NOUN
cte-761	211	12	in	in	ADP
cte-761	211	13	average	average	ADJ
cte-761	211	14	learning	learning	NOUN
cte-761	211	15	outcome	outcome	NOUN
cte-761	211	16	values	value	NOUN
cte-761	211	17	across	across	ADP
cte-761	211	18	all	all	DET
cte-761	211	19	groups	group	NOUN
cte-761	211	20	(	(	PUNCT
cte-761	211	21	f	f	NOUN
cte-761	211	22	=	=	SYM
cte-761	211	23	4.678693	4.678693	NUM
cte-761	211	24	;	;	PUNCT
cte-761	211	25	p	p	X
cte-761	211	26	=	=	SYM
cte-761	211	27	0.010613	0.010613	NUM
cte-761	211	28	)	)	PUNCT
cte-761	211	29	(	(	PUNCT
cte-761	211	30	table	table	NOUN
cte-761	211	31	3	3	NUM
cte-761	211	32	)	)	PUNCT
cte-761	211	33	.	.	PUNCT
cte-761	212	1	post	post	ADJ
cte-761	212	2	-	-	ADJ
cte-761	212	3	hoc	hoc	ADJ
cte-761	212	4	comparison	comparison	NOUN
cte-761	212	5	of	of	ADP
cte-761	212	6	means	mean	NOUN
cte-761	212	7	for	for	ADP
cte-761	212	8	groups	group	NOUN
cte-761	212	9	e	e	X
cte-761	212	10	vs.	vs.	ADP
cte-761	212	11	g	g	NOUN
cte-761	212	12	,	,	PUNCT
cte-761	212	13	e	e	X
cte-761	212	14	vs.	vs.	X
cte-761	212	15	eg	eg	NOUN
cte-761	212	16	,	,	PUNCT
cte-761	212	17	and	and	CCONJ
cte-761	212	18	g	g	PROPN
cte-761	212	19	vs.	vs.	X
cte-761	212	20	eg	eg	NOUN
cte-761	212	21	indicated	indicate	VERB
cte-761	212	22	that	that	SCONJ
cte-761	212	23	the	the	DET
cte-761	212	24	difference	difference	NOUN
cte-761	212	25	between	between	ADP
cte-761	212	26	means	mean	NOUN
cte-761	212	27	for	for	ADP
cte-761	212	28	groups	group	NOUN
cte-761	212	29	e	e	NOUN
cte-761	212	30	and	and	CCONJ
cte-761	212	31	g	g	PROPN
cte-761	212	32	fell	fall	VERB
cte-761	212	33	within	within	ADP
cte-761	212	34	statistical	statistical	ADJ
cte-761	212	35	error	error	NOUN
cte-761	212	36	margins	margin	NOUN
cte-761	212	37	(	(	PUNCT
cte-761	212	38	table	table	NOUN
cte-761	212	39	4	4	NUM
cte-761	212	40	)	)	PUNCT
cte-761	212	41	.	.	PUNCT
cte-761	213	1	pairwise	pairwise	NOUN
cte-761	213	2	post	post	ADJ
cte-761	213	3	-	-	ADJ
cte-761	213	4	hoc	hoc	ADJ
cte-761	213	5	comparison	comparison	NOUN
cte-761	213	6	results	result	NOUN
cte-761	213	7	confirmed	confirm	VERB
cte-761	213	8	statistical	statistical	ADJ
cte-761	213	9	247	247	NUM
cte-761	213	10	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	213	11	cte	cte	NOUN
cte-761	213	12	workshop	workshop	NOUN
cte-761	213	13	proceedings	proceeding	NOUN
cte-761	213	14	,	,	PUNCT
cte-761	213	15	2025	2025	NUM
cte-761	213	16	,	,	PUNCT
cte-761	213	17	vol	vol	NOUN
cte-761	213	18	.	.	PROPN
cte-761	213	19	12	12	NUM
cte-761	213	20	,	,	PUNCT
cte-761	213	21	pp	pp	ADJ
cte-761	213	22	.	.	PUNCT
cte-761	214	1	235–252	235–252	NUM
cte-761	214	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	214	3	figure	figure	NOUN
cte-761	214	4	12	12	NUM
cte-761	214	5	:	:	PUNCT
cte-761	214	6	visual	visual	ADJ
cte-761	214	7	analysis	analysis	NOUN
cte-761	214	8	in	in	ADP
cte-761	214	9	geogebra	geogebra	NOUN
cte-761	214	10	:	:	PUNCT
cte-761	214	11	flight	flight	NOUN
cte-761	214	12	duration	duration	NOUN
cte-761	214	13	comparison	comparison	NOUN
cte-761	214	14	for	for	ADP
cte-761	214	15	different	different	ADJ
cte-761	214	16	mathematical	mathematical	ADJ
cte-761	214	17	models	model	NOUN
cte-761	214	18	with	with	ADP
cte-761	214	19	clockwise	clockwise	NOUN
cte-761	214	20	and	and	CCONJ
cte-761	214	21	counterclockwise	counterclockwise	NOUN
cte-761	214	22	rotation	rotation	NOUN
cte-761	214	23	(	(	PUNCT
cte-761	214	24	rot=+1	rot=+1	NOUN
cte-761	214	25	and	and	CCONJ
cte-761	214	26	rot=-1	rot=-1	NOUN
cte-761	214	27	)	)	PUNCT
cte-761	214	28	.	.	PUNCT
cte-761	215	1	table	table	NOUN
cte-761	215	2	1	1	NUM
cte-761	215	3	final	final	ADJ
cte-761	215	4	learning	learn	VERB
cte-761	215	5	outcomes	outcome	NOUN
cte-761	215	6	of	of	ADP
cte-761	215	7	students	student	NOUN
cte-761	215	8	after	after	ADP
cte-761	215	9	the	the	DET
cte-761	215	10	experiment	experiment	NOUN
cte-761	215	11	.	.	PUNCT
cte-761	216	1	score	score	VERB
cte-761	216	2	50	50	NUM
cte-761	216	3	-	-	SYM
cte-761	216	4	55	55	NUM
cte-761	216	5	55	55	NUM
cte-761	216	6	-	-	SYM
cte-761	216	7	60	60	NUM
cte-761	216	8	60	60	NUM
cte-761	216	9	-	-	SYM
cte-761	216	10	65	65	NUM
cte-761	216	11	65	65	NUM
cte-761	216	12	-	-	SYM
cte-761	216	13	70	70	NUM
cte-761	216	14	70	70	NUM
cte-761	216	15	-	-	SYM
cte-761	216	16	75	75	NUM
cte-761	216	17	75	75	NUM
cte-761	216	18	-	-	SYM
cte-761	216	19	80	80	NUM
cte-761	216	20	80	80	NUM
cte-761	216	21	-	-	SYM
cte-761	216	22	85	85	NUM
cte-761	216	23	85	85	NUM
cte-761	216	24	-	-	SYM
cte-761	216	25	90	90	NUM
cte-761	216	26	90	90	NUM
cte-761	216	27	-	-	PUNCT
cte-761	216	28	95	95	NUM
cte-761	216	29	group	group	NOUN
cte-761	216	30	e	e	NOUN
cte-761	216	31	1	1	NUM
cte-761	216	32	2	2	NUM
cte-761	216	33	6	6	NUM
cte-761	216	34	9	9	NUM
cte-761	216	35	14	14	NUM
cte-761	216	36	10	10	NUM
cte-761	216	37	7	7	NUM
cte-761	216	38	4	4	NUM
cte-761	216	39	group	group	NOUN
cte-761	216	40	g	g	NOUN
cte-761	216	41	3	3	NUM
cte-761	216	42	6	6	NUM
cte-761	216	43	10	10	NUM
cte-761	216	44	13	13	NUM
cte-761	216	45	15	15	NUM
cte-761	216	46	8	8	NUM
cte-761	216	47	3	3	NUM
cte-761	216	48	1	1	NUM
cte-761	216	49	group	group	NOUN
cte-761	216	50	eg	eg	NOUN
cte-761	216	51	1	1	NUM
cte-761	216	52	3	3	NUM
cte-761	216	53	5	5	NUM
cte-761	216	54	10	10	NUM
cte-761	216	55	15	15	NUM
cte-761	216	56	9	9	NUM
cte-761	216	57	7	7	NUM
cte-761	216	58	5	5	NUM
cte-761	216	59	table	table	NOUN
cte-761	216	60	2	2	NUM
cte-761	216	61	primary	primary	ADJ
cte-761	216	62	statistical	statistical	ADJ
cte-761	216	63	data	datum	NOUN
cte-761	216	64	processing	processing	NOUN
cte-761	216	65	results	result	NOUN
cte-761	216	66	.	.	PUNCT
cte-761	217	1	group	group	NOUN
cte-761	217	2	valid	valid	PROPN
cte-761	217	3	n	n	CCONJ
cte-761	217	4	mean	mean	VERB
cte-761	217	5	conf.-95	conf.-95	NOUN
cte-761	217	6	%	%	NOUN
cte-761	217	7	conf.+95	conf.+95	NUM
cte-761	217	8	%	%	NOUN
cte-761	217	9	median	median	ADJ
cte-761	217	10	mode	mode	NOUN
cte-761	217	11	e	e	NOUN
cte-761	217	12	53	53	NUM
cte-761	217	13	73.36	73.36	NUM
cte-761	217	14	71.16	71.16	NUM
cte-761	217	15	75.55	75.55	NUM
cte-761	217	16	74	74	NUM
cte-761	217	17	multiple	multiple	NOUN
cte-761	217	18	g	g	PROPN
cte-761	217	19	59	59	NUM
cte-761	217	20	73.85	73.85	NUM
cte-761	217	21	71.74	71.74	NUM
cte-761	217	22	75.95	75.95	NUM
cte-761	217	23	74	74	NUM
cte-761	217	24	76	76	NUM
cte-761	217	25	eg	eg	NOUN
cte-761	217	26	55	55	NUM
cte-761	217	27	78.25	78.25	NUM
cte-761	217	28	75.94	75.94	NUM
cte-761	217	29	80.57	80.57	NUM
cte-761	217	30	77	77	NUM
cte-761	217	31	76	76	NUM
cte-761	217	32	group	group	NOUN
cte-761	217	33	freq	freq	NOUN
cte-761	217	34	.	.	PUNCT
cte-761	218	1	mode	mode	PROPN
cte-761	218	2	min	min	PROPN
cte-761	218	3	max	max	PROPN
cte-761	218	4	sd	sd	PROPN
cte-761	218	5	shapiro	shapiro	PROPN
cte-761	218	6	–	–	PUNCT
cte-761	218	7	wilk	wilk	NOUN
cte-761	218	8	test	test	NOUN
cte-761	218	9	e	e	NOUN
cte-761	218	10	5	5	NUM
cte-761	218	11	55	55	NUM
cte-761	218	12	90	90	NUM
cte-761	218	13	7.96	7.96	NUM
cte-761	218	14	w=0.98823	w=0.98823	NOUN
cte-761	218	15	p=0.87840	p=0.87840	PROPN
cte-761	218	16	g	g	NOUN
cte-761	218	17	5	5	NUM
cte-761	218	18	56	56	NUM
cte-761	218	19	91	91	NUM
cte-761	218	20	8.08	8.08	NUM
cte-761	218	21	w=0.98749	w=0.98749	PROPN
cte-761	218	22	p=0.80486	p=0.80486	PROPN
cte-761	218	23	eg	eg	VERB
cte-761	218	24	6	6	NUM
cte-761	218	25	60	60	NUM
cte-761	218	26	95	95	NUM
cte-761	218	27	8.56	8.56	NUM
cte-761	218	28	w=0.98158	w=0.98158	NOUN
cte-761	218	29	p=0.55809	p=0.55809	PROPN
cte-761	218	30	significance	significance	NOUN
