Recently, I played my first game of squash. Before one starts a competitive game, one must &lsquo;warm up&rsquo; the squash ball. Before this act, the ball barely bounces. After multiple impacts, the ball begins to get warm. At this point the ball bounces two to three times as much as it did previously, implying collisions go from inelastic to more elastic. The increase in bounce height can be mathematically represented through the coefficient of restitution (COR). This is a ratio between the velocity after a collision and the velocity before a collision. When COR = 1, the collision is perfectly elastic: all kinetic energy is conserved. When COR = 0, the collision is perfectly inelastic: no kinetic energy is conserved (no bounce). The COR for a ball varies with a number of factors including the elasticity of the ball, the pressure inside the ball, and the hardness of the surface on which the ball is dropped.

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In Topic 3 of the IB, students are taught about gas concepts and the relationship between pressure, volume, and temperature. In Topic 2, students are taught about forces and energy. This investigation aims to combine knowledge from Topic 2 and 3 to comprehensively answer the research question.

If an object is dropped from rest and there is negligible air resistance, the loss in gravitational potential energy is equal to the gain in kinetic energy.

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\(mgh_1 = \frac{1}{ 2} mu^2\)

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Where m is the mass of the object (kg),

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g is the acceleration due to gravity on earth (ms -2),

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u is the velocity of the object upon impact (ms -1)

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and h1 is the initial drop height of the ball (m).

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\(∴ u = \sqrt{2gh_1}\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Equation 1)

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By the same principle, the loss in kinetic energy after the collision is equal to the gain in gravitational potential energy, therefore the velocity after the collision is given by

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\(v = \sqrt{2gh}_2\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Equation 2)

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Where v is the velocity of the object after collision (ms-1),

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and h2 is the maximum height reached by the ball (m)

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The coefficient of restitution is a ratio that describes the velocity of an object before an impact and after an impact and is defined by the following equation

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\(e=\frac{v}{u}\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Equation 3)

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Equations 1 and 2 can be substituted into Equation 3

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\(e=\frac{\sqrt{2gh_2}}{\sqrt{2gh_1}}\)

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\(∴ e =\sqrt\frac{h_2}{h_1}\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Equation 4)

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Although the gas in the ball cannot be assumed to be ideal, there is a positive relationship between temperature and pressure. As temperature increases, gas molecules have greater kinetic energy and thus move faster. When molecules collide with the walls of the container, there is a greater change in momentum. By Newton&rsquo;s Second Law, a greater change in momentum applies a greater force onto the surface. Thus the force exerted on the inner walls of the ball increases. When the ball collides with the ground, the force due to pressure acts to restore the ball to its original shape, thus it exerts a force on the walls of the ball and therefore the ground. Newton&rsquo;s third law of motion states that when object A exerts a force on object B, object B will exert a force equal in magnitude and opposite in direction onto A. Thus the ground exerts a force equal in magnitude and opposite in direction onto the ball causing a resultant upward force. Newton&rsquo;s Second Law of motion states that the change in momentum of an object is directly proportional to the resultant force acting on the object, thus a greater force causes a greater change in momentum, giving the ball a greater initial post-collision velocity. Referring to Equation 1, the maximum height will be greater. Thus the COR will approach 1.

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Temperature also affects the elastic properties of the squash ball. As temperature increases, the ball experiences greater deformation when an equal force is applied. Hooke&rsquo;s Law (F = kx) states that force is the product of k, the spring constant, and x, the length of deformity. If x increases, k must decrease as force stays constant. Elastic potential energy is given by the equation

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\(E_p=\frac{1}{2}kx^2\)

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Since x is proportional to \(\frac{1}{k}\), this can be substituted in for x.

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\(Ep∝ k(\frac{1}{k})^2\)

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\(∴ E_p ∝\frac{1}{k}\)

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Since a greater amount of energy is stored in the ball, less is lost to the surrounding in the form of heat. This energy become kinetic energy as the ball expands to its original shape. This increases its rebound height, increasing COR.

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Thus theory predicts a graph of COR versus temperature to have a decreasing positive gradient and asymptotic towards COR = 1.

A range of preliminary trials were conducted to investigate control variables, test the range of temperatures that will be used, and to develop a method.

