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

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1 Davis DeSantis -- Oughton 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. 1

2 Davis DeSantis -- Oughton 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. 2

3 Davis DeSantis -- Oughton 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: 3

4 Davis DeSantis -- Oughton 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 4

5 Davis DeSantis -- Oughton 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 3 rd 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. 5

6 Davis DeSantis -- Oughton 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 6

7 Davis DeSantis -- Oughton 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. 7

8 Davis DeSantis -- Oughton 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). 8

9 Davis DeSantis -- Oughton Materials Soccer Ball Pump Pressure Gauge Meter Stick Camera Cooler Heating Pad Thermometer, C Refrigerator Duct Tape 9

10 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. 10

11 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. 11

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. 12

13 Data and Observations Table 1 Raw Data for Trials DOE(Temp,Surface Type) (+,+) (-,+) (+,-) (-,-) Drop Height Drop Height Average Rebound Height Rebound Height Average 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. 13

14 Table 2 Level of Factors Factors High Low (- Standard (0) (+) ) Temperature ( F) 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 Surface Coefficient of Restitution ( F) Material Run 1 Run 2 Run 3 1 Standard Standard Standard Standard Standard Standard Table 3 shows the order of trials and the resulting coefficient of restitution of the first three runs of the experiment. 14

15 Table 4 Data for DOE s 4, 5, and 6 Trial Temperature Surface Coefficient of Restitution ( F) Material Run 4 Run 5 Run 6 1 Standard Standard Standard Standard Standard Standard 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 (+, +) (+, -) ***** Standard Standard (-, +) (-, -) ***** Standard Standard Grand Average 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. 15

16 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 2 5 Grass was wet; cooler was shared with another 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 3 6 Grass was wet; cooler was shared with another group. 4 1,4,7 Trials ran smoothly. 4 2 Trial ran smoothly; cooler was shared with another 4 3,5 group. Cooler was shared with another group, didn't have anywhere to place the cold ball (-) 4 6 It was windy outside. 5 1,4,7 Trials ran smoothly. 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 1,4,7 Trials ran smoothly. 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. 16

17 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. 17

18 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. 18

19 Coefficient of Restitution Davis DeSantis Oughton Table 6 Table of Averages (Temp, Surf) Average C.O.R. ( +, + ) ( -, + ) ( +, - ) ( -, - ) 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 Average Temperature + 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 , which mean that, on average, as the temperature increased, the coefficient of restitution increased by

20 Coefficient of Restitution Davis DeSantis Oughton Table 8 Factor: Surface Type Surface Type - Grass + Turf Average Surface Type + 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 , which means that, on average, as the surface type became more firm, the coefficient of restitution increased by

21 Coefficient of Restitution Temperature Davis DeSantis Oughton Table 9 Interaction of Temperature and Surface Type Surface Type - + Slope Dashed Line Solid Line 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. Interaction Effect (Temperature and Surface Type) (+,+) (+,-) (-,+) (-,-) Surface Type Figure 8. Interaction Between Temperature and Surface Type The interaction effect between temperature and surface type was found to be , as minus is Since the two lines are not parallel, the graph suggests that there is an interaction between surface type and temperature of the soccer ball. 21

22 Coefficient of Restitution Davis DeSantis Oughton Standards Standards 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 = Overall effects of single factors: Effect of Temperature (T) = Effect of Surface Type (S) = Interaction between factors: Temperature and Surface Type = 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 22

23 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. Interaction Effect Dot Plot of Effects Effect of Temperature Effect of Surface 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 , 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 23

24 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. 24

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 , which means that on average, as temperature increases from low to high, the coefficient of restitution increases by The test also found that the effect of surface type was , which means that as the surface material becomes more firm (more resistant to force), the coefficient of restitution of the soccer ball increases by Because the range of standards (the minimum number that indicates a significant effect) is , 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 ; 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 25

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. 26

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. 27

28 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. 28

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. 29

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. 30

31 Works Cited Allen, Tom, Dr. "Extreme Temperature Could Influence Ball Properties at the 2022 World Cup." Web log post. Engineering Sport. N.p., 1 Sept Web. 8 Apr < Brosnan, James T., Andrew S. McNitt, and Maxim J. Schlossberg. "An Apparatus to Evaluate the Pace of Baseball Field Playing Surfaces." College of Agriculture Sciences. Penn State, 20 Jan Web. 8 Apr < Chapman, and Zuyderhoff. "Squash Ball Mechanics and Implications for Play." National Center for Biotechnology Information. U.S. National Library of Medicine, n.d. Web. 07 Apr < Drane, P. J., and J. A. Sherwood. "Characterization of the Effect of Temperature on Baseball COR Performance." UMass Lowell. UMass Lowell, n.d. Web. 8 Apr < Kip, M. K. "The Science of Soccer." Department of Physics. The University of Hong Kong, 21 May Web. 07 Apr < Kirby, Lucas R. Turf vs. Grass: The Ultimate Comparison. California State Science Fair. 31

32 University of Southern California, 2 Apr Web. 8 Apr < McGinnis, Peter M. (2005). Biomechanics of sport and exercise Biomechanics of sport and exercise (2nd ed.). Champaign, IL [u.a.]: Human Kinetics. p. 85. ISBN Moreno, Carolina. "Costa Rica Asks For US Rematch Of World Cup Qualifier Snow Game, Ball Movement Was 'Impossible'" The Huffington Post. TheHuffingtonPost.com, 25 Mar Web. 08 Apr < Wesson, John. "The Bounce." The Science of Soccer. New York: Taylor & Francis Group, Print. < oefficient+of+restitution+soccer+ball&source=bl&ots=0gg0wpqj1i&sig=xuf1 y6wapxztlxxmmsnocv- IEF4&hl=en&sa=X&ei=Kd9hUbP2BrPC4AOQ6oH4Bw&ved=0CDkQ6AEwAjg K#v=onepage&q=coefficient%20of%20restitution%20soccer%20ball&f=false>. 8 32