cte-761	218	31	in	in	ADP
cte-761	218	32	the	the	DET
cte-761	218	33	difference	difference	NOUN
cte-761	218	34	between	between	ADP
cte-761	218	35	the	the	DET
cte-761	218	36	mean	mean	NOUN
cte-761	218	37	for	for	ADP
cte-761	218	38	group	group	NOUN
cte-761	218	39	eg	eg	NOUN
cte-761	218	40	and	and	CCONJ
cte-761	218	41	the	the	DET
cte-761	218	42	means	mean	NOUN
cte-761	218	43	for	for	ADP
cte-761	218	44	groups	group	NOUN
cte-761	218	45	e	e	NOUN
cte-761	218	46	and	and	CCONJ
cte-761	218	47	g.	g.	PROPN
cte-761	218	48	4	4	NUM
cte-761	218	49	.	.	PUNCT
cte-761	219	1	discussion	discussion	NOUN
cte-761	219	2	this	this	DET
cte-761	219	3	study	study	NOUN
cte-761	219	4	compared	compare	VERB
cte-761	219	5	our	our	PRON
cte-761	219	6	earlier	early	ADJ
cte-761	219	7	findings	finding	NOUN
cte-761	219	8	when	when	SCONJ
cte-761	219	9	students	student	NOUN
cte-761	219	10	initially	initially	ADV
cte-761	219	11	used	use	VERB
cte-761	219	12	excel	excel	NOUN
cte-761	219	13	spreadsheets	spreadsheet	NOUN
cte-761	219	14	for	for	ADP
cte-761	219	15	learning	learn	VERB
cte-761	219	16	exercises	exercise	NOUN
cte-761	219	17	before	before	ADP
cte-761	219	18	incorporating	incorporate	VERB
cte-761	219	19	geogebra	geogebra	NOUN
cte-761	219	20	.	.	PUNCT
cte-761	220	1	throughout	throughout	ADP
cte-761	220	2	the	the	DET
cte-761	220	3	training	training	NOUN
cte-761	220	4	,	,	PUNCT
cte-761	220	5	248	248	NUM
cte-761	220	6	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	220	7	cte	cte	NOUN
cte-761	220	8	workshop	workshop	NOUN
cte-761	220	9	proceedings	proceeding	NOUN
cte-761	220	10	,	,	PUNCT
cte-761	220	11	2025	2025	NUM
cte-761	220	12	,	,	PUNCT
cte-761	220	13	vol	vol	NOUN
cte-761	220	14	.	.	PROPN
cte-761	220	15	12	12	NUM
cte-761	220	16	,	,	PUNCT
cte-761	220	17	pp	pp	ADJ
cte-761	220	18	.	.	PUNCT
cte-761	221	1	235–252	235–252	NUM
cte-761	221	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	221	3	figure	figure	NOUN
cte-761	221	4	13	13	NUM
cte-761	221	5	:	:	PUNCT
cte-761	221	6	dynamic	dynamic	ADJ
cte-761	221	7	vector	vector	NOUN
cte-761	221	8	diagram	diagram	NOUN
cte-761	221	9	of	of	ADP
cte-761	221	10	x	x	NOUN
cte-761	221	11	-	-	NOUN
cte-761	221	12	component	component	ADJ
cte-761	221	13	fx	fx	NOUN
cte-761	221	14	of	of	ADP
cte-761	221	15	the	the	DET
cte-761	221	16	resultant	resultant	NOUN
cte-761	221	17	forces	force	NOUN
cte-761	221	18	acting	act	VERB
cte-761	221	19	on	on	ADP
cte-761	221	20	the	the	DET
cte-761	221	21	projectile	projectile	NOUN
cte-761	221	22	in	in	ADP
cte-761	221	23	four	four	NUM
cte-761	221	24	flight	flight	NOUN
cte-761	221	25	cases	case	NOUN
cte-761	221	26	:	:	PUNCT
cte-761	221	27	ti	ti	X
cte-761	221	28	=	=	SYM
cte-761	221	29	0	0	NUM
cte-761	221	30	,	,	PUNCT
cte-761	221	31	rot	rot	VERB
cte-761	221	32	=	=	NOUN
cte-761	221	33	1	1	NUM
cte-761	221	34	;	;	PUNCT
cte-761	221	35	ti	ti	X
cte-761	221	36	=	=	SYM
cte-761	221	37	tm	tm	NOUN
cte-761	221	38	,	,	PUNCT
cte-761	221	39	rot	rot	VERB
cte-761	221	40	=	=	NOUN
cte-761	221	41	1	1	NUM
cte-761	221	42	;	;	PUNCT
cte-761	221	43	ti	ti	X
cte-761	221	44	=	=	SYM
cte-761	221	45	0	0	NUM
cte-761	221	46	,	,	PUNCT
cte-761	221	47	rot	rot	VERB
cte-761	221	48	=	=	SYM
cte-761	221	49	-1	-1	ADJ
cte-761	221	50	;	;	PUNCT
cte-761	221	51	ti	ti	X
cte-761	221	52	=	=	SYM
cte-761	221	53	tm	tm	NOUN
cte-761	221	54	,	,	PUNCT
cte-761	221	55	rot	rot	VERB
cte-761	221	56	=	=	SYM
cte-761	221	57	-1	-1	NOUN
cte-761	221	58	.	.	PUNCT
cte-761	221	59	table	table	NOUN
cte-761	221	60	3	3	NUM
cte-761	221	61	anova	anova	PROPN
cte-761	221	62	results	result	NOUN
cte-761	221	63	.	.	PUNCT
cte-761	222	1	ss	ss	PROPN
cte-761	222	2	df	df	PROPN
cte-761	222	3	ms	ms	PROPN
cte-761	222	4	ss_err	ss_err	PROPN
cte-761	222	5	df_err	df_err	PROPN
cte-761	222	6	ms_err	ms_err	PUNCT
cte-761	223	1	f	f	PROPN
cte-761	223	2	p	p	NOUN
cte-761	223	3	point	point	NOUN
cte-761	223	4	1249.818	1249.818	NUM
cte-761	223	5	2	2	NUM
cte-761	223	6	624.9091	624.9091	NUM
cte-761	223	7	10588.73	10588.73	NUM
cte-761	223	8	164	164	NUM
cte-761	223	9	64.56544	64.56544	NUM
cte-761	223	10	4.678693	4.678693	NUM
cte-761	223	11	0.010613	0.010613	NUM
cte-761	223	12	table	table	NOUN
cte-761	223	13	4	4	NUM
cte-761	223	14	pairwise	pairwise	NOUN
cte-761	223	15	post	post	ADJ
cte-761	223	16	-	-	ADJ
cte-761	223	17	hoc	hoc	ADJ
cte-761	223	18	comparison	comparison	NOUN
cte-761	223	19	results	result	NOUN
cte-761	223	20	.	.	PUNCT
cte-761	224	1	pairwise	pairwise	NOUN
cte-761	224	2	post	post	ADJ
cte-761	224	3	-	-	ADJ
cte-761	224	4	hoc	hoc	ADJ
cte-761	224	5	comparisons	comparison	NOUN
cte-761	224	6	of	of	ADP
cte-761	224	7	means	mean	NOUN
cte-761	224	8	e	e	X
cte-761	224	9	vs.	vs.	X
cte-761	224	10	g	g	PROPN
cte-761	224	11	e	e	PROPN
cte-761	224	12	vs.	vs.	X
cte-761	224	13	eg	eg	NOUN
cte-761	224	14	g	g	PROPN
cte-761	224	15	vs.	vs.	X
cte-761	224	16	eg	eg	PROPN
cte-761	224	17	lsd	lsd	NOUN
cte-761	224	18	-	-	PUNCT
cte-761	224	19	test	test	NOUN
cte-761	224	20	p>0.9468	p>0.9468	PROPN
cte-761	224	21	p<0.0262	p<0.0262	NOUN
cte-761	224	22	p<0.0120	p<0.0120	ADJ
cte-761	224	23	duncan	duncan	ADJ
cte-761	224	24	-	-	PUNCT
cte-761	224	25	test	test	NOUN
cte-761	224	26	p>0.9475	p>0.9475	NOUN
cte-761	224	27	p<0.0135	p<0.0135	PRON
cte-761	224	28	p<0.0151	p<0.0151	NOUN
cte-761	224	29	tukey	tukey	NOUN
cte-761	224	30	hsd	hsd	VERB
cte-761	224	31	for	for	ADP
cte-761	224	32	unequal	unequal	ADJ
cte-761	224	33	n	n	PRON
cte-761	224	34	test	test	VERB
cte-761	224	35	p>0.9977	p>0.9977	ADJ
cte-761	224	36	p<0.0061	p<0.0061	ADP
cte-761	224	37	0.0048	0.0048	NOUN
cte-761	224	38	we	we	PRON
cte-761	224	39	systematically	systematically	ADV
cte-761	224	40	examined	examine	VERB
cte-761	224	41	how	how	SCONJ
cte-761	224	42	various	various	ADJ
cte-761	224	43	visualisation	visualisation	NOUN
cte-761	224	44	methods	method	NOUN
cte-761	224	45	affected	affect	VERB
cte-761	224	46	educational	educational	ADJ
cte-761	224	47	outcomes	outcome	NOUN
cte-761	224	48	for	for	ADP
cte-761	224	49	agricultural	agricultural	ADJ
cte-761	224	50	university	university	NOUN
cte-761	224	51	students	student	NOUN
cte-761	224	52	.	.	PUNCT
cte-761	225	1	our	our	PRON
cte-761	225	2	findings	finding	NOUN
cte-761	225	3	indicate	indicate	VERB
cte-761	225	4	that	that	SCONJ
cte-761	225	5	visualising	visualise	VERB
cte-761	225	6	dynamic	dynamic	ADJ
cte-761	225	7	characteristics	characteristic	NOUN
cte-761	225	8	of	of	ADP
cte-761	225	9	mechanical	mechanical	ADJ
cte-761	225	10	movement	movement	NOUN
cte-761	225	11	using	use	VERB
cte-761	225	12	dynamic	dynamic	ADJ
cte-761	225	13	vector	vector	NOUN
cte-761	225	14	diagrams	diagram	NOUN
cte-761	225	15	in	in	ADP
cte-761	225	16	geogebra	geogebra	NOUN
cte-761	225	17	improved	improve	VERB
cte-761	225	18	educational	educational	ADJ
cte-761	225	19	achievements	achievement	NOUN
cte-761	225	20	of	of	ADP
cte-761	225	21	agricultural	agricultural	ADJ
cte-761	225	22	university	university	NOUN
cte-761	225	23	students	student	NOUN
cte-761	225	24	studying	study	VERB
cte-761	225	25	mathematical	mathematical	ADJ
cte-761	225	26	modelling	modelling	NOUN
cte-761	225	27	fundamentals	fundamental	NOUN
cte-761	225	28	.	.	PUNCT
cte-761	226	1	several	several	ADJ
cte-761	226	2	factors	factor	NOUN
cte-761	226	3	may	may	AUX
cte-761	226	4	explain	explain	VERB
cte-761	226	5	these	these	DET
cte-761	226	6	results	result	NOUN
cte-761	226	7	.	.	PUNCT
cte-761	227	1	firstly	firstly	ADV
cte-761	227	2	,	,	PUNCT
cte-761	227	3	implementing	implement	VERB
