A. Description of Study.......................................................................................................1
B. Sampling/Data Collection..............................................................................................2
C. Data Presentation.............................................................................................................9
F. Error Evaluation............................................................................................................15
H. Figure Section...............................................................................................................18
A. Description of Study
The goal of my study is to model how the bounce of a squash ball changes after
being rubbed under a shoe. The research question is what the change in the bounce
height of a squash ball is based on how many times it is rubbed under an apparatus
designed to mimic a shoe with part of a person’s weight on it. The observational units
are 10 new Dunlop Pro double yellow dot squash balls. The explanatory variable is the
number of times the ball is rubbed consecutively under the shoe apparatus. The response
variable is the height, in inches, that a given ball bounces to when dropped from a
constant height, measured immediately after the ball has been rubbed. I expect that as the
ball is rubbed more, it will bounce higher, a positive correlation. This is because a
squash ball bounces higher when the temperature of the ball is hotter. The rubbing of the
ball should cause friction that heats the ball to some extent, therefore making it bounce
more. This research may be valuable to a squash player. Often a player will rub the ball
under their shoe in order to warm it up and get it to bounce higher. This study could
Page 1 of 25
show them, with some margin of error, how many times they should rub the ball to get it
to a bounce height where it is playable. It could help reduce the process of trial and error
involved with rubbing a squash ball to warm it up and having to test it every once in a
while to se how high it bounces. Given the relationship from this project, a player could
save time and energy by rubbing a ball a specified number of times to make it playable.
B. Sampling/Data Collection
In order to take raw data, the study requires a set of apparatus to be created.
The primary apparatus is the device used to rub the squash ball in a consistent manner.
To make the apparatus, one needs the following:
• board from Home Depot, dimensions 24” x 9” x 1 5/8” (L x W x H), 2x10x08
• 2x : 12 In. x 8 In. x 16 In. Heavy Weight Block, Home Depot Model # 206001999
• 2x : Lehigh 2 In. Fast Eye Utility Pulley, Model # 7090-12
• 1x : ¼” Braided Nylon Rope
• 2x : Eyelet Hooks, Crown Bolt Zinc #14091
• 1x : Drill (capable of drilling a 3/8” diameter hole through concrete)
• 1x : Champion Size 11 Men’s Cruise Runner III shoe
Page 2 of 25
• 1x : Saw
• 1x : Pair of scissors
• 1x : Hammer
• 3x: 1” nails
• 1x: Adjustable wrench
• 3x : 3/8” Hex sleeve concrete anchors, Redhead #50115
• 1x: 35 Lb. Rubber Coated Hex Dumbbell, Model: SDR35
• 1x: Clock that ticks seconds aloud
1. Using drill, drill 4” deep, 3/8 inch diameter in a cinderblock (see fig 1)
2. Put the non-nut end of the anchor through the pulley hole (see fig 2)
3. Insert the anchor end of the anchor + pulley combination into the hole in the
cinderblock so that the pulley is in between the washer of the anchor and the side
of the cinderblock (see fig 3)
4. Using the hammer, lightly hammer in anchor until anchor sticks in block, while
the pulley is perpendicular to the top of the block
5. Using the wrench, twist the nut of the anchor until tight
6. Repeat steps 1-5 for the other cinderblock
7. Take the Champion shoe, and using scissors and the saw cut off most of the shoe
until just the sole is remaining (see fig 4)
8. Place the sole with the foot portion touching the board and the sole portion facing
outward in the position shown (see fig 5)
Page 3 of 25
9. Using the hammer, secure the sole into the wood by hammering nails into the
three points shown (see fig 6)
10. On other face of the piece of wood, mark out a point along the center of the board,
1 inch from the lengthwise end (see fig 7)
11. Take an eyelet hook and screw it in by hand into the wood at that point, pausing
when only one thread of the screw portion of the hook is visible, and the width of
the hook is parallel to the short edge of the board (see fig 7)
12. Repeat steps 7-8 on the other end of the board
13. Place the cinderblocks approx. 5 feet apart so that the attached pulleys face each
other and are collinear
14. Cut 108” of rope
15. Put one end of rope, end B through eyelet hook
16. Tie 7”of that end of rope into a double knot around the eye of the hook (see fig 8)
17. Put the other end of the rope, end A through the bottom part of the closest pulley,
going into the inside-facing part of the pulley (see fig 8)
18. Continue pulling that end of the rope and put it through the top hole of the same
pulley, going in through the outside-facing part (see fig 8)
19. Continue pulling the free end of the rope, putting it through the top part of the
pulley on the other cinderblock, going in through the inside-facing part (see fig 9)
20. Thread the free end of the rope through the bottom part of the same pulley, going
in through the outside-facing part (see fig 9)
