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Davis – DeSantis -- Oughton

1

Introduction

Sports and athleticism provide a way for people to challenge themselves against

others; a competition that tests skill and talent to see who is best. One of the biggest

problems in sports, however, is that there are a lot of factors other than skill that affect

the outcome of a game. These include personal factors, like physical and mental

condition, as well as environmental factors, like temperature, humidity, and the materials

involved. Because the personal factors cannot be measure and fixed in a traditional

sense, most of the research into the science of sports has focused on the environmental

factors. These factors require fixing because they add a random aspect to a game that is

about skill and strategy; the presence of these factors diminishes the value of skill,

instead favoring luck and chance. The experiment being done tests the effects of two

environmental factors – temperature and the surface material – on the bounce height of a

soccer ball.

The first factor being tested is temperature. Temperature is one of the most

influential factors on the outcome of a sporting event because it is not something that

only appears sometimes like humidity – there is always a temperature, and depending on

what it is, the physics of the game can change. Measuring the effects of temperature on

how a soccer ball bounces is important because it is a real problem in Soccer: there is a

lot of concern that temperature will have a very pronounced effect in the 2022 World

Cup, being held in Qatar, where temperatures can reach 50° Celsius (about 122°

Fahrenheit) (Allen). Finding out exactly how temperature affects soccer balls will lead to

developments and inventions that will reduce or eliminate the effects of temperature on

the game.


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The second factor being tested is the surface material. Traditionally, games were

played on natural grass; however, modern-day games are mostly played on turf, an

artificial material that simulates grass. The choice of which surface type to use has

significant effects on the performance of the soccer ball. This is because the different

properties of both types change the various properties of the ball, like the coefficient of

friction (Brosnan): real grass bends differently than turf, and the ground below turf is

rubber as opposed to the dirt below grass. Determining exactly how much the surface

material affects the ball will help sports temas and companies choose which material they

prefer and could lead to a the development of a surface material that blends the best traits

of natural grass and artificial turf.

The potential impact of this research, and research like it, is very important to the

game of soccer. Understanding exactly what affects the game will help increase

understanding of the physics behind the game, which will eventually lead to reducing the

effects of these outside variables. Additionally, this research can be used to make physics

a more approachable subject- by taking physics and applying it to the most widely-played

sport, it can reach more people and make them interested in not only the game, but the

physics behind it.


3

Review of Literature

Coefficient of restitution is defined as the ratio of an objects velocity following an

impact to its initial velocity (Kip). However, an objects coefficient of restitution may also

be measured as the square root of the ratio of its rebound height and its drop height. This

can be derived through the law of conservation of energy, where the balls energy prior to

being dropped is entirely gravitational potential energy, the equation for gravitational

potential energy being:

.

Where “m” is mass, “g” is acceleration due to gravity, and “h” is height. The maximum

kinetic energy of the ball is equal to its maximum gravitational potential, so can be

set as equal to kinetic energy, which is:

Where “v” is the velocity. Since the coefficient of restitution is defined as the ratio of the

two velocities, the coefficient of restitution can be written as:

In the side of the equation with gravitational potential energy, both the masses and

gravities cancel out, leaving only the heights. In the side of the equation with kinetic

energy, the masses and the

’s both cancel out, leaving only the velocities. This leaves the

equation of:


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The final step is to square root both sides, so that instead of having a ratio of velocities

squared, there is a ratio of the velocities, following the definition of coefficient of

restitution. The other side of the equation is left as:

√

the square root of the ratio of the heights, which is another equation to calculate the

coefficient of restitution of an object (LeClair).

Differences in temperature can affect the density of the air and therefore the air

resistance on an object, how well the players perform, and the coefficient of restitution.

The coefficient of restitution is the ratio of an object’s speed before and after an impact.

In perfectly elastic collisions, where the energy in the system is conserved, the coefficient

of restitution is 1; in perfectly inelastic collisions, where the energy of the system is not

conserved, the coefficient of restitution is 0 (McGinnis). The coefficient of restitution of

an object is affected by temperature because the coefficient of restitution is dependent on

the elasticity of the object, and at lower temperatures, the object becomes stiffer and less

elastic which causes the coefficient of restitution to decrease; likewise, at higher

temperatures, the object becomes more elastic and increases the coefficient of restitution

(Chapman and Zuyderhoff).

