The Research Experience for Teachers Program
http://www.cs.appstate.edu/ret
Subject Area(s): Physics, Physical Science, Mathematics
Computer Science Tools: Digital Camera, Movie Maker, ImageJ, Microsoft Excel (or equivalent
spreadsheet software)
Activity Title: “Analyzing the Coefficient of Restitution”
Grade Level: 9 - 12
Time Required: 60 min.
Recommended Group Size: 2-3
Summary: In this activity, students use Microsoft Movie Maker and ImageJ to analyze the coefficient of
restitution or “bounce ratio” of athletic balls. Coefficients of restitution are calculated by
measurements obtained from ImageJ comparing the height of drop versus the height of bounce for
several ball bounces. An Excel spreadsheet averaging of the coefficient of restitution per bounce should
determine the accurate projected height of a later bounce.
Computer Science Connection: Introduction to a digital camera, Movie Maker, ImageJ and Excel
spreadsheet for data analysis and curve fitting.
Keywords: coefficient of restitution, ratio, bounce, restitution, elastic collision, inelastic collision,
conservation of energy
Educational Standards: (optional)
Pre-Requisite Knowledge: Students should have experience working with ImageJ, Movie Maker,
graphing in Excel, and video recording. Students should also be familiar with solving and manipulating
algebraic equations.
Learning Objective: To calculate and compare the coefficient of restitution between different athletic
balls.
Materials List:
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Digital camcorder
Tripod for digital camera (suggested)
Meter stick, measuring tape, or reference size object (ie. 3x5 index card)
Several spherical athletic balls (suggested: tennis, (non-white)golf, handball, rubber ball)
Introduction/Motivation:
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The “coefficient of restitution” (CoR) is a measure of the elasticity of a collision. For a collision with
perfect conservation of mechanical energy (an “elastic collision”), CoR = 1. The coefficient of restitution
is a convenient way of specifying how much kinetic energy in a collision is transferred to internal energy.
An elastic collision occurs between two objects where the total kinetic energy of the two objects after
the collision is equal to their total kinetic energy before the collision. The coefficient of restitution is
defined as the ratio of the energy after a collision (bounce up) to the energy of the falling ball before the
impact of the collision: CoR =
𝐸𝑏𝑜𝑢𝑛𝑐𝑒 𝑢𝑝
𝐸𝑖𝑚𝑝𝑎𝑐𝑡 𝑎𝑡 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
. The impact for these collisions will be with the floor.
Since the floor does not move, all the kinetic energy is restricted to the velocity of the rebounding ball.
During the collision of small objects, kinetic energy is first converted to potential energy associated with
a repulsive force between the particles that make it up. This potential energy is then converted back to
kinetic energy and the ball bounces. The kinetic energy (KE) of the ball on impact and the gravitational
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potential energy (GPE) of the ball at each height will be equal. KE = GPE, so 2 mv2 = mgh, v = √2𝑔ℎ
In an inelastic collision, some of the kinetic energy of the ball is converted to light, heat or sound upon
impact. Because momentum (p) is conserved in all collisions, the coefficient of restitution can also be
𝑝𝑏𝑜𝑢𝑛𝑐𝑒 𝑢𝑝
𝑚𝑣𝑏𝑜𝑢𝑛𝑐𝑒 𝑢𝑝
𝑣𝑏𝑜𝑢𝑛𝑐𝑒 𝑢𝑝
written as CoR =
=
=
since the mass of the ball
𝑝𝑖𝑚𝑝𝑎𝑐𝑡 𝑎𝑡 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
𝑚𝑣𝑖𝑚𝑝𝑎𝑐𝑡 𝑎𝑡 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
𝑣𝑖𝑚𝑝𝑎𝑐𝑡 𝑎𝑡 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
cancels out. Unless appropriate technological equipment is available, the velocity of the ball before and
directly following the impact of collision is difficult to determine, however, the height of the original
drop and each successive bounce can be measured.
CoR = 𝑣
𝑣𝑏𝑜𝑢𝑛𝑐𝑒 𝑢𝑝
𝑖𝑚𝑝𝑎𝑐𝑡 𝑎𝑡 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛
=
√2𝑔ℎ𝑢𝑝
√2𝑔ℎ 𝑑𝑟𝑜𝑝
=
√ℎ𝑢𝑝
√ℎ𝑑𝑜𝑤𝑛
ℎ𝑢𝑝
= √ℎ
𝑑𝑜𝑤𝑛
, because the √2𝑔 cancels out.
The coefficient of restitution is the ratio of the square root of the heights of a falling object, from when
it hits a given surface to how high it rebounds. In laymen's terms, the coefficient of restitution is a
measure of bounciness.
Procedure
Background: Students should work together to evenly distribute the tasks and to develop a plan
for conducting the experiment.
Preparation: The instructor should ensure ImageJ, Movie Maker, and spreadsheet software are
available on all machines.
Lab Activity:
1. Place the reference item and ball in view of the camera to set the scale later in ImageJ.
2. Drop the ball and allow it to fall at least 4 times while recording bounces on the camera
Data Analysis Procedure:
Create a folder in your storage location to store the data and analysis for this lab.
Connect the camera to the computer with the USB cable.
Import your video into the folder you created.
Use Movie Maker to open the video file.
Use Movie Maker to take snapshots of find at the height for the original drop and height of the
first three bounces. You will use this to analyze the initial angle of each trial.
6. Use Image J to measure the height of the ball initially and for the first three bounces. (Make sure
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2.
3.
4.
5.
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7.
8.
9.
10.
to set your scale in Image J before you take measurements.)
Use spreadsheet software to make a data table with average results.
Calculate the average CoR for each ball and predict the height of the 4th bounce.
Repeat steps 5 and 6 to determine the actual height at bounce 4.
Be prepared to share your results with the class.
Assessment
Pre-Assessment
1.
2.
3.
4.
Would a ball bouncing on the floor be considered an elastic or an inelastic collision?
Give at least three examples of elastic collisions.
Give at least three examples of inelastic collisions.
Which of the balls will have the greatest CoR, or greatest “bounciness”?
Results/Conclusions
1.
2.
3.
4.
What was the average CoR for each ball?
Which ball had the largest CoR? Why do you think the CoR was large for this ball?
Which ball had the smallest CoR? Why do you think the CoR was small for this ball?
With what percent error were you able to estimate the height of the next bounce for each
ball?
5. What are the regulated CoR for soccer balls used in the World Cup? Tennis balls used at the
US Open, Wimbledon or the Olympics? Basketballs used in the NBA?
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