21. Take the free end and tie it to the other eyelet hook in the same way as step 16.
At this point the board should be in between the cinderblocks, with a rope tied to
Page 4 of 25
one end of the board, going through one pulley, back across the board, through
the other pulley, and then tied to the other end of the board (see fig 10)
22. Make sure that the ropes are not twisted and that the side with the sole is facing
down, resting on the ground. The front tip of the sole should be facing to the right
23. Place the dumbbell on top of the board so that it is centered on the board (see fig
24. Tie a piece of rope around the dumbbell and the board in order to secure the
dumbbell to the board (see fig 12)
25. Pull the concrete blocks apart so that they are still in a straight line and so that the
rope system is taut
26. Have the left edge the board 12” away from one cinderblock. Making sure that
the rope and middle line of the board are all in a single straight line, draw a line
12” long on the ground from the other edge of the board so that if the board is
pulled and the eyelet hook follows that line, the board will be moving perfectly
27. The apparatus is now fully set up (see fig 10)
The next step is to set up a simple apparatus to measure bounce height. Needed for this
• A video camera (Nikon Coolpix L18 used)
• A yardstick (Mill Stores yardstick used)
• A wood-paneled floor
Page 5 of 25
• A box of height 7” (New Balance shoe box used)
• A doorframe with molding in a straight line that juts out
• Salad tongs
1. Place the yardstick against the edge of the molding in the doorframe so that the 0”
end is fully against the floor, the side is against the molding, and the 36” end is
facing the ceiling (see fig 13)
2. Place the box on its side 10” from the yardstick, with the face of the box facing
the yardstick (see fig 14)
3. Place the camera on the box so that the lens of the camera faces the yardstick
4. Prepare the salad tongs to use immediately after rubbing the ball
This purpose of this apparatus is to measure the bounce height by film. As the ball is
dropped from the top of the yardstick, it hits the ground and bounces up parallel to and
next to the yardstick. By looking directly at the ball bouncing up alongside the yardstick,
frame-by-frame film analysis can determine exactly how far up on the yardstick the ball
bounced, giving a bounce height.
After setting up these two apparatus, the actual sampling begins.
For the rubbing portion:
Page 6 of 25
1. Place a squash ball under the sole of the shoe under the board so that the edge
of the ball is just under the tip of the front of the sole
2. Make sure that the entire apparatus is lined up as before
3. Standing facing the board so that the sole appears to point right, grasp the
section of the top rope connecting the two pulleys that is to the right of the
4. Pull the top rope to the left so that the board correspondingly moves to the
5. Stop after the board has moved 12”. This will be shown when the rightmost
edge of the board just comes to the end of the 12” line drawn previously.
6. Using the ticks of the clock for timing, make sure that one 12” motion of the
board with the ball under it takes 1 second. This single 1 foot/sec motion is
defined as a single rub.
A ball’s bounce height is measured after 0, 10, 20, 30, 40, 50, 60, and 70 rubs.
The number of rubs is done in a random order, based on the list sorting method in the
While doing a set of rubs, leave the video camera recording so that it does not
have to be turned on before every height measurement. This would cause the ball to cool
while the camera was being turned on.
