The Journal of Online Mathematics and Its Applications, Volume 7 (2007)

Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson

Note on notation: The coefficient of restitution is denoted COR in this paper; we have used `r` elsewhere in our article. In the term *R-squared* below, `R` is the correlation coefficient in statistical regression; elsewhere in our article, `R` denotes the elasticity coefficient.

The purpose of my experiment is to find if the coefficient of restitution (COR) is constant (the usual assumption), or if there is a correlation between the height the ball is dropped and the COR. The coefficient of restitution is the velocity of the ball after the bounce, divided by the velocity before the bounce.

I observed and recorded the bounce height of five different balls when dropped from 50 cm to 225 cm on two surfaces--linoleum and ceramic tile--and dropping a basketball as high as 520.7 cm on a gym floor. A microphone attached to a laptop computer recorded sounds for heights below 50 cm on these surfaces.

My graphs show that the CORs are not constant. A trend line helps to see if the CORs follows my hypothesis. The COR did follow my hypothesis; it became higher when the ball was dropped from a lower height in the range from 50 to 520.7 cm, except for the superball which was inconsistent. I also graphed the elasticity coefficient, which is the height bounced divided by the height dropped; the graphs were about the same as the COR graphs.

For heights below 50 cm, I determined the COR by entering the sound waves on a program that put them on a time line so I could find the bounce time, the height of the bounce, the velocity, and the COR. The CORs for the low heights did not follow my hypothesis as well; the CORs varied widely.

One day in a class my Dad taught five years ago on infinite series, a problem in the math book referred to bouncing balls. It was similar to one in my sister's pre-calculus book: "A ball is dropped from a height of 81 feet. After each bounce, it rebounds two thirds of the distance it fell. How far does the ball fall on its fourth fall?" He tried to illustrate the problem with a basketball but it did not work, conjecturally because the coefficient of restitution (COR) was not constant. The next day he used a superball and that seemed to work. He wanted to know what the COR was for the basketball and if the height you dropped it from mattered. He told me about it and I decided to use it for my experiment.

I first tried to find the height using the movie on a digital camera but the ball moved too much between frames of the movie so I could not tell when the ball was at the top of its bounce. I found that just watching was more accurate at least for the higher heights. I could not accurately see how high the ball bounced below fifty centimeters. So I used a microphone to record the sound.

My hypothesis was that the COR would be smaller the higher you dropped the ball and would get larger when you dropped it lower, because the further it has to drop the bigger percentage of its energy would be lost so it would not bounce as high percentage wise as at the lower heights. I thought it would take more energy because the ball would shake the floor more when dropped from a higher height and because the balls would bend in more. I thought the basketball and soccer ball would bend in more than the tennis ball and superball and ping-pong ball, so the COR would change more for the basketball and soccer ball than for the tennis ball and superball and ping-pong ball. I used different surfaces because in my research many web sites said that the surface matters a lot and I wanted to be able to prove my hypothesis on more than one surface.

- Pencil
- Paper
- Notebook
- Basketball
- Super ball
- Ping-pong ball
- Soccer ball
- Tennis ball
- Tape
- Two chairs
- Meter stick
- Linoleum kitchen floor
- Tile bathroom floor
- Villanova Pavilion gym floor
- Laptop computer
- Microphone
- Computer

On a length of paper 225 cm long I drew lines every 5 cm. I taped this up on the wall in the kitchen (linoleum floor) and got my logbook, a pencil and the basketball, super ball, ping-pong ball, and soccer ball. I brought in two chairs, one for the person who dropped the balls to use for the balls at the high heights, and one for me to sit on to be at eye level for the higher bounces. She dropped the balls while I got as close to eye level as possible and wrote how high they bounced in my logbook. I bounced the balls two times for each height and then repeated it so there were four bounces at each height for each ball. I then repeated the experiment in the bathroom (ceramic tile).

