31.1: The Coefficient of Restitution

We can compute a dimensionless number called the coefficient of restitution that measures how elastic a collision is. The coefficient of restitution \(\epsilon\) is defined as

\[\epsilon=\frac{p_{f}}{p_{i}},\]

where \(p_{f}\) is the final momentum of the body, and \(p_{i}\) is its initial momentum. For a perfectly elastic collision, \(\epsilon=1\); for a perfectly inelastic collision, \(\epsilon=0\); and for an inelastic collision, \(\epsilon\) is some number between 0 and 1 , being closer to 1 the more elastic it is.

An easy way to measure the coefficient of restitution is to drop a body on a flat surface. The height to which the body rebounds will determine the coefficient of restitution. Suppose the body is initially dropped from a height \(h_{i}\), and rebounds to a height \(h_{f}\). By conservation of energy, the kinetic energy of the body just before it hits the floor is \(m v_{i}^{2} / 2=m g h_{i}\), so its velocity is \(v_{i}=\sqrt{2 g h_{i}}\). Similarly, just after the collision the velocity is \(v_{f}=\sqrt{2 g h_{f}}\). Therefore the coefficient of restitution \(\epsilon\) is

\[\epsilon=\frac{p_{f}}{p_{i}}=\frac{m v_{f}}{m v_{i}}=\frac{v_{f}}{v_{i}}=\frac{\sqrt{2 g h_{f}}}{\sqrt{2 g h_{i}}}\]

or

\[\epsilon=\sqrt{\frac{h_{f}}{h_{i}}}\]

The coefficient of restitution is just the square root of the ratio of the rebound height to the initial height. \({ }^{1}\)

Now let's first look at a mathematical analysis of collisions in one dimension, where the anaysis is simpler. At the end of the chapter we'll examine collisions in two dimensions.