After a set of rubs:
Page 7 of 25
1. Hold the ball in the salad tongs, applying just enough pressure to stop it from
2. Position the held ball so that it is in front of the yardstick, being as close to it as
possible without touching it. The bottom tip of the ball should line up with the
36” mark (see fig)
3. Drop the ball by opening the tongs
4. Pause the film and go through the part where the ball drops frame by frame. The
maximum height that the bottom of the ball reaches is the bounce height. This
can be seen because the ball is in front of the yardstick and therefore a
measurement can be lined up
This entire process is repeated with each set amount of rubs, and then the ball is set aside
to cool for 10 minutes. That whole procedure is repeated for each of the 10 balls. This
gives a total of 80 data points, 8 heights per ball.
Page 8 of 25
C. Data Presentation
The graph of the data follows:
0 10 20 30 40 50 60 70
The window is:
X is between 0 and 76.9, inclusive
Y is between 3.25 and 11.12, inclusive
Least Squares Regression Line:
Page 9 of 25
ŷ = a + bx
b = r (sy/sx)
a = y − bx
b = .864 (1.236 / 23.057) = .0463
a = 7.793 – (.0463)(35) = 6.1725
The equation for the LSR is:
Y = 6.1725 + .0463x
This equates to:
Hb = 6.1725 + .0463(times rubbed)
This means that a typical squash ball dropped will have a bounce height of around
6.1725 inches before being rubbed at all. As it is rubbed, it is predicted to increase by .
0463 inches of bounce height for every one time it is rubbed.
There are other models as well that may be appropriate for this situation.
In the linear model, the data has an r = .864 and r2 = .747. This says that the data
has a strong linear association, and that the line matches the data reasonably well.
The linear model is shown here: (Window = 0<X<76.9, 3.25<Y<11.12)
Page 10 of 25
There are a few outliers down at the end where X = 0. Those values are
influential to some extent because they are on an extreme end of the data. Despite being
outliers however there were multiple values in that area all from different squash balls,
which could imply that they are actually reasonable data points.
I expect that bounce height should rise indefinitely as times rubbed increases. As
the friction keeps raising the heat of the ball, the pressure of the air inside the ball should
continue to increase, making it bounce higher. The ball may reach a point where it
explodes, or melts, or loses more heat during a rub than it gains. I feel however that
those events would be very large extrapolations of the data, and therefore believe that the
linear model is logically sound.
The residual of the linear model appears somewhat curved but still reasonable.
Window: -7<X<77, -2.54<Y<2.34
Page 11 of 25
Since the residuals appear curved, I decided to attempt a quadratic model, which would
account for heat being lost faster than rubbing would give. This is because the long-term
behavior of a quadratic has it falling back to zero. While I don’t think that is entirely
accurate, that could possibly straighten the curve in the residual plot somewhat.
The quadratic plot follows: (same window as linear model)
The r2 = .77, which means that the model is slightly more appropriate for the data,
although a very small difference.
The residual follows: (same window as linear residuals)
Page 12 of 25
The residual plot now appears more linear, which also supports that this plot is a
better fit for the data. However conceptually the quadratic graph appears to be leveling
off very soon after the data, meaning that at maybe 90 rubs onwards the ball would start
to bounce less, which I don’t think would happen at all.
The last model I attempted was natural log model. I reasoned that it was possible
that at some point rubbing the ball just didn’t make any more difference, and a natural log
that essentially flattens after some time seemed sensible.
The residual plot follows: (Window = -5.9<X<76.9, -2.01<Y<2.23)
This residual plot is visibly nonlinear, and does not appear to show a good
representation of the data.
From these three models, I conclude that the linear is the best option. Not only
does it make sense logically, but a good residual plot, a high r, and a good comparative r2
argue that it is the best-balanced model overall.
Page 13 of 25
Since there is a reasonable limit to how many times a squash player is willing to
rub a ball under his or her shoe, I want to see if a ball is playable at 100 rubs, after which
the player would most likely be bored or irritated.
According to the linear model, y = 6.158 + .0466x
So Hbounce = 6.158 + (.0466)(100)
This is a bounce of nearly 11 inches. If a ball bounces that high from a drop of
three feet, it is nearly in good playable condition, and would possibly require minimal
warmup of other forms. This shows that 100 rubs, an amount that is practical, would be
essentially sufficient to prepare a ball for play.