cte-761	227	4	mathematical	mathematical	ADJ
cte-761	227	5	models	model	NOUN
cte-761	227	6	in	in	ADP
cte-761	227	7	excel	excel	NOUN
cte-761	227	8	essentially	essentially	ADV
cte-761	227	9	involves	involve	VERB
cte-761	227	10	stepwise	stepwise	ADJ
cte-761	227	11	reproduction	reproduction	NOUN
cte-761	227	12	of	of	ADP
cte-761	227	13	mathematical	mathematical	ADJ
cte-761	227	14	operations	operation	NOUN
cte-761	227	15	,	,	PUNCT
cte-761	227	16	which	which	PRON
cte-761	227	17	enhances	enhance	VERB
cte-761	227	18	students	student	NOUN
cte-761	227	19	’	’	PART
cte-761	227	20	understanding	understanding	NOUN
cte-761	227	21	of	of	ADP
cte-761	227	22	the	the	DET
cte-761	227	23	technical	technical	ADJ
cte-761	227	24	aspects	aspect	NOUN
cte-761	227	25	of	of	ADP
cte-761	227	26	mathematical	mathematical	ADJ
cte-761	227	27	modelling	modelling	NOUN
cte-761	227	28	.	.	PUNCT
cte-761	228	1	this	this	DET
cte-761	228	2	approach	approach	NOUN
cte-761	228	3	allows	allow	VERB
cte-761	228	4	students	student	NOUN
cte-761	228	5	to	to	ADP
cte-761	228	6	self	self	NOUN
cte-761	228	7	-	-	PUNCT
cte-761	228	8	diagnose	diagnose	NOUN
cte-761	228	9	calculations	calculation	NOUN
cte-761	228	10	and	and	CCONJ
cte-761	228	11	provides	provide	VERB
cte-761	228	12	instructors	instructor	NOUN
cte-761	228	13	with	with	ADP
cte-761	228	14	additional	additional	ADJ
cte-761	228	15	didactic	didactic	ADJ
cte-761	228	16	advantages	advantage	NOUN
cte-761	228	17	,	,	PUNCT
cte-761	228	18	such	such	ADJ
cte-761	228	19	as	as	ADP
cte-761	228	20	the	the	DET
cte-761	228	21	ability	ability	NOUN
cte-761	228	22	to	to	PART
cte-761	228	23	identify	identify	VERB
cte-761	228	24	and	and	CCONJ
cte-761	228	25	249	249	NUM
cte-761	228	26	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	228	27	cte	cte	NOUN
cte-761	228	28	workshop	workshop	NOUN
cte-761	228	29	proceedings	proceeding	NOUN
cte-761	228	30	,	,	PUNCT
cte-761	228	31	2025	2025	NUM
cte-761	228	32	,	,	PUNCT
cte-761	228	33	vol	vol	NOUN
cte-761	228	34	.	.	PROPN
cte-761	228	35	12	12	NUM
cte-761	228	36	,	,	PUNCT
cte-761	228	37	pp	pp	ADJ
cte-761	228	38	.	.	PUNCT
cte-761	229	1	235–252	235–252	NUM
cte-761	229	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	229	3	figure	figure	NOUN
cte-761	229	4	14	14	NUM
cte-761	229	5	:	:	PUNCT
cte-761	229	6	dynamic	dynamic	ADJ
cte-761	229	7	vector	vector	NOUN
cte-761	229	8	diagram	diagram	NOUN
cte-761	229	9	of	of	ADP
cte-761	229	10	y	y	PROPN
cte-761	229	11	-	-	PUNCT
cte-761	229	12	component	component	NOUN
cte-761	229	13	fy	fy	PROPN
cte-761	229	14	of	of	ADP
cte-761	229	15	the	the	DET
cte-761	229	16	resultant	resultant	NOUN
cte-761	229	17	forces	force	NOUN
cte-761	229	18	acting	act	VERB
cte-761	229	19	on	on	ADP
cte-761	229	20	the	the	DET
cte-761	229	21	projectile	projectile	NOUN
cte-761	229	22	in	in	ADP
cte-761	229	23	four	four	NUM
cte-761	229	24	cases	case	NOUN
cte-761	229	25	(	(	PUNCT
cte-761	229	26	from	from	ADP
cte-761	229	27	top	top	NOUN
cte-761	229	28	to	to	ADP
cte-761	229	29	right	right	NOUN
cte-761	229	30	):	):	PUNCT
cte-761	229	31	ti	ti	X
cte-761	229	32	=	=	SYM
cte-761	229	33	0	0	NUM
cte-761	229	34	,	,	PUNCT
cte-761	229	35	rot	rot	VERB
cte-761	229	36	=	=	NOUN
cte-761	229	37	1	1	NUM
cte-761	229	38	;	;	PUNCT
cte-761	229	39	ti	ti	X
cte-761	229	40	=	=	SYM
cte-761	229	41	tm	tm	NOUN
cte-761	229	42	,	,	PUNCT
cte-761	229	43	rot	rot	VERB
cte-761	229	44	=	=	NOUN
cte-761	229	45	1	1	NUM
cte-761	229	46	;	;	PUNCT
cte-761	229	47	ti	ti	X
cte-761	229	48	=	=	SYM
cte-761	229	49	0	0	NUM
cte-761	229	50	,	,	PUNCT
cte-761	229	51	rot	rot	VERB
cte-761	229	52	=	=	SYM
cte-761	229	53	-1	-1	ADJ
cte-761	229	54	;	;	PUNCT
cte-761	229	55	ti	ti	X
cte-761	229	56	=	=	SYM
cte-761	229	57	tm	tm	NOUN
cte-761	229	58	,	,	PUNCT
cte-761	229	59	rot	rot	VERB
cte-761	229	60	=	=	SYM
cte-761	229	61	-1	-1	X
cte-761	229	62	.	.	PUNCT
cte-761	230	1	discuss	discuss	VERB
cte-761	230	2	intermediate	intermediate	ADJ
cte-761	230	3	effects	effect	NOUN
cte-761	230	4	and	and	CCONJ
cte-761	230	5	simulation	simulation	NOUN
cte-761	230	6	outcomes	outcome	NOUN
cte-761	230	7	that	that	PRON
cte-761	230	8	might	might	AUX
cte-761	230	9	remain	remain	VERB
cte-761	230	10	hidden	hidden	ADJ
cte-761	230	11	when	when	SCONJ
cte-761	230	12	using	use	VERB
cte-761	230	13	professional	professional	ADJ
cte-761	230	14	computer	computer	NOUN
cte-761	230	15	mathematics	mathematic	NOUN
cte-761	230	16	systems	system	NOUN
cte-761	230	17	like	like	ADP
cte-761	230	18	mathcad	mathcad	PROPN
cte-761	231	1	[	[	X
cte-761	231	2	3	3	NUM
cte-761	231	3	]	]	PUNCT
cte-761	231	4	,	,	PUNCT
cte-761	231	5	mathematica	mathematica	PROPN
cte-761	231	6	,	,	PUNCT
cte-761	231	7	matlab	matlab	PROPN
cte-761	231	8	,	,	PUNCT
cte-761	231	9	or	or	CCONJ
cte-761	231	10	maple	maple	NOUN
cte-761	231	11	.	.	PUNCT
cte-761	232	1	additionally	additionally	ADV
cte-761	232	2	,	,	PUNCT
cte-761	232	3	visualising	visualise	VERB
cte-761	232	4	mechanical	mechanical	ADJ
cte-761	232	5	motion	motion	NOUN
cte-761	232	6	trajectories	trajectory	NOUN
cte-761	232	7	through	through	ADP
cte-761	232	8	excel	excel	NOUN
cte-761	232	9	charts	chart	NOUN
cte-761	232	10	cultivates	cultivate	VERB
cte-761	232	11	intuitive	intuitive	ADJ
cte-761	232	12	understanding	understanding	NOUN
cte-761	232	13	of	of	ADP
cte-761	232	14	how	how	SCONJ
cte-761	232	15	movement	movement	NOUN
cte-761	232	16	characteristics	characteristic	NOUN
cte-761	232	17	change	change	VERB
cte-761	232	18	in	in	ADP
cte-761	232	19	response	response	NOUN
cte-761	232	20	to	to	ADP
cte-761	232	21	initial	initial	ADJ
cte-761	232	22	conditions	condition	NOUN
cte-761	232	23	and	and	CCONJ
cte-761	232	24	other	other	ADJ
cte-761	232	25	parameters	parameter	NOUN
cte-761	232	26	.	.	PUNCT
cte-761	233	1	this	this	PRON
cte-761	233	2	enables	enable	VERB
cte-761	233	3	students	student	NOUN
cte-761	233	4	to	to	PART
cte-761	233	5	consciously	consciously	ADV
cte-761	233	6	adjust	adjust	VERB
cte-761	233	7	input	input	NOUN
cte-761	233	8	parameters	parameter	NOUN
cte-761	233	9	to	to	PART
cte-761	233	10	achieve	achieve	VERB
cte-761	233	11	desired	desire	VERB
cte-761	233	12	simulation	simulation	NOUN
cte-761	233	13	results	result	NOUN
cte-761	233	14	.	.	PUNCT
cte-761	234	1	however	however	ADV
cte-761	234	2	,	,	PUNCT
cte-761	234	3	standard	standard	ADJ
cte-761	234	4	excel	excel	NOUN
cte-761	234	5	features	feature	NOUN
cte-761	234	6	do	do	AUX
cte-761	234	7	not	not	PART
cte-761	234	8	facilitate	facilitate	VERB
cte-761	234	9	visualisation	visualisation	NOUN
cte-761	234	10	of	of	ADP
cte-761	234	11	instantaneous	instantaneous	ADJ
cte-761	234	12	velocity	velocity	NOUN
cte-761	234	13	changes	change	NOUN
cte-761	234	14	and	and	CCONJ
cte-761	234	15	motion	motion	NOUN
cte-761	234	16	direction	direction	NOUN
cte-761	234	17	along	along	ADP
cte-761	234	18	trajectories	trajectory	NOUN
cte-761	234	19	.	.	PUNCT
cte-761	235	1	consequently	consequently	ADV
cte-761	235	2	,	,	PUNCT
cte-761	235	3	excel	excel	VERB
cte-761	235	4	usage	usage	NOUN
cte-761	235	5	alone	alone	ADV
cte-761	235	6	does	do	AUX
cte-761	235	7	not	not	PART
cte-761	235	8	create	create	VERB
cte-761	235	9	sufficient	sufficient	ADJ
cte-761	235	10	conditions	condition	NOUN
cte-761	235	11	for	for	ADP
cte-761	235	12	developing	develop	VERB
cte-761	235	13	skills	skill	NOUN
cte-761	235	14	in	in	ADP
cte-761	235	15	analysing	analyse	VERB
cte-761	235	16	dynamic	dynamic	ADJ
cte-761	235	17	movement	movement	NOUN