Another major factor on the physics of sports is the type of materials involved.

One example is the surface that the sport is played on. Games like soccer and football


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were traditionally played on grass, but many people and teams are switching to artificial

turf because it requires very little maintenance and generally looks better. In the case of

grass and turf, there would be two major factors which would affect the coefficient of

restitution, energy required to bend the blades of grass and the density of said grass. This

is because the coefficient of restitution of an object depends on the amount of energy lost

in a collision (Wesson) (Brosnan). In sports such as golf, where a balls bounce is one of

the most important aspects of playing the game skillfully, there are even benchmark

requirements of the balls coefficient of restitution. In golf, there is an upper limit to a

balls coefficient of restitution, which is 0.83 (Golf Today).

The bounce of a ball relies on the amount of energy lost in the form of thermal

energy when the ball hits the ground and compresses, changing from kinetic energy to

elastic potential energy. The loss of energy in the form of thermal energy depends on the

friction and heat of the surface. The ball, after being compressed, attempts to return to its

original shape, pushing down on the ground with a certain amount of force, the bounce of

the ball is caused by the force with which the ground pushes back on the ball, following

Newton’s 3rd

Law of Motion, which states that when one body exerts a force on another,

the second body simultaneously exerts a force equal in magnitude and opposite in

direction. Once the ball reaches the apex of its rebound, the kinetic energy changes to

gravitational potential energy; this process repeats itself until the ball has lost all of its

kinetic energy, as seen in Figure 1. In order to measure a ball’s coefficient of restitution,

the ball would be dropped onto a hard surface, and the square root of the quotient of the

rebound height and the drop height is the coefficient of restitution of that ball; this

equation can be seen in Figure 2.


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Figure 1. Soccer Ball’s Transfer of Energy

Figure 1 is a diagram of what happens to the energy in the ball as it rebounds. In

the first section, all of the ball’s energy is elastic potential energy. In the second section,

the ball has kinetic energy. In the third section, the ball’s kinetic energy is being turned

into gravitational potential energy.In the final section, all of the ball’s energy has become

gravitational potential energy.

Figure 2. Equation to find Coefficient of Restitution


7

Figure 2 is the equation used to find the ball’s coefficient of restitution from its

bounce and rebound heights, a sample calculation can be found in Appendix A.


8

Problem Statement

Problem:

What are the effects of different temperatures and the type of surface on the

coefficient of restitution on a soccer ball?

Hypothesis:

A soccer ball at lower temperatures and bounced on plain grass will have the lowest

coefficient of restitution.

Data Measured:

The height the soccer ball bounces was measured in meters (m).The temperature of the

soccer ball was measured in degrees Celsius (°C). The two types of surfaces that were tested

were grass and turf. The pressure inside the ball was measured in pounds per square inch (psi).


9

Materials

Soccer Ball

Pump

Pressure Gauge

Meter Stick

Camera

Cooler

Heating Pad

Thermometer, °C

Refrigerator

Duct Tape

Davis – DeSantis — Oughton

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Procedure

1. Connect temperature probes to Vernier Data Quest.

2. Using the TI-nspire calculator, randomize the trials within a 2-factor DOE.

3. Using a pump and pressure gauge, inflate the soccer balls to 7 psi.

4. Place a heating pad inside of the cooler at its highest setting.

5. Dependent on what temperature is decided, either place the soccer ball in a refrigerator

for at least 30 minutes to cool it to between 0° and 5° C (-), place the ball in the cooler

containing the heating pad for at least 30 minutes to heat the ball up to around 35° C

(+), or, if the trial is to be done at room temperature, leave the ball as is (standard).

6. If either the cooler or refrigerator is required for the trial, initiate Vernier Data Quest

and place temperature probes inside of the respective temperature control device.

7. Based on the trial being done, find a hard tile floor (+), a section of artificial grass turf

(standard), or a section of sod or grass (-) to drop the ball onto, make sure that

whichever surface is being used is as dry as possible.

