International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

Determining the Coefficient of Restitution

Through the “Bouncing Ball” Experiment using

Phyphox

Jesi Pebralia1, *

1Department of Physics, Universitas Jambi, Jambi, Indonesia

*Corresponding Author: jesipebralia@unja.ac.id

(Received 13-04-2022; Revised 26-04-2022; Accepted 26-04-2022)

Abstract

This study aims to determine the restitution coefficient based on the reflected

sound from the “bouncing ball” experiment. The experiment used a Phyphox-

based smartphone. The produced sound came from a reflection between

marble and the floor. Theoretically, the value of the coefficient of restitution

is obtained based on the square root of the final height of the object’s

reflection divided by its initial height. In this study, the determination of the

height of the bounce from the “bouncing ball” was measured using the

Phyphox application, which was analyzed based on the sound of the bouncing

ball and the time interval of the reflection. The results show that the value of

the coefficient of restitution for each marble were 0.93, 0.92, and 0.92, while

the average error were 0.65%, 0.85%, and 1.43%, respectively. Furthermore,

the average error value of the overall measurement is 0.97%. This error is

highly dependent on the shape of the object. The rounder a thing is, the higher

the level of accuracy will be. In this study, the determination of the coefficient

of restitution was carried out in two ways: by comparing the height of the

ball’s bounce and the time intervals for the n and n+1 bounce. The value of

the coefficient of restitution generated by these methods was identic. Thus,

This work is licensed under a Creative Commons Attribution 4.0 International License

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

this study had confirmed that the bounce ball experiment using the Phyphox

indicated valid data well so that it could be implemented for determining the

coefficient of restitution.

Keywords: bouncing ball, coecficient of restitution, Phyphox, smartphone

1 Introduction

The coefficient of restitution is a value that states the level of elasticity of objects in

the collision phenomenon. Particularly, the coefficient of restitution is a characterization

of the degrees of freedom in the inelastic collision and dimensionless [1]. The value of

the coefficient of restitution depends on the ratio of the final height and initial height of

the collision particles, which is mathematically expressed by the Equation (1):

𝑒 = √
ℎ2

ℎ1
. (1)

Determining the value of the coefficient of restitution is very useful for developing

various sub-fields of physics. The coefficient of restitution has become an essential part

of granular hydrodynamics and the kinetic theory of gas [2], [3], computation of granular

matter [4], and even in agriculture, especially for the development of agricultural

techniques [5]. The coefficient of restitution provides information on the energy lost

during the collision process [6]. It could be necessary for dry granular modelling and

multi-phase flow models.

Research in determining the value of the coefficient of restitution has been carried out

using different techniques. These are determining the coefficient of restitution using a

robot and piezoelectric sensor [7], determining the coefficient of restitution using a high-

speed camera [6], [8], determining the coefficient of restitution using the double

pendulum method [9], determining the coefficient of restitution using high-speed video

[10], and others [11]–[14].

One of the experiments that can be used to determine the value of the coefficient of

restitution is the “bouncing ball” experiment [15]. A bouncing ball is a bounce event from

a ball dropped without initial velocity from a certain height above the earth’s surface and

hits a particular surface. In the bouncing ball phenomenon, an inelastic collision occurs

where the ball will bounce up and until the ball stops at a specific time. The process of

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

the bouncing ball illustrates many aspects that could be observed from the principles of

mechanics, including the phenomenon of collisions when the ball hits the floor surface

[16].

According to the studies that have been carried out, the technique for determining the

value of the coefficient of restitution is expensive, complex, and challenging to carry out

independently by students. In this study, a cheap and practical technique for determining

the coefficient of restitution will be introduced using a smartphone. Generally, using

advanced technology in this era, smartphones have been equipped with sophisticated

sensors that can support the implementation of science practicums, especially physics. In

Addition, this is also supported by the existence of practical support applications that can

be downloaded and run freely on smartphones. One application that can be used for

physics experiments is the Phyphox.

Several studies using the Phyphox application include research on determining spring

constants on spring oscillation events [17], free-fall motion experiments using the

stopwatch acoustic feature [18], pendulum motion experiments [19], and others [20], [21].