I typed all the data into Excel and found the elasticity coefficient by dividing the height dropped by the height bounced. I learned the relationship between height and velocity and used the formula (`v` = `g` `t`, `h` = (1/2) `g` `t`^{2}, `v`^{2} = 2 `g` `h`, `v` = velocity, `h` = height dropped, `t` = time, `g` = acceleration of gravity 9.8 m/sec/sec) to find the speed before it hit and after it bounced. By dividing these I found the COR. I plotted both the elasticity coefficient, bounce height versus drop height, and the COR, bounce velocity versus drop velocity. They looked very similar (see the first four graphs). I added a trend line to each graph. I later redid the experiment with ten trials and added a tennis ball.

The lowest I could see them accurately was 50 cm so we had to use sound to do lower heights. People often use sounds to find the coefficient of restitution (see references). I used the microphone on a laptop to record the bounces. On a program called Cool Edit the sound waves were placed on a time line. With that information I could find the time for each bounce and use the equations mentioned earlier to get the height and velocity. Once the velocity was found it was easy to find the COR and graph the bounces. Three locations were used: linoleum kitchen floor, ceramic tile floor, and a gymnasium floor.

In a gym I put tape on the wall below stairs and bounced a basketball at three high heights: 520.7, 436.88, and 353.06 cm. I recorded how high it bounced in my logbook. I also used the movie on a digital camera to compare with where I observed it had bounced. I downloaded the movie on my computer at home and compared the movies with the numbers I had recorded.

I decided to see if the physics professor I talked to was right about a lot of the error being from the ridges on the balls. I used a marble because it is completely smooth and dropped it on marble because I wanted the surface I bounced it of to be smooth also. I only looked at the first bounce with the sound too because the statistician I talked to thought that that was why the sound was so varied and did not match up with the higher heights. I recorded ten bounces with a microphone on a lap top computer at 50 cm, 40 cm, 30 cm, 20 cm, and 10 cm. I then followed the "sound" procedure for the other low heights. I also recorded heights from 50 cm up to 190 cm using the "sight" procedure I used before for the middle heights with six trials at each height.

Many of the web sites and books assume it is a constant or is almost constant. The web site of the physicist Fu-Kwun Hwang said, "Scientists have found that, for most balls, this speed ratio remains constant over a wide range of collision speeds." The web site of physicist Porter W. Johnson said, "Each time the bounce height reduces by roughly the same factor, the coefficient of restitution." On the other hand, a physics student Paul Ryan experimentally showed that the coefficient of restitution does depend on the height; and his graphs look similar to mine (also with large variance in the data).

There is not a huge difference but there is definitely a difference for the different heights. The trend lines help me see the relationship between height and COR and prove my hypothesis correct for most balls because the COR for them does go up when the height dropped is lower. For example, with the basketball in the gym, the average COR at 50 cm was 0.767, but at 520.7 cm the average was only 0.688, a drop of ten percent.

As explained in the introduction, I thought the COR would change more for the basketball and soccer ball than for the tennis ball, superball, and ping-pong ball. My results show that the ping-pong ball had the most slopes for both the kitchen and bathroom floors, the basketball was generally second. The trend lines for the superball, tennis ball, and soccer ball were fairly flat.

I first found the CORs for each ball in the kitchen and bathroom with 4 trials each. In the kitchen the ping-pong ball had the most slope (the average COR dropped from 0.800 to 0.694), then the basketball (from 0.803 to 0.760), superball (0.797 to 0.765), and soccer ball (0.640 to 0.629). In the bathroom the ping-pong ball still had the most slope, the basketball second, and the superball and soccer ball the least.

I redid the experiment with ten trials because I hoped that more trials would give me better R-squared values. The ping pong and basket balls had okay R-squared values with four trials, but the soccer ball and superball had terrible R-squared values on the ceramic tile floor, and the soccer ball was also terrible on the kitchen floor. With ten trials the R-squared values were mostly better but the soccer ball and superball still did not have great R-squared values.