8. Place a meter stick on the surface so that it is completely vertical.

9. While recording with the camera, drop one of the soccer balls (depends on the trial)

with the center of the ball at the 1 meter mark.


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10. Import the video into Logger Pro and plot the points of the soccer ball as it falls and

rebounds.

11. Record the initial drop height and the rebound height given by Logger Pro and any

observations made during the trial being performed .

12. Repeat steps 3-11 until all trials have been completed

Diagram

Figure 3. Picture of Experimental Setup inside Cooler

Figure 3 is an image of how the experiment was set up inside the cooler. It includes the

soccer ball, the cooler, the heating pad, and the Data Quest with temperature probe.


12

Figure 4. Picture of the Experimental Setup outside Cooler

Figure 4 is an image of the experimental setup outside the cooler, where the soccer ball

was dropped. It includes the soccer ball, the meter stick, a roll of duct tape, and the pressure

gauge and air pump.


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Data and Observations

Table 1

Raw Data for Trials

DOE(Temp,Surface

Type)

Drop

Height

Drop Height

Average Rebound Height

Rebound Height

Average

(+,+)

1.068

1.054666667

0.8185

0.804033333

1.062 0.8049

1.031 0.7733

1.054 0.7943

1.082 0.8365

1.031 0.7967

(-,+)

1.081

1.059166667

0.6222

0.618733333

1.036 0.5861

1.074 0.6062

1.038 0.611

1.036 0.6312

1.09 0.6557

(+,-)

1.032

1.043833333

0.2811

0.321166667

1.022 0.3261

1.036 0.3105

1.05 0.3309

1.043 0.3369

1.08 0.3415

(-,-)

1.128

1.0845

0.2754

0.2876

1.094 0.3236

1.082 0.3133

1.083 0.3289

1.049 0.1941

1.071 0.2903

1 0.4481

Table 1 shows all of the raw data collected throughout the experiment, with the

drop height and the rebound height, with their corresponding level of factors.


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Table 2

Level of Factors

Factors High

(+) Standard (0)

Low (-

)

Temperature (°F) 95 70 5

Surface Material Tile Turf Grass

Table 2 shows the two factors being tested – temperature and surface material –

and their high, standard, and low values.

All calculations made in reference to the coefficient of restitution of the soccer

ball were found using this formula:

√

A sample calculation for this formula can be found in Appendix A.

Table 3

Data for DOE’s 1, 2, and 3

Trial Temperature

(°F)

Surface

Material

Coefficient of Restitution

Run 1 Run 2 Run 3

1 Standard Standard 0.67079 0.67550 0.66612

5 + + 0.87543 0.87058 0.86605

2 + - 0.52190 0.56487 0.54746


3 - + 0.75867 0.75215 0.75129

6 - - 0.49411 0.54387 0.53810


Table 3 shows the order of trials and the resulting coefficient of restitution of the

first three runs of the experiment.


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Table 4

Data for DOE’s 4, 5, and 6

Trial Temperature

(°F)

Surface

Material

Coefficient of Restitution

Run 4 Run 5 Run 6


2 + + 0.86810 0.87926 0.87906

6 + - 0.56138 0.56834 0.56232


5 - + 0.76722 0.78056 0.77560

3 - - 0.55108 0.43016 0.52063


Table 4 shows the order of trials run and the resulting coefficient of restitution of

the trials run for the last three runs on the experiment.

Table 5

Average Coefficient of Restitution

Run Temperature (°F) Surface Material Average CoR

***** Standard Standard 0.67558

(+ , +) + + 0.87308

(+ , -) + - 0.55438


(- , +) - + 0.76425

(- , -) - - 0.51299


Grand Average 0.67648

Table5 shows the average coefficient of restitution for each combination of

factors. It was calculated using the data in Tables 3 and 4. The grand average is also

shown.


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Table 5

Observations

DOE Trial Observations

1 1,2,3,4,

5, & 7 Trials ran smoothly.

1 6 Grass was wet; cooler was shared with another

group.

