Exercise 4: Coefficient of restitution
The last way that we will quantify the energy change as your ball bounces is by determining a parameter known as the coefficient of restitution. This term literally means “a number or ‘factor’ that measures how much of something is returned”. In this case, it measures how much speed is returned to an object after a collision. Mathematically, it is defined as the ratio of the speeds,
where 
is the speed of the ball right before it collides with the ground, and 
is the speed of the ball right after the collision, as it leaves the ground, and the | | represents taking the “absolute value” of the velocities. So if you calculate e = 0.5, that means the ball leaves the ground half as fast as it initially hit the ground. Half of the speed was “lost” during the collision with the ground.
Exercise 4.1 (2 marks)
i) What value of e would correspond to energy being completely conserved?
ii) What value of e would correspond to energy being completely lost?
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Exercise 4.2 (3 marks)
Use your data from Exercise 2 where you bounced your ball on the hard surface to plot a new graph of e vs Collision Number. Set your y-axis to go between 0 and 1.0.
Helpful hints: Be careful to use the correct speed for 
when calculating each new value of e. For example, the first e we can calculate is that of the 2nd bounce. The speed the ball hits the ground with before the second collision is the same speed the ball leaves the ground with after the first collision; the first speed recorded in the table from 2.2. The speed the ball leaves the ground with after the second collision is then the speed between the second and third collisions; the second speed in the table.
Submit your completed graph with the correct y-axis for your first experiment (hard surface).
Think about it: Choosing to compare the speed of your ball between collisions may seem like an odd choice, since earlier we were more concerned with how much energy the ball retains between bounces. However, this choice is a practical one since 1) speed is often relatively easy to calculate and 2) in many cases, the speed before and after a collision will be directly related to the energy returned to an object. However, you can equivalently write e directly in terms of the kinetic energy K.
Exercise 4.3 (1 mark)
Exercise 4.4 (1 mark)
Use the equation for e and your plot from 4.2 to determine how much (what fraction of) energy is lost from collision to collision. What is that fraction in terms of e?
Hint: For a constant e value, the same fraction of energy is lost on each collision. Was your plot from 4.2 relatively constant?
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Before you continue!
Before continuing, be sure you have completed (4.1) to (4.4), which will be graded and submitted through Crowdmark.
