Activity Title: “Analyzing the Coefficient of Restitution”


The Research Experience for Teachers Program /

Learning Objectives: After completing this lab you will be able to:

  • Use a digital camera, Microsoft Movie Maker and ImageJ to collect and record data
  • Analyze the data on Microsoft Excel and produce a graph of linear regression
  • Determine the coefficient of restitution from data and graphs
  • Accurately estimate the height of the next bounce of the ball

Materials List:

  • 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:

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 =

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 potential energy (GPE) of the ball at each height will be equal. (m is the mass of the ball in kilograms, v is the impact (downward) or thrust (upward) velocity of the ball in m/s, g is the gravitational field constant on the surface of the earth,

9.8 or 9.8 , and h is the height of the initial drop and/or maximum height of each bounce.

KE = GPE, so

mv2 = mgh,

mass divides out, solving for the velocity

v =

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 written as:

CoR =

CoR

Since the mass of the balls cancels out:

CoR

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

Replacing velocities with

CoR

Because the cancels out.

CoR

Through algebraic equality

CoR

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.

Pre-Assessment

  1. Would a ball bouncing on the floor be considered an elastic or an inelastic collision?
  2. Give at least three examples of elastic collisions.
  3. Give at least three examples of inelastic collisions.
  4. Which of the balls will have the greatest CoR, or greatest “bounciness”?

Procedure

Background: In your lab group, work together to develop a plan for conducting the experiment that evenly distributesthe tasks among the members of your group.

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:

  1. Create a folder in your storage location to store the data and analysis for this lab.
  2. Connect the camera to the computer with the USB cable.
  3. Import your video into the folder you created.
  4. Use Movie Maker to open the video file.
  5. 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

to set your scale in Image J before you take measurements.)

  1. Use spreadsheet software to make a data table with average results.
  2. Calculate the average CoR for each ball and predict the height of the 4th bounce.
  3. Repeat steps 5 and 6 to determine the actual height at bounce 4.
  4. Be prepared to share your results with the class.

Ball Identity / Height of Initial Drop
(m) / Height of
1st Bounce
(m) / Height of
2nd Bounce
(m) / Height of
3rd Bounce
(m) / Coefficient
of
Determination / Estimation Height of 4th Bounce (m) / Measured Height of 4th Bounce (m)

Assessment

Results/Conclusions

  1. What was the average CoR for each ball? This is recorded in your table.
  2. Which ball had the largest CoR? Why do you think the CoR was large for this ball?
  1. Which ball had the smallest CoR? Why do you think the CoR was small for this ball?
  1. With what percent error were you able to estimate the height of the next bounce for each ball?

% error = 100 x

  1. 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?

NSF, ASU-RET Coefficient of Restitution – Student Page 1