2 1,3,4,7 Trials ran smoothly; cooler was shared with another

group

2 2,6 Grass was wet; cooler was shared with another

group


group

3 1,4,7 Trials ran smoothly; cooler was shared with another

group.

3 2,3,5 Trials ran smoothly; cooler was shared with another

group


group.

4 1,4,7 Trials ran smoothly.

4 2 Trial ran smoothly; cooler was shared with another

group.

4 3,5 Cooler was shared with another group, didn't have

anywhere to place the cold ball (-)

4 6 It was windy outside.


5 3 Grass was wet; strangely low CoR

5 2,5,6 Cooler was shared with another group. No place to

store cold ball (-) between trials nearby.


6 5 Ball barely hit the meter stick, may have affected

the rebound height.

6 2 Ball was dropped at an angle

6 3,6 Grass was wet and it was windy outside.

Table 5 is a table of the observations made while the experiment was being done.


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Figure 5. Points of Measurement in Experiment

In the experiment, the ball was recorded being dropped from a height of about one

meter. For the coefficient of restitution to be measured accurately, the height always had

to be measured from the same point on the ball, which was always the center of the ball.

The rebound height of the ball was at the apex of the first bounce. All of the heights were

measured using a video input to LoggerPro.


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Data Analysis and Interpretation

In the experiment, two different independent variables were used, those being the

temperature of the ball and surface type. The purpose of the experiment was to see the

effects that both temperature and surface type had on the coefficient of restitution of a

soccer ball. Also, the experiment was used to find the interaction between the two factors,

temperature and surface type. By using a two-factor design of experiment, or a two-factor

DOE, both the effects of the two independent variables and the interaction between the

two factors could be seen. In a DOE, the coefficient of restitution under the combination

of a high (+) and low (-) factors was recorded. Below, it is shown how the effects and

interactions of the two factors were determined, the effects were determined by finding

the change from low to high, and the interaction effect was found by subtracting the slope

of the line with the interaction of the low factors from the line with the interaction of the

high factors. Randomization was used to reduce the effects of bias by allowing for

several groups of similar trials. Repetition was used because repetition reduces the

impacts of outliers by allowing for a larger number of total trials.


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Table 6

Table of Averages

(Temp, Surf) Average C.O.R.

( + , + ) 0.873083

( - , + ) 0.764248

( + , - ) 0.554378

( - , - ) 0.512993

Table 6 shows the average coefficient of restitution for each set of paired factors,

with temperature being the factor on the left and surface type being the factor on the left.

Table 7

Factor: Temperature

Temperature

(-) 5°F (+) 95°F

0.764248 0.873083

0.512993 0.554378

Average 0.638621 0.71373

Figure 6. Effect of Temperature

Table 7 and Figure 6, shown above, shows the effect of the temperature on the

coefficient of restitution of the soccer ball. The effect of temperature is 0.07511, which

mean that, on average, as the temperature increased, the coefficient of restitution

increased by 0.07511.

-

+

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 1

Co

eff

icie

nt

of

Re

stit

uti

on

Temperature


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Table 8

Factor: Surface Type

Surface Type

- Grass + Turf

0.554378 0.873083

0.512993 0.764248

Average 0.533686 0.818665

Figure 7. Effect of Surface Type

Table 8 and Figure 7, shown above, show the effect of surface type on the

coefficient of restitution of the soccer ball. The effect of surface type is 0.284979, which

means that, on average, as the surface type became more firm, the coefficient of

restitution increased by 0.284929.

- +

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 1

Co

eff

icie

nt

of

Re

stit

uti

on

Surface Type


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Table 9

Interaction of Temperature and Surface Type

Surface Type - + Slope T

emper

ature

Dashed Line

+ 0.554378 0.873083 0.159352

Solid Line

- 0.512993 0.764248 0.125628

The table above, Table 9, shows the average coefficient of restitution of all

trials at the high and low levels of temperature and surface type.

Figure 8. Interaction Between Temperature and Surface Type

The interaction effect between temperature and surface type was found to be

0.033724, as 0.159352 minus 0.125628 is 0.033724. Since the two lines are not parallel,

the graph suggests that there is an interaction between surface type and temperature of the

soccer ball.