cte-761	235	18	characteristics	characteristic	NOUN
cte-761	235	19	based	base	VERB
cte-761	235	20	on	on	ADP
cte-761	235	21	numerical	numerical	ADJ
cte-761	235	22	data	datum	NOUN
cte-761	235	23	.	.	PUNCT
cte-761	236	1	conversely	conversely	ADV
cte-761	236	2	,	,	PUNCT
cte-761	236	3	geogebra	geogebra	NOUN
cte-761	236	4	enables	enable	VERB
cte-761	236	5	students	student	NOUN
cte-761	236	6	to	to	PART
cte-761	236	7	develop	develop	VERB
cte-761	236	8	intuitive	intuitive	ADJ
cte-761	236	9	spatial	spatial	ADJ
cte-761	236	10	perceptions	perception	NOUN
cte-761	236	11	more	more	ADV
cte-761	236	12	250	250	NUM
cte-761	236	13	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	236	14	cte	cte	NOUN
cte-761	236	15	workshop	workshop	NOUN
cte-761	236	16	proceedings	proceeding	NOUN
cte-761	236	17	,	,	PUNCT
cte-761	236	18	2025	2025	NUM
cte-761	236	19	,	,	PUNCT
cte-761	236	20	vol	vol	NOUN
cte-761	236	21	.	.	PROPN
cte-761	236	22	12	12	NUM
cte-761	236	23	,	,	PUNCT
cte-761	236	24	pp	pp	ADJ
cte-761	236	25	.	.	PUNCT
cte-761	237	1	235–252	235–252	NUM
cte-761	237	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	237	3	rapidly	rapidly	ADV
cte-761	237	4	through	through	ADP
cte-761	237	5	visual	visual	ADJ
cte-761	237	6	analysis	analysis	NOUN
cte-761	237	7	of	of	ADP
cte-761	237	8	dynamic	dynamic	ADJ
cte-761	237	9	motion	motion	NOUN
cte-761	237	10	characteristics	characteristic	NOUN
cte-761	237	11	,	,	PUNCT
cte-761	237	12	primarily	primarily	ADV
cte-761	237	13	through	through	ADP
cte-761	237	14	dynamic	dynamic	ADJ
cte-761	237	15	visualisation	visualisation	NOUN
cte-761	237	16	of	of	ADP
cte-761	237	17	vector	vector	NOUN
cte-761	237	18	characteristics	characteristic	NOUN
cte-761	237	19	.	.	PUNCT
cte-761	238	1	we	we	PRON
cte-761	238	2	observed	observe	VERB
cte-761	238	3	that	that	SCONJ
cte-761	238	4	after	after	ADP
cte-761	238	5	completing	complete	VERB
cte-761	238	6	learning	learn	VERB
cte-761	238	7	exercises	exercise	NOUN
cte-761	238	8	using	use	VERB
cte-761	238	9	geogebra	geogebra	NOUN
cte-761	238	10	,	,	PUNCT
cte-761	238	11	students	student	NOUN
cte-761	238	12	more	more	ADV
cte-761	238	13	readily	readily	ADV
cte-761	238	14	formulated	formulate	VERB
cte-761	238	15	meaningful	meaningful	ADJ
cte-761	238	16	responses	response	NOUN
cte-761	238	17	to	to	ADP
cte-761	238	18	qualitative	qualitative	ADJ
cte-761	238	19	evaluation	evaluation	NOUN
cte-761	238	20	questions	question	NOUN
cte-761	238	21	.	.	PUNCT
cte-761	239	1	interactive	interactive	ADJ
cte-761	239	2	study	study	NOUN
cte-761	239	3	of	of	ADP
cte-761	239	4	dynamic	dynamic	ADJ
cte-761	239	5	plots	plot	NOUN
cte-761	239	6	,	,	PUNCT
cte-761	239	7	such	such	ADJ
cte-761	239	8	as	as	ADP
cte-761	239	9	figure	figure	NOUN
cte-761	239	10	11	11	NUM
cte-761	239	11	,	,	PUNCT
cte-761	239	12	allowed	allow	VERB
cte-761	239	13	them	they	PRON
cte-761	239	14	to	to	PART
cte-761	239	15	quickly	quickly	ADV
cte-761	239	16	and	and	CCONJ
cte-761	239	17	accurately	accurately	ADV
cte-761	239	18	answer	answer	VERB
cte-761	239	19	questions	question	NOUN
cte-761	239	20	like	like	ADP
cte-761	239	21	“	"	PUNCT
cte-761	239	22	how	how	SCONJ
cte-761	239	23	does	do	AUX
cte-761	239	24	rotation	rotation	NOUN
cte-761	239	25	direction	direction	NOUN
cte-761	239	26	affect	affect	VERB
cte-761	239	27	linear	linear	ADJ
cte-761	239	28	velocity	velocity	NOUN
cte-761	239	29	?	?	PUNCT
cte-761	239	30	”	"	PUNCT
cte-761	239	31	and	and	CCONJ
cte-761	239	32	“	"	PUNCT
cte-761	239	33	at	at	ADP
cte-761	239	34	what	what	DET
cte-761	239	35	points	point	NOUN
cte-761	239	36	will	will	AUX
cte-761	239	37	the	the	DET
cte-761	239	38	velocity	velocity	NOUN
cte-761	239	39	in	in	ADP
cte-761	239	40	airless	airless	ADJ
cte-761	239	41	space	space	NOUN
cte-761	239	42	equal	equal	ADJ
cte-761	239	43	the	the	DET
cte-761	239	44	velocity	velocity	NOUN
cte-761	239	45	in	in	ADP
cte-761	239	46	air	air	NOUN
cte-761	239	47	?	?	PUNCT
cte-761	239	48	”	"	PUNCT
cte-761	240	1	however	however	ADV
cte-761	240	2	,	,	PUNCT
cte-761	240	3	students	student	NOUN
cte-761	240	4	working	work	VERB
cte-761	240	5	exclusively	exclusively	ADV
cte-761	240	6	in	in	ADP
cte-761	240	7	excel	excel	NOUN
cte-761	240	8	demonstrated	demonstrate	VERB
cte-761	240	9	superior	superior	ADJ
cte-761	240	10	performance	performance	NOUN
cte-761	240	11	in	in	ADP
cte-761	240	12	addressing	address	VERB
cte-761	240	13	questions	question	NOUN
cte-761	240	14	about	about	ADP
cte-761	240	15	quantitative	quantitative	ADJ
cte-761	240	16	model	model	NOUN
cte-761	240	17	characteristics	characteristic	NOUN
cte-761	240	18	.	.	PUNCT
cte-761	241	1	as	as	ADP
cte-761	241	2	in	in	ADP
cte-761	241	3	our	our	PRON
cte-761	241	4	previous	previous	ADJ
cte-761	241	5	research	research	NOUN
cte-761	241	6	[	[	X
cte-761	241	7	2	2	NUM
cte-761	241	8	,	,	PUNCT
cte-761	241	9	4	4	NUM
cte-761	241	10	]	]	PUNCT
cte-761	241	11	,	,	PUNCT
cte-761	241	12	we	we	PRON
cte-761	241	13	maintain	maintain	VERB
cte-761	241	14	that	that	SCONJ
cte-761	241	15	using	use	VERB
cte-761	241	16	geogebra	geogebra	NOUN
cte-761	241	17	alone	alone	ADV
cte-761	241	18	for	for	ADP
cte-761	241	19	studying	study	VERB
cte-761	241	20	mathematical	mathematical	ADJ
cte-761	241	21	modelling	modelling	NOUN
cte-761	241	22	fundamentals	fundamental	NOUN
cte-761	241	23	creates	create	VERB
cte-761	241	24	difficulties	difficulty	NOUN
cte-761	241	25	in	in	ADP
cte-761	241	26	numerically	numerically	ADV
cte-761	241	27	evaluating	evaluate	VERB
cte-761	241	28	simulation	simulation	NOUN
cte-761	241	29	results	result	NOUN
cte-761	241	30	.	.	PUNCT
cte-761	242	1	consequently	consequently	ADV
cte-761	242	2	,	,	PUNCT
cte-761	242	3	it	it	PRON
cte-761	242	4	does	do	AUX
cte-761	242	5	not	not	PART
cte-761	242	6	develop	develop	VERB
cte-761	242	7	sufficient	sufficient	ADJ
cte-761	242	8	skills	skill	NOUN
cte-761	242	9	for	for	ADP
cte-761	242	10	numerically	numerically	ADV
cte-761	242	11	assessing	assess	VERB
cte-761	242	12	phenomenon	phenomenon	NOUN
cte-761	242	13	or	or	CCONJ
cte-761	242	14	process	process	NOUN
cte-761	242	15	characteristics	characteristic	NOUN
cte-761	242	16	.	.	PUNCT
cte-761	243	1	concurrently	concurrently	ADV
cte-761	243	2	,	,	PUNCT
cte-761	243	3	extensive	extensive	ADJ
cte-761	243	4	geogebra	geogebra	NOUN
cte-761	243	5	usage	usage	NOUN
cte-761	243	6	contributes	contribute	VERB
cte-761	243	7	to	to	ADP
cte-761	243	8	effective	effective	ADJ
cte-761	243	9	formation	formation	NOUN
cte-761	243	10	of	of	ADP
cte-761	243	11	intuitive	intuitive	ADJ
cte-761	243	12	spatial	spatial	ADJ
cte-761	243	13	representations	representation	NOUN
cte-761	243	14	,	,	PUNCT
cte-761	243	15	which	which	PRON
cte-761	243	16	are	be	AUX
cte-761	243	17	vital	vital	ADJ
cte-761	243	18	for	for	ADP
cte-761	243	19	engineering	engineering	NOUN
cte-761	243	20	professionals	professional	NOUN
cte-761	243	21	.	.	PUNCT
cte-761	244	1	since	since	SCONJ
cte-761	244	2	no	no	DET
cte-761	244	3	significant	significant	ADJ
cte-761	244	4	difference	difference	NOUN
cte-761	244	5	was	be	AUX
cte-761	244	6	observed	observe	VERB
cte-761	244	7	between	between	ADP
cte-761	244	8	final	final	ADJ
cte-761	244	9	learning	learning	NOUN
cte-761	244	10	outcomes	outcome	NOUN
cte-761	244	11	in	in	ADP
cte-761	244	12	groups	group	NOUN
cte-761	244	13	e	e	NOUN
cte-761	244	14	and	and	CCONJ
cte-761	244	15	g	g	PROPN
cte-761	244	16	,	,	PUNCT
cte-761	244	17	the	the	DET
cte-761	244	18	simultaneous	simultaneous	ADJ
cte-761	244	19	use	use	NOUN