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Two squash balls at different temperatures (5&deg;C and 60&deg;C) were dropped from different heights (1.00m and 1.70m). This was recorded with the slow-motion feature on a smartphone camera. The footage was analysed using LoggerPro to determine the maximum height reached by the squash ball after the first bounce. 60&deg;C was chosen as a maximum temperature as this 15&deg;C above a squash ball after a usual game of squash (Sportscentaur). 5&deg;C was chosen as it was the lowest temperature the ball could get to with the means available (ice). Even if the ball was initially at 0&deg;C, some heat energy was gained in exposure to air before the ball could be dropped. At 5 &plusmn; 0.5 &deg;C and an initial drop height of 1.00 &plusmn; 0.02 m, the rebound height was 0.143 &plusmn; 0.005 m.

The accuracy of the rebound height, &plusmn; 0.005 m, is based on the uncertainty of the ruler used for measurements. But the ball&#39;s position against the ruler was observed by the camera, hence the accuracy.

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Upon further testing, it was found that the squash ball can be brought up to a temperature of 70&deg;C safely. Thus a range of temperatures from 5&deg;C to 65&deg;C was chosen.

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The error in measuring bounce height was 0.005m. For the lowest achieved bounce height (highlighted in blue), the percentage error is&nbsp;\(\frac{0.005}{ 0.143} × 100 = 3. 5\%\) . Although this is reasonably low, a drop from a larger distance would give a larger range of data. The drop height was limited by safety. A drop height above 1.85m became unsafe due to it&rsquo;s height in comparison to the person dropping the ball, especially considering the ball may be at 60&deg;C. Thus a height of 1.80m was chosen.

Independent variable - The temperature of the squash ball in &deg;C &plusmn; 0.5, changed using a water bath 5, 15, 25, 35, 45, 55, and 65.

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Dependent variable - The maximum height the squash ball reaches after one bounce in metres &plusmn; 0.005 This was measured using video analysis software. A slow-motion recording of the experiment was done. Within the frame was a one-meter ruler in order for the video analysis software to calibrate distances. A frame-by-frame analysis of the recording was done to determine the point of change in direction of the ball. The displacement between this point and the floor was found and recorded.

Exposure to 65&deg;C water for two seconds or more can cause third-degree burns (CPSC). When dealing with this temperature, caution was taken to avoid contact with the water and the ball. This was done by using tongs to remove squash balls from the water bath. Furthermore, eye protection was worn in case of splashing. Temperatures above 50&deg;C can still cause severe burns if in contact with skin, thus caution was taken with all temperatures above 50&deg;C. Another safety hazard was the retort stand. Due to its heavy base, knocking it over would likely lead to damage of property or harm to an individual. This was especially important considering it was at the edge of a table. The experiment was done with minimal interference from peers to ensure no accidents occured.

Water baths were turned off when not in use and were set to the minimum temperature possible in order to reduce energy consumption. All squash balls used in the process were reused. There were no other environmental issues.

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Due to the small scale of the experiment, there were no ethical issues.

Apparatus Setup.svg

<ul>
	<li>Apparatus was set up as shown in Figure 4.</li>
	<li>5 squash balls were placed in a water bath set to 65&deg;C.</li>
	<li>The slow-motion camera was turned on and started recording.</li>
	<li>When the balls reached 65&deg;C (checked with a digital infrared thermometer), they were taken out of the water bath one at a time with tongs and placed on the retort clamp as quickly as possible. The ball retort clamp was quickly loosened until the ball fell.</li>
	<li>Step 4 was repeated four more times to obtain a more accurate mean value of h2</li>
	<li>Steps 2-5 were repeated at temperatures of 55&deg;C, 45&deg;C, 35&deg;C, 25&deg;C, 15&deg;C, and 5&deg;C. With temperatures below 25&deg;C, squash balls were placed in an ice bath with their temperatures being monitored.</li>
</ul>

Table Of Results.svg

Consider T = 5.0 &plusmn; 0.5&nbsp;&deg;C

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h2 mean

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\(\frac{h _{2(1)} +h_{ 2(2)} +h_{ 2(3)} +h_{ 2(4)} +h_{ 2(5) }}{5}\)

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\(=\frac{ 0.175 + 0.190 + 0.182 + 0.173 + 0.191)}{5}\)

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= 0.182m

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h2 mean uncertainty

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= max|h (1, 2, 3, 4, 5) -h mean |

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= 0.191 - 0.182

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= 0.009m

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COR

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\(COR = \sqrt\frac{h_{2}}{h_1}\)

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Where h1 = 1.80 &plusmn; 0.01 m

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Consider h2 =0.182 &plusmn; 0.009.

