I played squash for 4 years when I was younger, and at the beginning of every lesson my teacher told me to warm up the squash ball we’d be using that day. I would either roll it underneath my shoe or hit it against the walls of the court. I noticed that the ball would bounce higher after I did so. I always wondered why this would occur, and my teacher would simply say that it was because the ball was warmer, however that was never a good enough answer for me. I want to explore how the bounce height of squash balls change as temperature of the ball changes. There are different types of squash balls which have different initial bounce heights and I would like to see how they compare to each other under the same temperatures. I will be comparing how temperature changes the bounce height of a double yellow dotted squash ball (professional level) and a red dotted squash ball (intermediate level). To investigate this, I have formed the basic research topic of how does the elasticity of different squash balls change with temperature. My specific research question is how does the coefficient of restitution of different squash balls change with temperature. By looking specifically at the coefficient of restitution, I will be able to form a more wholesome and accurate conclusion and analysis of my findings.
I hypothesize that the red ball will on average bounce higher than the yellow ball and that the coefficient of restitution will increase as the temperatures of the squash balls increase.
The purpose of my research is to examine how temperature affects the bounce height and coefficient of restitution of balls used in the game of squash. From this topic I have proposed the following research question: How does temperature affect the elasticity and coefficient of restitution of squash balls?
The basic physics that is involved in this experiment is elastic collision. When a ball is bounced, a number of energy conversions occur. Initially, the ball has gravitational potential energy which is the energy of any object that is held above ground level and is determined by its height and mass. The equation for gravitational potential energy is displayed below:
As the ball falls, this gravitational potential energy is converted to kinetic energy.
When the ball hits the ground, it experiences an elastic collision in which it losses some energy because no squash ball is perfectly elastic. The maintained kinetic energy converts to elastic potential energy as the ball reforms. Once the ball is fully formed and moving upwards, the elastic potential energy converts back to kinetic energy and then once again to gravitational potential energy as the ball increases in height. However, because the ball is not perfectly elastic, the ball will bounce to a lower height than the drop height. This can be analyzed with the coefficient of restitution.
The coefficient of restitution is a numerical value which indicates how much kinetic energy remains after a collision of two objects¹, in this experiment the squash ball and floor.
If the coefficient is high (close to 1.00) it means that very little kinetic energy was lost during the collision. For instance, the coefficient of restitution is said to be precisely 1.00 if a collision was perfectly elastic. If the coefficient is low (close to zero) it suggests that a large fraction of the kinetic energy was converted into heat or was otherwise absorbed through deformation.
One factor that can influence the bounce of a ball is the temperature of the ball. As I have seen in during my squash lessons, a warmer ball will bounce higher than a colder ball. Squash balls are hollow, so when the temperature of the squash ball changes the air pressure of the squash ball changes as well. Decreasing the temperature of a squash ball lowers the air pressure inside the squash ball, similar to the deflating of a ball. Increasing the temperature and in turn the air pressure can be compared to over-inflating a ball. The other way in which temperature influences the height a ball bounces is by impacting its elasticity. Elasticity is a measure of how a ball deforms and then reforms as its kinetic energy is converted to elastic potential energy. The balls used in this experiment, squash balls, are made of rubber. When the squash ball impacts the floor, the bonds of the rubber stretch. When the ball is heated, it becomes more elastic, so the bonds are able to move more freely. Therefore, the rubber is able to stretch more than a cooler ball. This then means that the ball bounces higher. On the other hand, when a squash ball is made colder, the bonds of the rubber tighten making the ball more rigid. Thus, when the ball hits the floor it absorbs energy rather than transferring it.
Double yellow dot
Set up a meter stick against a wall, and a camera facing the meter stick.
Place either a double yellow or blue dotted squash ball in a water bath, and leave it there for 15 minutes.
After 15 minutes, use an infrared thermometer to measure the temperature of the squash ball and remove the squash ball.
Make sure to remove the squash ball with a spoon as to lower the chances of a burn.
Place the squash ball at the top of the meter stick, and make sure the camera is recording.
Drop the squash ball and wait until it has stopped bouncing.
Repeat Step 5 four more times, so that you have five readings at your first temperature.