(-,+)

(+,+)

(-,-)

(+,-)

00.10.20.30.40.50.60.70.80.9

1

-1 1

Co

eff

icie

nt

of

Re

stit

uti

on

Surface Type

Interaction Effect (Temperature and Surface Type)


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Figure 9. Standards

Figure 9, shows the standards for 6 DOE’s and 18 standard trials, since 3 standard

trials were run per DOE. The standards were used as a control in the DOE, and since

there was only a very small variation in standards, which shows that the data is very

consistent and that the experiment was most likely run correctly.

Grand Average = 0.676176

Overall effects of single factors:

Effect of Temperature (T) = 0.07511

Effect of Surface Type (S) = 0.28498

Interaction between factors:

Temperature and Surface Type = 0.033724

Figure 10. Prediction Equation

The prediction equation is shown in Figure 10. The temperature is displayed as

variable T and the surface type is shown as variable S. The prediction equation shows the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

Co

eff

icie

nt

of

Re

stit

uti

on

Standards

Standards


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grand average added to both the individual effects and the interaction effect, with both

the individual effects and interaction effects divided by two. To predict the coefficient of

restitution in a trial, one can insert the high and/or values.

Figure 11. Dot Plot of all three Effects

Figure 11, shows a dot plot of both effects and the interaction effect in the design

of experiment. Because the coefficient of restitution of an object must between zero and

one the values for effects are close to zero, so it is hard to tell from the dot plot what can

be considered significant; however, since the effect of surface type is much larger than

that of the other two effects, it is likely that it had a significant effect.

| |

| | | | | | Figure 12. Test of Significance

Using a test of significance, it can be seen whether or not an effect was significant

on the soccer ball’s coefficient of restitution. If the effect of a factor is greater than or

equal to 0.051188, the effect is significant. In figure 12, it is shown how this value was

found and calculated. Since both the effect of temperature and surface type were greater

0 0.05 0.1 0.15 0.2 0.25 0.3

Dot Plot of Effects Interaction Effect 0.033724

Effect of Surface 0.28498

Effect of Temperature 0.07511


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than this value, they can both be said to be significant, however, the interaction effect was

not significant.

Figure 13. Parsimonious Prediction Equation

In the figure above, Figure 13, the parsimonious prediction equation is displayed.

The Parsimonious prediction equation is a prediction equation which only includes the

effects that were considered significant, those being surface type (S) and temperature (T).

According to this DOE and test of significance, all effects were positive, and both the

effects of temperature and surface type were considered significant; however, the

interaction between these two factors was not considered significant.


25

Conclusion

The hypothesis that the soccer ball at the low temperature and bounced on natural

grass will have the lowest coefficient of restitution was accepted. In order to reach this

conclusion, an experiment was designed and conducted where a soccer ball was either heated

or cooled and then bounced on one of three different surface materials, after which the

coefficient of restitution was calculated with the aid of the LoggerPro software.

The data was analyzed using a 2-Factor Design of Experiment (DOE), which determines

whether or not one or both of the factors had a significant effect on the coefficient of restitution

of the soccer ball. The test found that the effect of temperature was 0.07511, which means that

on average, as temperature increases from low to high, the coefficient of restitution increases

by 0.07511. The test also found that the effect of surface type was 0.28498, which means that as

the surface material becomes more firm (more resistant to force), the coefficient of restitution

of the soccer ball increases by 0.28498. Because the range of standards (the minimum number

that indicates a significant effect) is 0.05119, both factors can be said to have a significant effect

on the soccer ball’s coefficient of restitution. The analysis also gave the interaction effect of

temperature and the surface type as 0.03372; this implies that there is no interaction between

the temperature of the ball and the hardness of the floor, which makes sense because they are

two independent factors – the firmness of the floor is a constant value and cannot be changed

just because it was impacted by an object of a different temperature. All of this agrees with

current science: the temperature of an object is known to affect the object’s elasticity because

as the object cools, it becomes stiffer and less elastic (meaning it will transfer less energy in a

collision), and therefore will have an effect on its coefficient of restitution, which measures

elasticity (Chapman); surface material is also known to have an effect on coefficient of


26

restitution because harder surfaces are less elastic, meaning it will transfer less energy to the

ball in a collision (Brosnan).

