The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson

Infinitely Bouncing Balls

A golf ball or a super ball bounces rather nicely; the height of each bounce is a fraction of the height of the previous bounce. For instance, suppose we drop a golf ball from a height of 64 centimeters. Say it recovers three-quarters of its height on the next bounce. Then the next bounce will be (3 / 4) ∗ 64 = 48 cm high. The next bounce will be (3 / 4) ∗ 48 = 36 cm high, and the next (3 / 4) ∗ 36 = 27 cm high, etc.

In the interactive mathlet below, you can vary both parameters: the initial height of the ball and the recovery coefficient--the ratio of the height of a bounce to the height of the previous bounce. Enter the initial height in the textbox (for instance, 300), select the recovery coefficient with the scroll bar, and press Set to set the values of the parameters. Then press Start to drop the ball. Try experimenting with different values of the parameters.

Surprisingly, if each bounce height is a fraction of the previous bounce height, then the time it takes for one bounce is a different fraction of the time taken by the previous bounce. We will investigate this relationship in the next several pages. For more information on the underlying physics, please digress to the discussion of conservation of energy and momentum or the discussion of how balls actually bounce.