Repeat Steps 2 to 6 for the other squash ball. If you used a blue dot, then now use the double yellow dot.
Repeat steps 2 to 7 for a different temperature.
Place either the blue dot or double yellow dot squash ball in a freezer.
Leave the squash ball in the freezer for 15 minutes.
After 15 minutes, remove the squash ball and use an infrared thermometer to measure its temperature.
Repeat steps 5 to 9.
Once you have taken sufficient readings, export the videos you have taken to a laptop and measure the bounce heights of the squash balls.
The results have been organized by squash ball. The uncertainties were calculated individually for each data set, as there was a difference in average results between the balls. An uncertainty in the mean was measured as there were 25 trials in total for each squash ball, so an uncertainty in the mean allowed for a more accurate measurement in error.
The equation ∆x_avg=∆x/√N=R/(2√N) was used to calculate the uncertainty in the mean, with R being the maximum data point subtracted by the minimum data point, and N being the amount of data points.
Below is the data for the red squash ball:
Below is the data for the yellow squash ball:
Analysis of Results
Both graphs display an accurate linear trend line with high uncertainties, especially Graph 1. Both graphs demonstrate an R value close to 1, meaning near perfect correlation, however due to the high uncertainties a closer correlation could not be reached. The linear trend line is visually more accurate on Graph 1, the graph of the red dot squash ball. The trend demonstrates that there is a relationship between temperature and elasticity, with an increase in temperature causing an increase in bounce height as well as an increase in the coefficient of restitution. The R-values also demonstrate that the red dotted squash ball is more perfectly elastic than a double yellow dotted squash ball as the R value of the red dotted ball is .019 higher than the R value of the double yellow ball. The trend equations of Graph 1 and 2 are inaccurate, however, as they both suggest that at 0˚C the coefficient of restitutions of the balls are higher than what they were. The average coefficient of restitution at 1˚C for the red ball was 0.252, compared to a suggested value of 0.309. The average coefficient of restitution at 1˚C for the yellow ball was 0.248, compared to a suggested value of 0.307. This inaccuracy may have occurred due to the intervals chosen for this experiment, as they were quite spread.
The results prove the theory that increasing the temperature of a hollow ball increases the bounce height of the ball as pressure inside the cavity of the ball increases. The results also demonstrate that red dotted squash balls consistently bounce slightly higher than yellow dotted squash balls, even at different temperatures. The red dotted squash ball has a lower pressure than the yellow dotted squash ball, so the ball’s bounce height is effected more by heat than the yellow dotted squash ball.
One major error that was identified during the experiment was that drop height was not kept constant. The average drop height had a range of 0.05 meters, which meant that sometimes one squash ball hit the ground with a greater velocity than the other. The results which were predicted still occurred, however, showing that the varying drop height did not show a major effect on the data. If this experiment was to be reattempted in the future an apparatus could be made, such as a small ruler placed perpendicular to the top of a meter stick, to keep drop height constant at 1 meter.
Another limitation was that the balls were heated in a water bath. This created several possible issues. Firstly, the balls had to be transported from the water bath to the meter stick. Transportation of the balls meant that heat could’ve possibly been lost. To amend this issue, the heating apparatus could be placed nearer to the meter stick or vice versa. Secondly, the temperature of the balls could not be accurately determined. To improve on the readings, an infrared thermometer should be used in the future. However, the results were still accurate to theory so the temperature readings may not have needed changing. Thirdly, the bounce of the balls may have been effected as three out of the five temperatures involved a water bath. The fact that the balls were enveloped in a thin layer of water may have altered bounce height. If the experiment was to be repeated, drying the balls quickly after they were removed from the water bath may have made results more accurate.
Furthermore, measuring the bounce height was found to be difficult as no programs such as Logger Pro were used. Height measurements were made purely by judgement on a laptop. Using more a precise video camera, or a clearer measuring stick may have allowed for more accurate results. However, the theory still held.
The hypothesis was proven correct as an increase in temperature caused an increase in the coefficient of restitution. This is demonstrated in both equations; y = 0.0046x + 0.3077 and y = 0.0055x + 0.3095.
...(download the rest of the essay above)