cte-761	244	20	of	of	ADP
cte-761	244	21	excel	excel	NOUN
cte-761	244	22	and	and	CCONJ
cte-761	244	23	geogebra	geogebra	NOUN
cte-761	244	24	appears	appear	VERB
cte-761	244	25	to	to	PART
cte-761	244	26	compensate	compensate	VERB
cte-761	244	27	for	for	ADP
cte-761	244	28	individual	individual	ADJ
cte-761	244	29	limitations	limitation	NOUN
cte-761	244	30	,	,	PUNCT
cte-761	244	31	thus	thus	ADV
cte-761	244	32	yielding	yield	VERB
cte-761	244	33	superior	superior	ADJ
cte-761	244	34	educational	educational	ADJ
cte-761	244	35	achievements	achievement	NOUN
cte-761	244	36	.	.	PUNCT
cte-761	245	1	moreover	moreover	ADV
cte-761	245	2	,	,	PUNCT
cte-761	245	3	incorporating	incorporate	VERB
cte-761	245	4	dynamic	dynamic	ADJ
cte-761	245	5	vector	vector	NOUN
cte-761	245	6	diagrams	diagram	NOUN
cte-761	245	7	in	in	ADP
cte-761	245	8	geogebra	geogebra	NOUN
cte-761	245	9	further	far	ADV
cte-761	245	10	enhanced	enhance	VERB
cte-761	245	11	these	these	DET
cte-761	245	12	results	result	NOUN
cte-761	245	13	.	.	PUNCT
cte-761	246	1	5	5	X
cte-761	246	2	.	.	X
cte-761	246	3	conclusion	conclusion	NOUN
cte-761	246	4	analysis	analysis	NOUN
cte-761	246	5	of	of	ADP
cte-761	246	6	our	our	PRON
cte-761	246	7	collected	collect	VERB
cte-761	246	8	data	datum	NOUN
cte-761	246	9	demonstrates	demonstrate	VERB
cte-761	246	10	that	that	SCONJ
cte-761	246	11	students	student	NOUN
cte-761	246	12	who	who	PRON
cte-761	246	13	simultaneously	simultaneously	ADV
cte-761	246	14	utilised	utilise	VERB
cte-761	246	15	geogebra	geogebra	NOUN
cte-761	246	16	and	and	CCONJ
cte-761	246	17	excel	excel	VERB
cte-761	246	18	at	at	ADP
cte-761	246	19	the	the	DET
cte-761	246	20	highest	high	ADJ
cte-761	246	21	learning	learning	NOUN
cte-761	246	22	difficulty	difficulty	NOUN
cte-761	246	23	level	level	NOUN
cte-761	246	24	with	with	ADP
cte-761	246	25	dynamic	dynamic	ADJ
cte-761	246	26	vector	vector	NOUN
cte-761	246	27	diagram	diagram	NOUN
cte-761	246	28	visualisation	visualisation	NOUN
cte-761	246	29	displayed	display	VERB
cte-761	246	30	enhanced	enhance	VERB
cte-761	246	31	understanding	understanding	NOUN
cte-761	246	32	of	of	ADP
cte-761	246	33	mathematical	mathematical	ADJ
cte-761	246	34	modelling	modelling	NOUN
cte-761	246	35	problems	problem	NOUN
cte-761	246	36	and	and	CCONJ
cte-761	246	37	improved	improve	VERB
cte-761	246	38	ability	ability	NOUN
cte-761	246	39	to	to	PART
cte-761	246	40	apply	apply	VERB
cte-761	246	41	mathematical	mathematical	ADJ
cte-761	246	42	modelling	modelling	NOUN
cte-761	246	43	in	in	ADP
cte-761	246	44	problem	problem	NOUN
cte-761	246	45	-	-	PUNCT
cte-761	246	46	solving	solving	NOUN
cte-761	246	47	.	.	PUNCT
cte-761	247	1	we	we	PRON
cte-761	247	2	found	find	VERB
cte-761	247	3	no	no	DET
cte-761	247	4	statistically	statistically	ADV
cte-761	247	5	significant	significant	ADJ
cte-761	247	6	difference	difference	NOUN
cte-761	247	7	in	in	ADP
cte-761	247	8	learning	learn	VERB
cte-761	247	9	outcomes	outcome	NOUN
cte-761	247	10	between	between	ADP
cte-761	247	11	student	student	NOUN
cte-761	247	12	groups	group	NOUN
cte-761	247	13	who	who	PRON
cte-761	247	14	exclusively	exclusively	ADV
cte-761	247	15	used	use	VERB
cte-761	247	16	either	either	CCONJ
cte-761	247	17	excel	excel	VERB
cte-761	247	18	or	or	CCONJ
cte-761	247	19	geogebra	geogebra	NOUN
cte-761	247	20	during	during	ADP
cte-761	247	21	mathematical	mathematical	ADJ
cte-761	247	22	modelling	modelling	NOUN
cte-761	247	23	instruction	instruction	NOUN
cte-761	247	24	.	.	PUNCT
cte-761	248	1	furthermore	furthermore	ADV
cte-761	248	2	,	,	PUNCT
cte-761	248	3	we	we	PRON
cte-761	248	4	observed	observe	VERB
cte-761	248	5	that	that	SCONJ
cte-761	248	6	students	student	NOUN
cte-761	248	7	who	who	PRON
cte-761	248	8	employed	employ	VERB
cte-761	248	9	excel	excel	VERB
cte-761	248	10	for	for	ADP
cte-761	248	11	computer	computer	NOUN
cte-761	248	12	modelling	modelling	NOUN
cte-761	248	13	responded	respond	VERB
cte-761	248	14	more	more	ADV
cte-761	248	15	effectively	effectively	ADV
cte-761	248	16	to	to	ADP
cte-761	248	17	quantitative	quantitative	ADJ
cte-761	248	18	examination	examination	NOUN
cte-761	248	19	questions	question	NOUN
cte-761	248	20	,	,	PUNCT
cte-761	248	21	while	while	SCONJ
cte-761	248	22	students	student	NOUN
cte-761	248	23	who	who	PRON
cte-761	248	24	exclusively	exclusively	ADV
cte-761	248	25	used	use	VERB
cte-761	248	26	geogebra	geogebra	NOUN
cte-761	248	27	demonstrated	demonstrate	VERB
cte-761	248	28	superior	superior	ADJ
cte-761	248	29	performance	performance	NOUN
cte-761	248	30	on	on	ADP
cte-761	248	31	qualitative	qualitative	ADJ
cte-761	248	32	questions	question	NOUN
cte-761	248	33	.	.	PUNCT
cte-761	249	1	our	our	PRON
cte-761	249	2	study	study	NOUN
cte-761	249	3	concludes	conclude	VERB
cte-761	249	4	that	that	SCONJ
cte-761	249	5	the	the	DET
cte-761	249	6	simultaneous	simultaneous	ADJ
cte-761	249	7	use	use	NOUN
cte-761	249	8	of	of	ADP
cte-761	249	9	excel	excel	NOUN
cte-761	249	10	and	and	CCONJ
cte-761	249	11	geogebra	geogebra	NOUN
cte-761	249	12	with	with	ADP
cte-761	249	13	dynamic	dynamic	ADJ
cte-761	249	14	vector	vector	NOUN
cte-761	249	15	diagram	diagram	NOUN
cte-761	249	16	visualisation	visualisation	NOUN
cte-761	249	17	enhanced	enhance	VERB
cte-761	249	18	students	student	NOUN
cte-761	249	19	’	'	PUNCT
cte-761	249	20	academic	academic	ADJ
cte-761	249	21	achievement	achievement	NOUN
cte-761	249	22	in	in	ADP
cte-761	249	23	mathematical	mathematical	ADJ
cte-761	249	24	modelling	modelling	NOUN
cte-761	249	25	.	.	PUNCT
cte-761	250	1	this	this	PRON
cte-761	250	2	is	be	AUX
cte-761	250	3	evidenced	evidence	VERB
cte-761	250	4	by	by	ADP
cte-761	250	5	the	the	DET
cte-761	250	6	statistically	statistically	ADV
cte-761	250	7	significant	significant	ADJ
cte-761	250	8	difference	difference	NOUN
cte-761	250	9	between	between	ADP
cte-761	250	10	average	average	ADJ
cte-761	250	11	academic	academic	ADJ
cte-761	250	12	achievement	achievement	NOUN
cte-761	250	13	results	result	NOUN
cte-761	250	14	shown	show	VERB
cte-761	250	15	in	in	ADP
cte-761	250	16	table	table	NOUN
cte-761	250	17	2	2	NUM
cte-761	250	18	and	and	CCONJ
cte-761	250	19	confirmed	confirm	VERB
cte-761	250	20	by	by	ADP
cte-761	250	21	the	the	DET
cte-761	250	22	results	result	NOUN
cte-761	250	23	in	in	ADP
cte-761	250	24	tables	table	NOUN
cte-761	250	25	3	3	NUM
cte-761	250	26	and	and	CCONJ
cte-761	250	27	4	4	NUM
cte-761	250	28	.	.	PUNCT
cte-761	251	1	this	this	DET
cte-761	251	2	research	research	NOUN
cte-761	251	3	demonstrates	demonstrate	VERB
cte-761	251	4	that	that	SCONJ
cte-761	251	5	appropriate	appropriate	ADJ
cte-761	251	6	software	software	NOUN
cte-761	251	7	utilisation	utilisation	NOUN
cte-761	251	8	for	for	ADP
cte-761	251	9	visualising	visualise	VERB
cte-761	251	10	dynamic	dynamic	ADJ
cte-761	251	11	characteristics	characteristic	NOUN
cte-761	251	12	of	of	ADP
cte-761	251	13	mathematical	mathematical	ADJ
cte-761	251	14	models	model	NOUN
cte-761	251	15	in	in	ADP
cte-761	251	16	teaching	teach	VERB
cte-761	251	17	mathematical	mathematical	ADJ
cte-761	251	18	modelling	modelling	NOUN
cte-761	251	19	at	at	ADP
cte-761	251	20	agricultural	agricultural	ADJ
cte-761	251	21	universities	university	NOUN
cte-761	251	22	effectively	effectively	ADV
cte-761	251	23	improves	improve	VERB
cte-761	251	24	student	student	NOUN
cte-761	251	25	performance	performance	NOUN
cte-761	251	26	.	.	PUNCT
cte-761	252	1	we	we	PRON
cte-761	252	2	confirm	confirm	VERB
cte-761	252	3	that	that	SCONJ