I used sound to find the CORs for heights below 50 cm because I could not see accurately that low. I hoped that the CORs would continue following my hypothesis, but the CORs were very varied. In the kitchen the ping-pong ball had the best R-squared value, which was only 0.112, and the superball (R-squared = 0.011) and tennis ball (R-squared = 0.026) slopes actually went the wrong way. In the bathroom the basketball, the soccer ball, and the superball had good R-squared values (basketball R-squared value 0.721, superball R-squared value 0.719, soccer ball 0.445) and had good slopes, which supported my hypothesis, but the tennis ball (R-squared = 0.089) and superball (R-squared = 0.024) had slopes going the wrong way.

There are many things that I wish I could have done better. The tiles in the bathroom might have affected results because the tiles have an uneven texture. Also I might not have been as accurate in there because there was not much room to adjust positions for the different heights. One of the web sites I read was on how balls spin so this might have affected the results, but I tried dropping the balls several times to see how much they spun and they did not spin enough to make a difference. I talked to a physics professor named Dr. Phares and he thought that air friction was a big factor, but I do not think that could have caused the variance in my data because air friction would make more difference for the higher heights and less for the lower heights whereas my results varied with no regard to height. He also thought that while the spin may not make a big difference in how high it bounces it would make a difference in which part of the ball hits the surface, so I should try using a smooth ball. So I used a marble bouncing off a marble surface because it would not matter where it hits on the floor or what part of the ball hits the floor. Dr. Phares also thought that the COR would eventually reach one and fit an exponential line which I disagree with. I did think that the COR might fit an exponential line when I started but I tried putting exponential lines in some of my graphs and they did not fit them better then the linear (straight) line I used. I also talked to a statistician named Dr. Pigeon and he agreed with me and said that there was no indication in my data that an exponential line would fit it better than a linear. The statistician thought most of my error was experimental error and I was going to have some no matter what I did. He thought that the lower heights varied more than the higher heights because I recorded the sound from all the bounces so I had no control over the height dropped. So when I redid my experiment with a marble on marble, I just took the first bounce. I recorded up to 50 cm with sound and started eying the bounces at 50cm. The observed heights and the sound heights at 50 cm were about 5cm different. I do not believe I could be 5cm off so I do not know what might have been wrong. Maybe the sound was not as accurate as I thought or somehow the marble was dropped differently for the sound than for the observed.

Scientists are studying how balls bounce, roll, or spin for many reasons. Practically every game has balls in it so anybody really interested in sports is interested in how balls bounce or roll. So even if my experiment was not directly related to any sport it still helped me learn about how balls bounce.

My results show there is definitely some correlation between the height dropped and the COR. My hypothesis was that the COR would be smaller the higher the balls are dropped and a bigger the lower they are dropped.

I collected over 1200 COR values for five balls on three surfaces. The data mostly followed my hypothesis. The ping-pong ball and basketball followed my hypothesis well on all the surfaces. My conjecture that the basketball and soccer ball would have the steepest slopes was not true as the ping-pong ball had the steepest slope.

My Mom and my brother dropped the balls for me and helped me look for material for my research report, and encouraged me. My Dad helped me understand the math part of my project and taught me how to use Excel. Dr. Phares, a physics professor at Villanova, talked with me about my experiment and let me play around with a motion sensor. Dr. Pigeon, a statistics professor at Villanova, talked with me about my data.

- Typical Data
- Complete Data (Excel)

- Hwang, Fu-Kwun. Bouncing Balls
- Johnson, Porter W. Bouncing Balls"
- Kagan, David and Atkinson, David. The Coefficient of Restitution of Baseballs as a Function of Relative Humidity
- Pasachoff, Jay M and Wolfson, Richard. Physics Toronto Little, Brown and Company, 1987
- Ryan, Paul. Computerized Technique to Determine and Analyze the Coefficient of Restitution
- Saxon, John, Advanced Mathematics: An Incremental Development, second edition, Saxon Publishers, 2003, pg. 573.