This study essentially stems from causation. A squash ball is a hollow rubber
ball. It traps heat in the rubber walls effectively, which causes the air inside to heat up.
As shown by the ideal gas law, an increase in temperature of a gas proportionally
increases its pressure. Since internal pressure translates to the force that a ball rebounds
with, higher heat means higher pressure, which should create a larger bounce height. The
rubbing action, using weight on the ball and the tread of a shoe, produces friction.
Friction, when affecting the ball, gives off proportional heat energy. This heat energy is
Page 14 of 25
what pressurizes the ball. The more rubbing of the ball, the more heat from friction, and
therefore a higher bounce height.
F. Error Evaluation
There were many sources of error in this study. For example, when the ball was
rubbed by the moving board, it most likely deviated from a straight path. This deviation
would cause a higher speed of rub because the distance over time would increase. A
higher speed means a higher force of friction, making the ball bounce more than it ideally
Another source of error was the time measurement itself. The timing was done by
listening to a clock ticking because using a stopwatch while working the apparatus would
be difficult for one person to do. However a human cannot accurately and consistently
time themselves by hearing ticking noises. This means that the time per rub could be
either higher or lower than one second, which could lower or raise the speed, changing
friction and bounce height.
When transitioning from rubbing the ball to measuring its height, heat was lost in
the exchange. Although I tried to move the ball from being rubbed to being dropped as
fast as possible, it definitely did cool to some extent, which lowered the bounce height.
Page 15 of 25
The tongs used to hold the ball were also a source of error. The tongs may not
have opened consistently every time, which could impart spin on the ball or slow it down
somehow by neutralizing some gravitational force. The arm holding the tongs was
unlikely to be perfectly steady as well. Both of these factors could either cause the ball to
bounce less or more by not bouncing straight up, or by “kicking” off the ground with
spin, or just accelerating less.
The floor that the ball bounced on sometime appeared to have higher bounces for
no reason, or to impart its own spin on the ball due to traction, or to have dead spots.
This floor definitely changed the heights of some of the bounces in a random manner.
The yardstick used to measure the bounce height was most likely not perfectly
straight up. This means that the height measurements would be higher than they should
be, because an angled yardstick would make a measurement of, for example, 10”, have an
actual height of less than 10” because of the tilt.
The hammering of the concrete anchors into the cinderblocks caused the
cinderblocks to chip away around the hole to some extent. This made it so the pulleys
could not be anchored perfectly and therefore were not perpendicular but rather slightly
slanted. Slanted pulleys would change the way that the rope moved, which would affect
the motion of the board, changing rubbing strength or speed
Pulling the rope itself to move the board meant that vertical force could be
accidentally applied on the board, which would change the weight on it. Changing the
weight would affect friction and therefore bounce height.
Page 16 of 25
When the balls were left for 10 minutes, they probably did not return to their
former temperature, which would mean that further sets of rubs would build of earlier
ones, increasing height.
Confounding variables were also present in the study. Major examples would be
temperature and humidity, which would directly affect the ball. The amount of heat lost
in transition and through inactivity was another. The inherent qualities of each ball could
vary, although all the balls were new. The way that the shape of the sole of the shoe
affected the ball is also very difficult to measure. These and others confounded the study
to the effect that while theoretically causation seemed likely, it could not be practically
The goal of this project was to observe the change in bounce height of a squash
ball as it was rubbed by a simulated shoe with weight of it. A closed rope system with
two anchored pulleys attached to a weighted board was created in order to rub the ball
consistently. The bounce height was measured by dropping a ball against a yardstick and
finding the maximum height that it reached by analyzing film taken of the drop. There
were problems with the setup, especially loss of heat, the other effects of friction on the
coating of the ball, the chipping of the cinderblocks, and the composition of the floor.
The results showed that there is a strong linear correlation between the number of times a
squash ball is rubbed versus how high it bounces. This study would be much more
effective if the rubbing was done by a machine of some sort, if the ball was dropped
immediately after rubbing, and if a laser system was used to measure height instead.
Page 17 of 25