cte-761	252	4	visualising	visualise	VERB
cte-761	252	5	mechanical	mechanical	ADJ
cte-761	252	6	movement	movement	NOUN
cte-761	252	7	characteristics	characteristic	NOUN
cte-761	252	8	using	use	VERB
cte-761	252	9	dynamic	dynamic	ADJ
cte-761	252	10	vector	vector	NOUN
cte-761	252	11	diagrams	diagram	NOUN
cte-761	252	12	enhances	enhance	VERB
cte-761	252	13	student	student	NOUN
cte-761	252	14	educational	educational	ADJ
cte-761	252	15	achievements	achievement	NOUN
cte-761	252	16	.	.	PUNCT
cte-761	253	1	however	however	ADV
cte-761	253	2	,	,	PUNCT
cte-761	253	3	the	the	DET
cte-761	253	4	hypothesis	hypothesis	NOUN
cte-761	253	5	that	that	SCONJ
cte-761	253	6	combining	combine	VERB
cte-761	253	7	geogebra	geogebra	NOUN
cte-761	253	8	visualisation	visualisation	NOUN
cte-761	253	9	with	with	ADP
cte-761	253	10	a	a	DET
cte-761	253	11	differentiated	differentiated	ADJ
cte-761	253	12	learning	learning	NOUN
cte-761	253	13	approach	approach	NOUN
cte-761	253	14	creates	create	VERB
cte-761	253	15	additional	additional	ADJ
cte-761	253	16	conditions	condition	NOUN
cte-761	253	17	for	for	ADP
cte-761	253	18	improving	improve	VERB
cte-761	253	19	students	student	NOUN
cte-761	253	20	’	'	PUNCT
cte-761	253	21	knowledge	knowledge	NOUN
cte-761	253	22	,	,	PUNCT
cte-761	253	23	accounting	account	VERB
cte-761	253	24	for	for	ADP
cte-761	253	25	their	their	PRON
cte-761	253	26	vocational	vocational	ADJ
cte-761	253	27	training	training	NOUN
cte-761	253	28	specifics	specific	NOUN
cte-761	253	29	,	,	PUNCT
cte-761	253	30	requires	require	VERB
cte-761	253	31	further	further	ADJ
cte-761	253	32	verification	verification	NOUN
cte-761	253	33	.	.	PUNCT
cte-761	254	1	declaration	declaration	NOUN
cte-761	254	2	on	on	ADP
cte-761	254	3	generative	generative	ADJ
cte-761	254	4	ai	ai	PROPN
cte-761	254	5	:	:	PUNCT
cte-761	254	6	the	the	DET
cte-761	254	7	authors	author	NOUN
cte-761	254	8	have	have	AUX
cte-761	254	9	not	not	PART
cte-761	254	10	employed	employ	VERB
cte-761	254	11	any	any	DET
cte-761	254	12	generative	generative	ADJ
cte-761	254	13	ai	ai	NOUN
cte-761	254	14	tools	tool	NOUN
cte-761	254	15	.	.	PUNCT
cte-761	255	1	references	reference	NOUN
cte-761	255	2	[	[	X
cte-761	255	3	1	1	NUM
cte-761	255	4	]	]	X
cte-761	255	5	bobyliev	bobyliev	ADV
cte-761	255	6	,	,	PUNCT
cte-761	255	7	d.y	d.y	PROPN
cte-761	255	8	.	.	PROPN
cte-761	255	9	and	and	CCONJ
cte-761	255	10	vihrova	vihrova	PROPN
cte-761	255	11	,	,	PUNCT
cte-761	255	12	o.v	o.v	PROPN
cte-761	255	13	.	.	PROPN
cte-761	255	14	,	,	PUNCT
cte-761	255	15	2021	2021	NUM
cte-761	255	16	.	.	PUNCT
cte-761	256	1	problems	problem	NOUN
cte-761	256	2	and	and	CCONJ
cte-761	256	3	prospects	prospect	NOUN
cte-761	256	4	of	of	ADP
cte-761	256	5	distance	distance	NOUN
cte-761	256	6	learning	learn	VERB
cte-761	256	7	in	in	ADP
cte-761	256	8	teaching	teach	VERB
cte-761	256	9	fundamental	fundamental	ADJ
cte-761	256	10	disciplines	discipline	NOUN
cte-761	256	11	to	to	ADP
cte-761	256	12	future	future	ADJ
cte-761	256	13	mathematics	mathematic	NOUN
cte-761	256	14	teachers	teacher	NOUN
cte-761	256	15	.	.	PUNCT
cte-761	257	1	journal	journal	PROPN
cte-761	257	2	of	of	ADP
cte-761	257	3	physics	physics	PROPN
cte-761	257	4	:	:	PUNCT
cte-761	257	5	conference	conference	NOUN
cte-761	257	6	series	series	NOUN
cte-761	257	7	,	,	PUNCT
cte-761	257	8	1840	1840	NUM
cte-761	257	9	,	,	PUNCT
cte-761	257	10	p.012002	p.012002	NOUN
cte-761	257	11	.	.	PUNCT
cte-761	258	1	available	available	ADJ
cte-761	258	2	from	from	ADP
cte-761	258	3	:	:	PUNCT
cte-761	258	4	https	https	NOUN
cte-761	258	5	:	:	PUNCT
cte-761	258	6	//doi.org/10.1088/1742	//doi.org/10.1088/1742	NOUN
cte-761	258	7	-	-	PUNCT
cte-761	258	8	6596/1840/1/012002	6596/1840/1/012002	NOUN
cte-761	258	9	.	.	PUNCT
cte-761	259	1	251	251	NUM
cte-761	260	1	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	260	2	https://doi.org/10.1088/1742-6596/1840/1/012002	https://doi.org/10.1088/1742-6596/1840/1/012002	PROPN
cte-761	260	3	https://doi.org/10.1088/1742-6596/1840/1/012002	https://doi.org/10.1088/1742-6596/1840/1/012002	PROPN
cte-761	260	4	cte	cte	PROPN
cte-761	260	5	workshop	workshop	NOUN
cte-761	260	6	proceedings	proceeding	NOUN
cte-761	260	7	,	,	PUNCT
cte-761	260	8	2025	2025	NUM
cte-761	260	9	,	,	PUNCT
cte-761	260	10	vol	vol	NOUN
cte-761	260	11	.	.	PROPN
cte-761	260	12	12	12	NUM
cte-761	260	13	,	,	PUNCT
cte-761	260	14	pp	pp	ADJ
cte-761	260	15	.	.	PUNCT
cte-761	261	1	235–252	235–252	NUM
cte-761	262	1	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	263	1	[	[	X
cte-761	263	2	2	2	NUM
cte-761	263	3	]	]	SYM
cte-761	263	4	flehantov	flehantov	PROPN
cte-761	263	5	,	,	PUNCT
cte-761	263	6	l.	l.	PROPN
cte-761	263	7	and	and	CCONJ
cte-761	263	8	ovsiienko	ovsiienko	NOUN
cte-761	263	9	,	,	PUNCT
cte-761	263	10	y.	y.	NOUN
cte-761	263	11	,	,	PUNCT
cte-761	263	12	2019	2019	NUM
cte-761	263	13	.	.	PUNCT
cte-761	264	1	the	the	DET
cte-761	264	2	simultaneous	simultaneous	ADJ
cte-761	264	3	use	use	NOUN
cte-761	264	4	of	of	ADP
cte-761	264	5	excel	excel	NOUN
cte-761	264	6	and	and	CCONJ
cte-761	264	7	geogebra	geogebra	NOUN
cte-761	264	8	to	to	ADP
cte-761	264	9	training	train	VERB
cte-761	264	10	the	the	DET
cte-761	264	11	basics	basic	NOUN
cte-761	264	12	of	of	ADP
cte-761	264	13	mathematical	mathematical	ADJ
cte-761	264	14	modeling	modeling	NOUN
cte-761	264	15	.	.	PUNCT
cte-761	265	1	ceur	ceur	PROPN
cte-761	265	2	workshop	workshop	NOUN
cte-761	265	3	proceedings	proceeding	NOUN
cte-761	265	4	,	,	PUNCT
cte-761	265	5	2393	2393	NUM
cte-761	265	6	,	,	PUNCT
cte-761	265	7	pp.864–879	pp.864–879	PROPN
cte-761	265	8	.	.	PUNCT
cte-761	265	9	available	available	ADJ
cte-761	265	10	from	from	ADP
cte-761	265	11	:	:	PUNCT
cte-761	265	12	http://ceur-ws.org/vol-2393/	http://ceur-ws.org/vol-2393/	PROPN
cte-761	265	13	paper_258.pdf	paper_258.pdf	PROPN
cte-761	265	14	.	.	PUNCT
cte-761	266	1	[	[	X
cte-761	266	2	3	3	NUM
cte-761	266	3	]	]	X
cte-761	266	4	flehantov	flehantov	PROPN
cte-761	266	5	,	,	PUNCT
cte-761	266	6	l.o	l.o	PROPN
cte-761	266	7	.	.	PROPN
cte-761	266	8	and	and	CCONJ
cte-761	266	9	antonets	antonet	NOUN
cte-761	266	10	,	,	PUNCT
cte-761	266	11	a.v	a.v	PROPN
cte-761	266	12	.	.	PROPN
cte-761	266	13	,	,	PUNCT
cte-761	266	14	2017	2017	NUM
cte-761	266	15	.	.	PUNCT
cte-761	267	1	computer	computer	NOUN
cte-761	267	2	simulation	simulation	NOUN
cte-761	267	3	of	of	ADP
cte-761	267	4	the	the	DET
cte-761	267	5	body	body	NOUN
cte-761	267	6	movement	movement	NOUN
cte-761	267	7	,	,	PUNCT
cte-761	267	8	which	which	PRON
cte-761	267	9	was	be	AUX
cte-761	267	10	thrown	throw	VERB
cte-761	267	11	at	at	ADP
cte-761	267	12	an	an	DET
cte-761	267	13	angle	angle	NOUN
cte-761	267	14	to	to	ADP
cte-761	267	15	the	the	DET
cte-761	267	16	horizon	horizon	NOUN
cte-761	267	17	,	,	PUNCT
cte-761	267	18	taking	take	VERB
cte-761	267	19	into	into	ADP
cte-761	267	20	account	account	NOUN
cte-761	267	21	the	the	DET
cte-761	267	22	magnus	magnus	PROPN
cte-761	267	23	effect	effect	NOUN
cte-761	267	24	.	.	PUNCT
cte-761	268	1	information	information	NOUN
cte-761	268	2	technologies	technology	NOUN
cte-761	268	3	in	in	ADP
cte-761	268	4	education	education	NOUN
cte-761	268	5	,	,	PUNCT
cte-761	268	6	30	30	NUM
cte-761	268	7	,	,	PUNCT
cte-761	268	8	pp.97–109	pp.97–109	VERB
cte-761	268	9	.	.	PUNCT
cte-761	269	1	available	available	ADJ
cte-761	269	2	from	from	ADP
cte-761	269	3	:	:	PUNCT
cte-761	269	4	https://doi.org/10.14308/ite000624	https://doi.org/10.14308/ite000624	X
