How does the temperature of a squash ball affect its rebound height when dropped from rest, from a

Research Question
How does the temperature of a squash ball affect its rebound height when dropped from rest, from a fixed height
onto a fixed surface?
Introduction
Lately, I have played my first game of squash. Before one starts a competitive game, one must ‘warm up’ 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.
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.
Background Information
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. 𝑚𝑔ℎ1=1
2𝑚𝑢2
Where mis the mass of the object (kg),
gis the acceleration due to gravity on earth (ms-2),
uis the velocity of the object upon impact (ms-1)
and h1is the initial drop height of the ball (m).
∴ 𝑢= 2𝑔ℎ1(Equation 1)
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 𝑣= 2𝑔ℎ2(Equation 2)
Where vis the velocity of the object after collision (ms-1),
and h2is the maximum height reached by the ball (m)
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 𝑒= 𝑣
𝑢(Equation 3)
Equations 1 and 2 can be substituted into Equation 3
𝑒= 2𝑔ℎ2
2𝑔ℎ1
∴ 𝑒= ℎ2
ℎ1
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(Equation 4)
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’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’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’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.
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’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 𝐸𝑝=1
2𝑘𝑥2
Since x is proportional to , this can be substituted in for x.
1
𝑘𝐸𝑝 ∝ 𝑘(1
𝑘)2
∴ 𝐸𝑝 ∝ 1
𝑘
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.
Thus theory predicts a graph of COR versus temperature to have a decreasing positive gradient and asymptotic towards
COR = 1.
Preliminary Trials
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.
Two squash balls at different temperatures (5°C and 60°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°C was chosen as a maximum
temperature as this 15°C above a squash ball after a usual game of squash (Sportscentaur). 5°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°C, some heat
energy was gained in exposure to air before the ball could be dropped. At 5 ± 0.5 °C and an initial drop height of 1.00 ±
0.02 m, the rebound height was 0.143 ± 0.005 m.
Temperature (°C) ± 0.5
Initial drop height (m) ± 0.02
Rebound height (m) ± 0.005
5
1.00
0.143
5
1.70
0.250
60
1.00
0.477
60
1.70
0.823
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The accuracy of the rebound height, ± 0.005 m, is based on the uncertainty of the ruler used for measurements. But the
ball's position against the ruler was observed by the camera, hence the accuracy.
Upon further testing, it was found that the squash ball can be brought up to a temperature of 70°C safely. Thus a range of
temperatures from 5°C to 65°C was chosen.
The error in measuring bounce height was 0.005m. For the lowest achieved bounce height (highlighted in blue), the
percentage error is . Although this is reasonably low, a drop from a larger distance would give a
0.005
0.143 ×100=3.5%
larger range of data. The drop height was limited by safety. A drop height above 1.85m became unsafe due to it’s height in
comparison to the person dropping the ball, especially considering the ball may be at 60°C. Thus a height of 1.80m was
chosen.
Variables
Independent variable - The temperature of the squash ball in °C ± 0.5, changed using a water bath
5, 15, 25, 35, 45, 55, and 65.
Dependent variable - The maximum height the squash ball reaches after one bounce in metres ± 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.
Control Variable
Value
Method/Justification
The initial height of the
ball
1.80m ± 0.02
A retort stand was set up on top of a desk. A retort clamp was set at a
height of 1.80m above the ground. The retort clamp was opened to a
width slightly smaller than the width of the squash ball. The ball was
then placed on the arm of the clamp, and the width was increased to
allow the ball to fall through. This ensured that the ball was dropped
from the same point in each trial and was dropped from rest at each trial.
The same retort stand and clamp were used for all trials. Different drop
heights would have changed the velocity on impact due to acceleration
due to gravity.
The initial velocity of the
ball
0 ms-1
A retort clamp was used to position the ball 1.80 metres above the
ground. The clamp was opened to about 3.5 cm. This was the minimum
width at which the squash ball could rest upon the surface of the clamp.
The clamp was then opened to allow the ball to drop through.
Because the change in velocity is constant depending on the height, an
initial velocity would have affected the final velocity. Although terminal
velocity would have an effect on the change in velocity, at low velocities
and surface areas, this factor is negligible.
The elasticity of the ball
One red dot
Squash balls come in various standards, these are identified by coloured
dots on the balls. The standard varies in quality, durability, and elasticity.
10 single red dot balls were selected for the investigation. This ensured
there were no systematic errors in data due to a difference in the ball.
Elasticity and
consistency of ground
---
The balls were dropped onto the same flat surface for each trial. This
ensured there were no errors with balls bouncing to the side or
differences in the change in momentum. An elastic surface would have
increased force exerted onto the ball, increasing its kinetic energy and
therefore maximum height. . Since this would affect the dependent
variable, it was kept constant.
Safety
Exposure to 65°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°C
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can still cause severe burns if in contact with skin, thus caution was taken with all temperatures above 50°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.
Environmental and Ethical issues
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.
Due to the small scale of the experiment, there were no ethical issues.
Apparatus
Name
Quantity
Uncertainty
Retort stand
1
---
Retort clamp
1
---
Squash ball
10
---
Ruler (1.000m)
1
± 0.001m
Smartphone slow-motion camera
1
---
Water bath
1
± 0.1°C
Digital infrared thermometer
1
± 0.1°C
Fig 1 - Apparatus setup
Method
1. Apparatus was set up as shown in Figure 1.
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2. 5 squash balls were placed in a water bath set to 65°C.
3. The slow-motion camera was turned on and started recording.
4. When the balls reached 65°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.
5. Step 4 was repeated four more times to obtain a more accurate mean value of h2
6. Steps 2-5 were repeated at temperatures of 55°C, 45°C, 35°C, 25°C, 15°C, and 5°C. With temperatures below
25°C, squash balls were placed in an ice bath with their temperatures being monitored.
Table of Results
T ± 0.5
°C
h2± 0.005 m
h2mean (m)
COR
h2(1)
h2(2)
h2(3)
h2(4)
h2(5)
5.0
0.175
0.190
0.182
0.173
0.191
0.182 (± 0.009)
0.318 (± 0.009)
15.0
0.174
0.218
0.189
0.231
0.199
0.202 (± 0.029)
0.335 (± 0.024)
25.0
0.368
0.368
0.371
0.356
0.373
0.367 (± 0.009)
0.452 (± 0.005)
35.0
0.597
0.641
0.643
0.593
0.624
0.620 (± 0.025)
0.587 (± 0.012)
45.0
0.796
0.756
0.785
0.802
0.811
0.790 (± 0.027)
0.662 (± 0.012)
55.0
0.834
0.865
0.882
0.890
0.869
0.868 (± 0.028)
0.694 (± 0.011)
65.0
0.983
0.941
0.918
0.928
0.913
0.937 (± 0.035)
0.721 (± 0.014)
h1= 1.80 ± 0.01 m
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