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\(COR = \sqrt\frac{0.182}{1.80} =0.318\)

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COR Uncertainty

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Consider h2 =0.182&nbsp;&plusmn; 0.009

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\(=\frac {\% unc\ h_{2}}{2} + \frac{\%unc\ h_1}{2}\)

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\(\%unc\ h_2 = 100×\frac{0.009}{0.182}\)

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= 4.95%

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\(\%unc\ h_1 = 100× \frac{0.01}{1.80}\)

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= 0.556%

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&nbsp;\(\%unc \ COR =\frac{4.95}{2}\frac{0.556}{2}\)

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%unc COR = 2.48+0.28

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%unc COR = 2.76%

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\(unc\ COR = \frac{2.76}{ 100} × 0.318\)

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unc COR = 0.009

A graph of h2 avg versus T to show.svg

A graph Of COR Versus T To Establish The Relationship.svg

The graph shown in Fig 2 has an increasing positive gradient between 5 and 25&deg;C. Between the temperatures of 35 and 65&deg;C, there is a positive relationship with a decreasing gradient.

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To investigate the mathematical relationship between COR and T, the assumption was made that

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T &prop; CORn

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&there4; T = k &times; CORn

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log(T) = log(k &times; CORn)

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log(T) = nlog(COR)+ log(k)

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Thus, assuming there is a mathematical relationship between T and COR, a graph of log(T) versus log(COR) will be a straight line graph with gradient n, where n determines the relationship between the two variables.

The graph shown in Fig 2 has an increasing positive gradient between 5 and 25&deg;C. Between the temperatures of 35 and 65&deg;C, there is a positive relationship with a decreasing gradient.

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To investigate the mathematical relationship between COR and T, the assumption was made that

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T &prop; CORn

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&there4; T = k &times; CORn

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log(T) = log (k &times; CORn)

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log(T) = nlog(COR)+ log (k)

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Thus, assuming there is a mathematical relationship between T and COR, a graph of log(T) versus log(COR) will be a straight line graph with gradient n, where n determines the relationship between the two variables.

log(T) uncertainty

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Consider T = 5.0 &plusmn; 0.5 &deg;C

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log(t) unc =

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max[log(T) &minus; log(T &minus; 0. 5), log(T + 0. 5) &minus; log(T)]

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= max[0. 699 &minus; 0. 653, 0. 740 &minus; 0. 699]

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= 0. 046

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log(COR) uncertainty

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Consider COR = 0.318 &plusmn; 0.009

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log(COR) unc =

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max[log(COR) &minus; log(COR &minus; 0.009), log(COR + 0.009) &minus; log(COR)]

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= max[&minus; 0. 498 &minus;&minus; 0. 510, &minus; 0. 485 &minus;&minus; 0. 498]

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= 0. 013

A graph of log(COR) versus log(T) to determine.svg

<ul>
	<li>Fig 3 plots COR versus T. Between T = 5&deg;C and T = 25&deg;C, COR increases at an increasing rate. Between T = 35&deg;C and T = 65&deg;C, COR increases at a decreasing rate.</li>
	<li>Theory predicts a graph of COR versus T to have a decreasing positive gradient and asymptotic towards COR = 1 (pg&nbsp;2). Experimentally, a graph with a positive gradient, decreasing gradient was obtained when T &gt; 25&deg;C. The graph appears to be asymptotic, however, a greater range of data is needed to confirm this.</li>
	<li>All points follow the predicted theory other than the point at (5.0, 0.182). Although this may be an anomaly, since five trials were taken, it is unlikely. Thus the point suggests that when T &lt; 15.0&deg;C, the ball loses its elasticity at a decreasing rate.</li>
</ul>

Although temperature affects the pressure inside the ball, this investigation cannot investigate the direct relationship between pressure and COR mostly because the gas cannot be assumed to be ideal but also because temperature also affects the elasticity of the ball. One way to measure the effect of pressure on the COR is through the use of a different type of ball. A basketball can be used as its pressure can easily be adjusted using an air pump. The method could stay relatively the same, using a slow-motion camera to measure the rebound height. The usual pressure of a basketball is around 51,000 to 58,000 pascals, thus a range of pressures from 50,000 to 60,000 pascals can be used (NBA).

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Pressure can be increased in a few ways: decreasing the volume, increasing the number of moles of gas, or increasing the temperature. This method would increase the number of moles of gas using a pump. Due to a greater number of gas molecules, there would be more collisions with the inner wall, thus a greater force would be applied.

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The pressure inside the ball can be measured using a digital pressure gauge. This has an accuracy of 0.01 kPa, for a minimum pressure, 50kPa, the percentage uncertainty would be \(\frac{0.01}{ 5} × 100 = 0. 02\%\), therefore extremely small.