cte-761	269	5	.	.	PUNCT
cte-761	270	1	[	[	X
cte-761	270	2	4	4	NUM
cte-761	270	3	]	]	X
cte-761	270	4	flehantov	flehantov	PROPN
cte-761	270	5	,	,	PUNCT
cte-761	270	6	l.o	l.o	PROPN
cte-761	270	7	.	.	PROPN
cte-761	270	8	and	and	CCONJ
cte-761	270	9	ovsiienko	ovsiienko	PROPN
cte-761	270	10	,	,	PUNCT
cte-761	270	11	y.i	y.i	PROPN
cte-761	270	12	.	.	PROPN
cte-761	270	13	,	,	PUNCT
cte-761	270	14	2016	2016	NUM
cte-761	270	15	.	.	PUNCT
cte-761	270	16	differentiated	differentiated	ADJ
cte-761	270	17	approach	approach	NOUN
cte-761	270	18	in	in	ADP
cte-761	270	19	teaching	teach	VERB
cte-761	270	20	the	the	DET
cte-761	270	21	basics	basic	NOUN
cte-761	270	22	of	of	ADP
cte-761	270	23	mathematical	mathematical	ADJ
cte-761	270	24	modeling	modeling	NOUN
cte-761	270	25	to	to	ADP
cte-761	270	26	agricultural	agricultural	ADJ
cte-761	270	27	engineers	engineer	NOUN
cte-761	270	28	.	.	PUNCT
cte-761	271	1	information	information	NOUN
cte-761	271	2	technologies	technology	NOUN
cte-761	271	3	in	in	ADP
cte-761	271	4	education	education	NOUN
cte-761	271	5	,	,	PUNCT
cte-761	271	6	29	29	NUM
cte-761	271	7	,	,	PUNCT
cte-761	271	8	pp.23–36	pp.23–36	NOUN
cte-761	271	9	.	.	PUNCT
cte-761	272	1	available	available	ADJ
cte-761	272	2	from	from	ADP
cte-761	272	3	:	:	PUNCT
cte-761	272	4	https://doi.org/10	https://doi.org/10	PROPN
cte-761	272	5	.	.	PROPN
cte-761	272	6	14308	14308	NUM
cte-761	272	7	/	/	SYM
cte-761	272	8	ite000610	ite000610	NOUN
cte-761	272	9	.	.	PUNCT
cte-761	273	1	[	[	X
cte-761	273	2	5	5	NUM
cte-761	273	3	]	]	SYM
cte-761	273	4	flehantov	flehantov	PROPN
cte-761	273	5	,	,	PUNCT
cte-761	273	6	l.o	l.o	PROPN
cte-761	273	7	.	.	PROPN
cte-761	273	8	and	and	CCONJ
cte-761	273	9	volchkova	volchkova	PROPN
cte-761	273	10	,	,	PUNCT
cte-761	273	11	m.i	m.i	PROPN
cte-761	273	12	.	.	PROPN
cte-761	273	13	,	,	PUNCT
cte-761	273	14	2012	2012	NUM
cte-761	273	15	.	.	PUNCT
cte-761	274	1	evaluation	evaluation	NOUN
cte-761	274	2	of	of	ADP
cte-761	274	3	homogeneity	homogeneity	NOUN
cte-761	274	4	of	of	ADP
cte-761	274	5	academic	academic	ADJ
cte-761	274	6	groups	group	NOUN
cte-761	274	7	by	by	ADP
cte-761	274	8	the	the	DET
cte-761	274	9	criterion	criterion	NOUN
cte-761	274	10	of	of	ADP
cte-761	274	11	academic	academic	ADJ
cte-761	274	12	performance	performance	NOUN
cte-761	274	13	.	.	PUNCT
cte-761	275	1	information	information	NOUN
cte-761	275	2	technologies	technology	NOUN
cte-761	275	3	in	in	ADP
cte-761	275	4	education	education	NOUN
cte-761	275	5	,	,	PUNCT
cte-761	275	6	11	11	NUM
cte-761	275	7	,	,	PUNCT
cte-761	275	8	pp.146–152	pp.146–152	PROPN
cte-761	275	9	.	.	PUNCT
cte-761	275	10	available	available	ADJ
cte-761	275	11	from	from	ADP
cte-761	275	12	:	:	PUNCT
cte-761	275	13	https://doi.org/10.14308/ite000284	https://doi.org/10.14308/ite000284	NOUN
cte-761	275	14	.	.	PUNCT
cte-761	276	1	[	[	X
cte-761	276	2	6	6	NUM
cte-761	276	3	]	]	PUNCT
cte-761	276	4	geogebra	geogebra	NOUN
cte-761	276	5	wiki	wiki	NOUN
cte-761	276	6	,	,	PUNCT
cte-761	276	7	2025	2025	NUM
cte-761	276	8	.	.	PUNCT
cte-761	276	9	nsolveode	nsolveode	NOUN
cte-761	276	10	command	command	NOUN
cte-761	276	11	.	.	PUNCT
cte-761	277	1	available	available	ADJ
cte-761	277	2	from	from	ADP
cte-761	277	3	:	:	PUNCT
cte-761	277	4	https://wiki	https://wiki	PROPN
cte-761	277	5	.	.	PROPN
cte-761	277	6	geogebra.org/en/nsolveode_command	geogebra.org/en/nsolveode_command	NOUN
cte-761	277	7	.	.	PUNCT
cte-761	278	1	[	[	X
cte-761	278	2	7	7	NUM
cte-761	278	3	]	]	SYM
cte-761	278	4	hall	hall	NOUN
cte-761	278	5	,	,	PUNCT
cte-761	278	6	j.	j.	PROPN
cte-761	278	7	and	and	CCONJ
cte-761	278	8	lingefjärd	lingefjärd	PROPN
cte-761	278	9	,	,	PUNCT
cte-761	278	10	t.	t.	PROPN
cte-761	278	11	,	,	PUNCT
cte-761	278	12	2016	2016	NUM
cte-761	278	13	.	.	PUNCT
cte-761	279	1	mathematical	mathematical	ADJ
cte-761	279	2	modeling	modeling	NOUN
cte-761	279	3	:	:	PUNCT
cte-761	279	4	applications	application	NOUN
cte-761	279	5	with	with	ADP
cte-761	279	6	geogebra	geogebra	NOUN
cte-761	279	7	.	.	PUNCT
cte-761	280	1	hoboken	hoboken	PROPN
cte-761	280	2	:	:	PUNCT
cte-761	280	3	john	john	PROPN
cte-761	280	4	wiley	wiley	PROPN
cte-761	280	5	&	&	CCONJ
cte-761	280	6	sons	son	NOUN
cte-761	280	7	.	.	PUNCT
cte-761	281	1	[	[	X
cte-761	281	2	8	8	NUM
cte-761	281	3	]	]	X
cte-761	281	4	horda	horda	PROPN
cte-761	281	5	,	,	PUNCT
cte-761	281	6	i.m	i.m	PROPN
cte-761	281	7	.	.	PROPN
cte-761	281	8	and	and	CCONJ
cte-761	281	9	flehantov	flehantov	PROPN
cte-761	281	10	,	,	PUNCT
cte-761	281	11	l.o	l.o	PROPN
cte-761	281	12	.	.	PROPN
cte-761	281	13	,	,	PUNCT
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cte-761	281	15	.	.	PUNCT
cte-761	282	1	teaching	teach	VERB
cte-761	282	2	the	the	DET
cte-761	282	3	basics	basic	NOUN
cte-761	282	4	of	of	ADP
cte-761	282	5	mathematical	mathematical	ADJ
cte-761	282	6	modeling	modeling	NOUN
cte-761	282	7	in	in	ADP
cte-761	282	8	agricultural	agricultural	ADJ
cte-761	282	9	universities	university	NOUN
cte-761	282	10	using	use	VERB
cte-761	282	11	ms	ms	PROPN
cte-761	282	12	excel	excel	NOUN
cte-761	282	13	spreadsheets	spreadsheet	NOUN
cte-761	282	14	.	.	PUNCT
cte-761	283	1	information	information	NOUN
cte-761	283	2	technologies	technology	NOUN
cte-761	283	3	in	in	ADP
cte-761	283	4	education	education	NOUN
cte-761	283	5	,	,	PUNCT
cte-761	283	6	22	22	NUM
cte-761	283	7	,	,	PUNCT
cte-761	283	8	pp.167–177	pp.167–177	NOUN
cte-761	283	9	.	.	PUNCT
cte-761	284	1	[	[	X
cte-761	284	2	9	9	NUM
cte-761	284	3	]	]	X
cte-761	284	4	international	international	ADJ
cte-761	284	5	geogebra	geogebra	PROPN
cte-761	284	6	institute	institute	NOUN
cte-761	284	7	,	,	PUNCT
cte-761	284	8	2025	2025	NUM
cte-761	284	9	.	.	PUNCT
cte-761	284	10	geogebra	geogebra	NOUN
cte-761	284	11	math	math	NOUN
cte-761	284	12	calculator	calculator	NOUN
cte-761	284	13	.	.	PUNCT
cte-761	285	1	available	available	ADJ
cte-761	285	2	from	from	ADP
cte-761	285	3	:	:	PUNCT
cte-761	285	4	https://www.geogebra.org	https://www.geogebra.org	X
cte-761	285	5	.	.	PUNCT
cte-761	286	1	[	[	X
cte-761	286	2	10	10	NUM
cte-761	286	3	]	]	X
cte-761	286	4	kalitkin	kalitkin	X
cte-761	286	5	,	,	PUNCT
cte-761	286	6	n.n	n.n	PROPN
cte-761	286	7	.	.	PROPN
cte-761	286	8	and	and	CCONJ
cte-761	286	9	koryakin	koryakin	PROPN
cte-761	286	10	,	,	PUNCT
cte-761	286	11	p.v	p.v	PROPN
cte-761	286	12	.	.	PROPN
cte-761	286	13	,	,	PUNCT
cte-761	286	14	2013	2013	NUM
cte-761	286	15	.	.	PUNCT
cte-761	286	16	numerical	numerical	ADJ
cte-761	286	17	methods	method	NOUN
cte-761	286	18	:	:	PUNCT
cte-761	286	19	in	in	ADP
cte-761	286	20	2	2	NUM
cte-761	286	21	books	book	NOUN
cte-761	286	22	.	.	PUNCT
cte-761	287	1	book	book	NOUN
cte-761	287	2	2	2	NUM
cte-761	287	3	.	.	PUNCT
cte-761	287	4	methods	method	NOUN
cte-761	287	5	of	of	ADP
cte-761	287	6	mathematical	mathematical	ADJ
cte-761	287	7	physics	physics	NOUN
cte-761	287	8	.	.	PUNCT
cte-761	288	1	moscow	moscow	PROPN
cte-761	288	2	:	:	PUNCT
cte-761	288	3	academy	academy	NOUN
cte-761	288	4	.	.	PUNCT
cte-761	289	1	[	[	X
cte-761	289	2	11	11	NUM
cte-761	289	3	]	]	X
cte-761	289	4	klochko	klochko	NOUN
cte-761	289	5	,	,	PUNCT
cte-761	289	6	v.i	v.i	PROPN
cte-761	289	7	.	.	PROPN
cte-761	289	8	and	and	CCONJ
cte-761	289	9	bondarenko	bondarenko	PROPN
cte-761	289	10	,	,	PUNCT
cte-761	289	11	z.v	z.v	PROPN
cte-761	289	12	.	.	PROPN
cte-761	289	13	,	,	PUNCT
cte-761	289	14	2013	2013	NUM
cte-761	289	15	.	.	PUNCT
cte-761	290	1	some	some	DET
cte-761	290	2	aspects	aspect	NOUN
cte-761	290	3	of	of	ADP
