Modeling the bounce of a gas-filled ball

The coefficient of restitution $\epsilon$ characterizes the energy retained when a ball bounces, and can easily be measured in an ``at home'' experiment. For thin-walled gas-filled balls such as basketballs, we construct a simple two parameter model to describe how $\epsilon$ changes as a function of the ball's internal pressure. A comparison with data shows good agreement. With additional assumptions, the model also predicts how $\epsilon$ changes as a function of temperature. A comparison with tennis ball data shows surprisingly good agreement.