The experimental design was fairly sound, but there were some flaws that could have

affected the results of the experiment. The one that could have the largest potential effect was

the fact that the soccer balls, especially the ones that had the high or low temperature, changed

temperature during the experiment. After the ball was taken out of the cooler with the heating

pad, for example, it started to cool back to room temperature, meaning that the temperature of

the ball would have dropped with each trial; if temperature does have a significant affect, this

would cause the elasticity – and therefore the coefficient of restitution – to change slightly with

each trial. Another potential source of error was the inconsistency present in utilizing the

LoggerPro software. Because the LoggerPro is not an automatic software, it relies on the person

using it to be as accurate and consistent as possible each time; obviously, someone cannot be

perfectly precise every time, so the heights gathered from a video can be a slightly different

value each time they are measured. Various factors that cannot be controlled may also have

affected the experiment, like the fact that the temperature outside cannot be set like the

temperature inside the cooler or refrigerator which could cause the ball to reach temperature

equilibrium at a different rate than inside, or that the grass was wet when the experiment was

being done which could have affected how the ball bounces; ideally, the tests would have been

done in an environment where these outside factors can be controlled. These errors in the

design of the experiment and in its execution were mostly incontrollable given the time and

resources at hand; fortunately, their effects were not significant enough to jeopardize the

conclusions made. Furthermore, because these factors affected every trial instead of just one or

two, their effects should not cause one trial to be radically different from the others.


27

The results of this experiment, while not world-changing, are of significant importance

to the game of soccer. They give insight into exactly what affects the game and to what degree

and allow other researchers to determine the best way to reduce these effects. However, this

experiment is not the end of research into this topic; further research involving these particular

factors may test a wider range of temperatures or use multiple types of soccer balls. Further

research can also be done to see how factors other than temperature or surface material affect

the performance of a soccer ball; factors like humidity, air pressure inside and outside the ball,

wind speed, and even grass height are all examples of possible avenues for future research. In

addition to changing the factors being tested, future research can also change what is being

affected, so that instead of seeing how the factors affect the coefficient of restitution, they can

see how they affect attributes like ball spin, ball force, or the travel arc. Testing more of these

outside factors is important to not only increase understanding of the physics behind the game,

but to also help determine the best methods to decrease the effects that these outside factors

have on the performance and ensuring that soccer remains a game of skill and athleticism rather

than luck and chance.


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Appendix A

The response variable measured in these experiments was the soccer balls

coefficient of restitution had to be calculated using the drop height of the ball and its

rebound height. The following formula was used to calculate the balls coefficient of

restitution.

√

A sample calculation using this formula is shown below.

√

√

Figure 14. Sample Calculation of Coefficient of Restitution

Figure 14 shows an example calculation of coefficient of restitution using data

from a (+,+) trial.

The effect of a factor is calculated using the following formula:

With C.O.R.+ being the coefficient of restitution of the ball in a trial with the high value

of the factor and C.O.R.- being the coefficient of restitution of the ball in a trial with the

low value of the factor.


29

Figure 15. Sample Calculation of Effect

Figure 15 above displays a sample calculation of the effect of a single factor.

After calculating the factors of both individual factors, the interaction effect was

calculated using the formula below:

with S+ being the slope of the line using high values and S- being the slope of the line

with the low values.

Figure 16. Sample Calculation of Interaction Effect

Figure 16 above shows a sample calculation of the interaction effect between the

two factors.

After measuring both the effects and the interaction effect, a test of significance

was done in order to find out which factors or the interaction which had a significant

effect on the balls coefficient of restitution. For an effect or interaction to be significant,

the effect must be larger than twice the range of standards. Below is the formula used,

followed by a sample calculation.


30

Figure 17. Sample Calculation of Test of Significance

If the effect of a factor or the interaction effect between two factors is greater than

the value calculated using the test of significance shown above in figure 17, then the

factor or interaction would be considered significant.


31

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8

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