The value of the coefficient of restitution can be determined by using the Phyphox

application, by finding the ratio of the object’s speed between two adjacent bounce [22].

In this study, we provide a method to determine the value of the coefficient of restitution

of bouncing ball by using Phyphox based on two approaches. The first approach is

through the ratio of the height of marbles in two adjacent bounce. While the second

approach is through the ratio of time intervals between two adjacent bounce.

2 Research Methodology

In this study, the value of the coefficient of restitution between the marble and the floor

would be calculated through the bouncing ball experiment. Based on equation (1), the

value of the restitution coefficient could be determined if the initial height and final height

of the following bounce process were known. The object used in this study were three

marbles with different diameters. The purpose is to evaluate the effect of the size of

marble.

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

The experiment design is illustrated in Figure 1. The first process was setting the initial

height of the marbles using a ruler. It was 15 cm from the floor. The smartphone was

placed on the floor in a position close to the bounce of the marbles. To find the value of

the bounce marbles’ height, a smartphone was installed with the Phyphox. After that, the

ball was dropped without initial velocity and allowed to bounce. The sound produced by

the marble’s bounce would be detected and recorded by the smartphone sensor. Then it

would be processed and converted to generate data on interval time, height, and energy

of the bounce. Furthermore, the data would be displayed on the smartphone’s LCD. The

number of bounces produced in this experiment was five times.

Figure 1. Experiment design in determining the coefficient of restitution

In the next stage, the accuracy of the obtained data needs to be declared because it is

related to the error value of a measurement. The smaller the measurement error value, the

greater the level of research accuracy. So, it might be stated that the data experiment was

valid. The measurement error value is calculated through the Equation (2),

𝑒𝑟𝑟𝑜𝑟 = |
𝑚𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑−𝑚𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡

𝑚𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑
| × 100%, (2)

where 𝑚𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 represents the actual value measured through standard measuring

instruments and 𝑚𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 represents the value displayed by the smartphone.

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

The error value was attained from the initial height measurement read by the

smartphone compared to the actual height. In this study, the arisen error should be under

2%. Therefore, if the error value is less than 2%, the data collection process could be

continued, while others would repeat the experiment process.

Figure 2. Flowchart of Detecting the coefficient of restitution by Using Phyphox

After determining the error value, the data generated by the smartphone would be

analyzed to determine the coefficient of restitution and standard deviations. The

calculation of the standard deviation value follows the Equation (3),

𝐷𝑠 =
1

𝑛
√

𝑛 ∑𝑛𝑖=1 𝑥𝑖
2−(∑𝑛𝑖=1 𝑥𝑖)

(𝑛−1
, (3)

where n represents the number of measurements and 𝑥𝑖 represents the measurement

results in the 𝑖𝑡ℎ experiment.

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

3 Results and Discussion

In this study, three types of marbles with different diameters were used in the

experiment. They were labelled with marble 1, having a diameter of 14.1 mm, marble 2,

having a diameter of 15.2 mm, and marble 3, having a diameter of 27.4 mm. To obtain

the valid data, the experiment was done and repeated five times. The experiment results

are shown in Figure 3.

(a)

(b) (c)

Figure 3. The experiment results of the coefficient of restitution for (a) marble 1, (b)

marble 2, and (c) marble 3

Figure 3 shows the bounce ball experiment results using marble with various diameters.

The initial height of the marble is set at 15 cm from the floor. In this experiment, the

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

coefficient of restitution for marble 1 was 0.90 to 0.95, while for marble 2, the coefficient

of restitution was 0.90 to 0.94, as well as the coefficient of restitution for marble 3 was in

the range of 0.84 to 0.95. The dash line in each figure indicate the average value of the

coefficient of restitution. The average value of the three marbles respectively are 0.9282,

0.9247, and 0.9237. Based on equation (2), average error value of the three marbles

respectively are 0.65%, 0.85%, and 1.43%, respectively. Moreover, the average

coefficient of restitution for collisions in this experiment was displayed in Table 1.