cte-761	290	4	the	the	DET
cte-761	290	5	use	use	NOUN
cte-761	290	6	of	of	ADP
cte-761	290	7	information	information	NOUN
cte-761	290	8	and	and	CCONJ
cte-761	290	9	communication	communication	NOUN
cte-761	290	10	technologies	technology	NOUN
cte-761	290	11	in	in	ADP
cte-761	290	12	teaching	teach	VERB
cte-761	290	13	differential	differential	ADJ
cte-761	290	14	equations	equation	NOUN
cte-761	290	15	.	.	PUNCT
cte-761	291	1	scientific	scientific	ADJ
cte-761	291	2	notes	note	NOUN
cte-761	291	3	,	,	PUNCT
cte-761	291	4	5	5	NUM
cte-761	291	5	,	,	PUNCT
cte-761	291	6	pp.64–70	pp.64–70	NOUN
cte-761	291	7	.	.	PUNCT
cte-761	292	1	[	[	X
cte-761	292	2	12	12	NUM
cte-761	292	3	]	]	X
cte-761	292	4	kramarenko	kramarenko	PROPN
cte-761	292	5	,	,	PUNCT
cte-761	292	6	t.h	t.h	PROPN
cte-761	292	7	.	.	PROPN
cte-761	292	8	,	,	PUNCT
cte-761	292	9	pylypenko	pylypenko	PROPN
cte-761	292	10	,	,	PUNCT
cte-761	292	11	o.s	o.s	PROPN
cte-761	292	12	.	.	PROPN
cte-761	292	13	and	and	CCONJ
cte-761	292	14	zaselskiy	zaselskiy	PROPN
cte-761	292	15	,	,	PUNCT
cte-761	292	16	v.i	v.i	PROPN
cte-761	292	17	.	.	PROPN
cte-761	292	18	,	,	PUNCT
cte-761	292	19	2020	2020	NUM
cte-761	292	20	.	.	PUNCT
cte-761	293	1	prospects	prospect	NOUN
cte-761	293	2	of	of	ADP
cte-761	293	3	using	use	VERB
cte-761	293	4	the	the	DET
cte-761	293	5	augmented	augment	VERB
cte-761	293	6	reality	reality	NOUN
cte-761	293	7	application	application	NOUN
cte-761	293	8	in	in	ADP
cte-761	293	9	stem	stem	NOUN
cte-761	293	10	-	-	PUNCT
cte-761	293	11	based	base	VERB
cte-761	293	12	mathematics	mathematic	NOUN
cte-761	293	13	teaching	teaching	NOUN
cte-761	293	14	.	.	PUNCT
cte-761	294	1	ceur	ceur	PROPN
cte-761	294	2	workshop	workshop	NOUN
cte-761	294	3	proceedings	proceeding	NOUN
cte-761	294	4	,	,	PUNCT
cte-761	294	5	2547	2547	NUM
cte-761	294	6	,	,	PUNCT
cte-761	294	7	pp.130–144	pp.130–144	PROPN
cte-761	294	8	.	.	PUNCT
cte-761	294	9	available	available	ADJ
cte-761	294	10	from	from	ADP
cte-761	294	11	:	:	PUNCT
cte-761	294	12	http://ceur-ws.org/	http://ceur-ws.org/	DET
cte-761	294	13	vol-2547	vol-2547	NOUN
cte-761	294	14	/	/	SYM
cte-761	294	15	paper10.pdf	paper10.pdf	PROPN
cte-761	294	16	.	.	PUNCT
cte-761	295	1	[	[	X
cte-761	295	2	13	13	NUM
cte-761	295	3	]	]	X
cte-761	295	4	what	what	PRON
cte-761	295	5	was	be	AUX
cte-761	295	6	that	that	DET
cte-761	295	7	chinese	chinese	ADJ
cte-761	295	8	proverb	proverb	NOUN
cte-761	295	9	that	that	PRON
cte-761	295	10	goes	go	VERB
cte-761	295	11	something	something	PRON
cte-761	295	12	like	like	ADP
cte-761	295	13	“	"	PUNCT
cte-761	295	14	i	i	PRON
cte-761	295	15	hear	hear	VERB
cte-761	295	16	and	and	CCONJ
cte-761	295	17	i	i	PRON
cte-761	295	18	forget	forget	VERB
cte-761	295	19	,	,	PUNCT
cte-761	295	20	i	i	PRON
cte-761	295	21	see	see	VERB
cte-761	296	1	and	and	CCONJ
cte-761	296	2	i	i	PRON
cte-761	296	3	remember	remember	VERB
cte-761	296	4	,	,	PUNCT
cte-761	296	5	i	i	PRON
cte-761	296	6	do	do	VERB
cte-761	297	1	and	and	CCONJ
cte-761	297	2	i	i	PRON
cte-761	297	3	understand	understand	VERB
cte-761	297	4	”	"	PUNCT
cte-761	297	5	?	?	PUNCT
cte-761	297	6	,	,	PUNCT
cte-761	297	7	2025	2025	NUM
cte-761	297	8	.	.	PUNCT
cte-761	298	1	available	available	ADJ
cte-761	298	2	from	from	ADP
cte-761	298	3	:	:	PUNCT
cte-761	298	4	https://english.stackexchange.com/questions/226886/	https://english.stackexchange.com/questions/226886/	PROPN
cte-761	298	5	what	what	PRON
cte-761	298	6	-	-	PUNCT
cte-761	298	7	was	be	AUX
cte-761	298	8	-	-	PUNCT
cte-761	298	9	that	that	PRON
cte-761	298	10	-	-	PUNCT
cte-761	298	11	chinese	chinese	ADJ
cte-761	298	12	-	-	PUNCT
cte-761	298	13	proverb	proverb	NOUN
cte-761	298	14	-	-	PUNCT
cte-761	298	15	that	that	PRON
cte-761	298	16	-	-	PUNCT
cte-761	298	17	goes	go	VERB
cte-761	298	18	-	-	PUNCT
cte-761	298	19	something	something	PRON
cte-761	298	20	-	-	PUNCT
cte-761	298	21	like	like	ADP
cte-761	298	22	-	-	PUNCT
cte-761	298	23	i	i	PRON
cte-761	298	24	-	-	PUNCT
cte-761	298	25	hear	hear	NOUN
cte-761	298	26	-	-	PUNCT
cte-761	298	27	and	and	CCONJ
cte-761	298	28	-	-	PUNCT
cte-761	298	29	i	i	PRON
cte-761	298	30	-	-	PUNCT
cte-761	298	31	forget	forget	VERB
cte-761	298	32	-	-	PUNCT
cte-761	298	33	i	i	PRON
cte-761	298	34	-	-	PUNCT
cte-761	298	35	s.	s.	PROPN
cte-761	299	1	252	252	NUM
cte-761	299	2	https://doi.org/10.55056/cte.761	https://doi.org/10.55056/cte.761	PROPN
cte-761	299	3	http://ceur-ws.org/vol-2393/paper_258.pdf	http://ceur-ws.org/vol-2393/paper_258.pdf	ADJ
cte-761	299	4	http://ceur-ws.org/vol-2393/paper_258.pdf	http://ceur-ws.org/vol-2393/paper_258.pdf	PROPN
cte-761	299	5	https://doi.org/10.14308/ite000624	https://doi.org/10.14308/ite000624	INTJ
cte-761	299	6	https://doi.org/10.14308/ite000610	https://doi.org/10.14308/ite000610	NOUN
cte-761	299	7	https://doi.org/10.14308/ite000610	https://doi.org/10.14308/ite000610	INTJ
cte-761	299	8	https://doi.org/10.14308/ite000284	https://doi.org/10.14308/ite000284	PROPN
cte-761	299	9	https://wiki.geogebra.org/en/nsolveode_command	https://wiki.geogebra.org/en/nsolveode_command	NUM
cte-761	299	10	https://wiki.geogebra.org/en/nsolveode_command	https://wiki.geogebra.org/en/nsolveode_command	NUM
cte-761	299	11	https://www.geogebra.org	https://www.geogebra.org	X
cte-761	299	12	http://ceur-ws.org/vol-2547/paper10.pdf	http://ceur-ws.org/vol-2547/paper10.pdf	PROPN
cte-761	299	13	http://ceur-ws.org/vol-2547/paper10.pdf	http://ceur-ws.org/vol-2547/paper10.pdf	NOUN
cte-761	299	14	https://english.stackexchange.com/questions/226886/what-was-that-chinese-proverb-that-goes-something-like-i-hear-and-i-forget-i-s	https://english.stackexchange.com/questions/226886/what-was-that-chinese-proverb-that-goes-something-like-i-hear-and-i-forget-i-s	X
cte-761	299	15	https://english.stackexchange.com/questions/226886/what-was-that-chinese-proverb-that-goes-something-like-i-hear-and-i-forget-i-s	https://english.stackexchange.com/questions/226886/what-was-that-chinese-proverb-that-goes-something-like-i-hear-and-i-forget-i-s	X
cte-761	299	16	1	1	NUM
cte-761	299	17	introduction	introduction	NOUN
cte-761	299	18	2	2	NUM
cte-761	299	19	experiment	experiment	NOUN
cte-761	299	20	description	description	NOUN
cte-761	299	21	2.1	2.1	NUM
cte-761	299	22	general	general	ADJ
cte-761	299	23	design	design	VERB
cte-761	299	24	2.2	2.2	NUM
cte-761	299	25	mathematical	mathematical	ADJ
cte-761	299	26	model	model	NOUN
cte-761	299	27	development	development	NOUN
cte-761	299	28	2.3	2.3	NUM
cte-761	299	29	implementation	implementation	NOUN
cte-761	299	30	in	in	ADP
cte-761	299	31	excel	excel	PROPN
cte-761	299	32	(	(	PUNCT
cte-761	299	33	group	group	NOUN
cte-761	299	34	e	e	NOUN
cte-761	299	35	)	)	PUNCT
cte-761	299	36	2.4	2.4	NUM
cte-761	299	37	implementation	implementation	NOUN
cte-761	299	38	in	in	ADP
cte-761	299	39	geogebra	geogebra	NOUN
cte-761	299	40	(	(	PUNCT
cte-761	299	41	group	group	NOUN
cte-761	299	42	g	g	NOUN
cte-761	299	43	)	)	PUNCT
cte-761	299	44	2.5	2.5	NUM
cte-761	299	45	combined	combine	VERB
cte-761	299	46	implementation	implementation	NOUN
cte-761	299	47	with	with	ADP
cte-761	299	48	excel	excel	NOUN
cte-761	299	49	and	and	CCONJ
cte-761	299	50	geogebra	geogebra	NOUN
cte-761	299	51	(	(	PUNCT
cte-761	299	52	group	group	NOUN
cte-761	299	53	eg	eg	NOUN
cte-761	299	54	)	)	PUNCT
cte-761	299	55	3	3	NUM
cte-761	299	56	results	result	VERB
cte-761	299	57	4	4	NUM
cte-761	299	58	discussion	discussion	NOUN
cte-761	299	59	5	5	NUM
cte-761	299	60	conclusion	conclusion	NOUN