Bouncing ball impacts

In a recent paper in this journal, Paul Hatchell used a simplified bounce model to describe experimental observations of multiple impacts of a bouncing table tennis ball.¹ The model predicts that the ball will bounce an infinite number of times before it comes to a stop. The same model was used previously, where other balls were predicted to take a finite time to bounce an infinite number of times.^2,3 There are several problems with the model. The first is that the impact of a ball on a horizontal surface takes a finite time, typically a few milliseconds. If a ball bounces an infinite number of times then it will make an infinite number of collisions with the surface so it will take an infinite time to stop bouncing.

Given that a ball will stop bouncing after a few seconds, it is clear that it bounces only a finite number of times. For example, suppose that the coefficient of restitution is 0.5. If the ball is dropped from a height of 1 m, then the height of the bounce will decrease to about 0.01 mm after a few seconds. If the impact time for each bounce is neglected then the bounce height one second later would be $3.7×10−46$ mm, 34 orders of magnitude smaller than the diameter of a proton. Well before that happens, the impact force on the ball will decrease to a value smaller than the weight of the ball, at which point the ball will stop bouncing.

The fate of a bouncing ball can be observed clearly by dropping it on a piezoelectric disk to measure the impact force.⁴ If the ball is dropped from a very small height then the ball will bounce about ten times before it stops bouncing. The force on the disk drops to zero each time the ball bounces in the air, so the flight time in the air can be measured accurately between each collision. When the bounce height is less than about 0.01 mm, the ball no longer becomes airborne between collisions. Instead, the ball vibrates up and down while maintaining contact with the piezoelectric disk, the average force on the disk being equal to the weight of the ball. In less than one second, the vibrations become too small to measure and the force on the disk settles to a constant value equal to the weight of the ball.