Analytical calculation

The coefficient of restitution is a significant empirical parameter in any physical

modelling where there is energy loss caused by particle collisions [23]. One of the

essential factors that influence the factor determining the value of the restitution

coefficient is the velocity value immediately after the 𝑛𝑡ℎ reflection [24],

𝑣𝑛 = 𝑣0𝑒
𝑛, (4)

where 𝑣0 is the velocity of the ball just before the collision. The time interval between

adjacent collisions (𝑛𝑡ℎ to (𝑛 + 1)𝑡ℎ) is expressed by the Equation (5),

𝑇𝑛 =
2𝑣𝑛

𝑔

𝑇𝑛 =
2𝑣0𝑒

𝑛

𝑔

𝑇𝑛 = 𝑇0𝑒
𝑛. (5)

where g is the gravitational acceleration and 𝑇0 = 2𝑣0/𝑔. Then, from the equation 5, it

could be obtained that the coefficient of restitution could also be determined through the

time interval of the bouncing ball,

𝑒 𝑛 =
𝑇𝑛
𝑇0

. (6)

Time interval (𝑇𝑛) of bouncing marble was show in Figure 4.

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

Figure 4. The time interval vs n-bounce for each marble

From Figure 4, it is obtained that for marble 1, when 𝑛 = 1, then the value of ℎ =

0,1497 𝑚, dan 𝑇𝑛 = 0,322𝑠. So, substituting those data to 𝑇0, it could be determined the

coefficient of restitution,

𝑇0 = (2𝑣0/𝑔)

𝑇0 = (2√2𝑔ℎ/𝑔)

𝑇0 = (√
8ℎ

𝑔
)

𝑇0 = (√
8(0,1497𝑚)

9,8 𝑚/𝑠2
)

𝑇0 = 0,3496𝑠, (7)

Thus, for 𝑛 = 1 the coefficient of restitution of marble 1 is

𝑒1 =
0,322𝑠

0,3496 𝑠

𝑒1 ≈ 0,92. (8)

The coefficient of restitution of marble 2 dan marble 3 could be found by applying the

same method. Those are 𝑒2 ≈ 0,93 and 𝑒3 ≈ 0,92.

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

The last step in this study was comparing values of the coefficient of restitution. The

purpose was to see the validity of the data between the experimental and analytical

methods. The data is shown in Table 1.

Table 1. Comparison of the coefficient of restitution between analytical and

experimental method

Marble Coefficient of Restitution

Experimental Analytical

d1 = 14,1 mm 𝑒 ≈ 0,93 𝑒 ≈ 0,92
d2 = 15,2 mm 𝑒 ≈ 0,92 𝑒 ≈ 0,93
d3 = 27,4 mm 𝑒 ≈ 0,92 𝑒 ≈ 0,92

Table 1 shows that the coefficient of restitution between the experiment and analytical

approach is not the same but very identic. Many factors could cause this. The

measurement of the restitution coefficient value is highly dependent on the shape and

material of the object, the level of surface roughness of the reflection, the level of

sphericity of the thing, and the measurement error. However, the small measurement error

may cause a shift in energy to the translational or rotational components [23].

Furthermore, since there was no change in the coefficient of restitution for three kinds of

marble, it could be stated that there is no effect from the diameter of the marble to

determine the coefficient of restitution. Thus, this study confirmed that bounce ball

experiment using the Phyphox indicates valid data well so that it could be implemented

to determine the coefficient of restitution.

4 Conclusion

This research has succeeded in determining the coefficient of restitution. The method

was cheap and practical through the phenomenon of bouncing balls and smartphones

integrated with the Phyphox application. The value of the coefficient of restitution for

each marble was 0.93, 0.92, and 0.92, while the average error was 0.65%, 0.85%, and

1.43%, respectively. Moreover, the average error value of the overall measurement is

0.97%. This error is highly dependent on the shape of the object. The rounder an object

is, the higher the level of accuracy will be. In this study, the determination of the

coefficient of restitution was carried out in two ways: by comparing the height of the

International Journal of Applied Sciences and Smart Technologies

Volume 4, Issue 1, pages 35-46

p-ISSN 2655-8564, e-ISSN 2685-9432

ball’s bounce and the time intervals for the n and n+1 bounce. The value of the coefficient

of restitution generated by these methods was identic. Thus, this study had confirmed that

the bouncing ball experiment using the Phyphox indicated valid data well so that it could

be implemented for determining the coefficient of restitution

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