Zeno Bouncing

If you drop a rigid ball onto a rigid surface, it eventually stops bouncing without ever bouncing the last time! This is called Zeno behavior, after Zeno’s most famous paradox (Achilles and the tortoise).

Suppose we drop the ball from a height of 1m, and the coefficient of restitution is 0.9. So every bounce is followed by another bounce of 90% its height, ad-infinitum. Yet after 16.7 seconds the ball comes to a complete rest! Yes, there is always another bounce after every bounce, and yes, there are no more bounces after 16.7 seconds.

Proof

Under the conditions described above, the height attained by the ball after the n’th bounce is

$\large{h_0 = 1}$

$\large{h_{n>0} = 0.9h_{n-1}}$

Solving the recursive equation above, we find that

$\large{h_n = 0.9^n}$

after the n’th bounce. Falling freely from rest in gravity g, the ball travels a distance of

$\large{h = \frac{1}{2} g t^2}$

in time t. Therefore it takes the ball a time of

$\large{t = \sqrt{2h/g}}$

to fall to the ground from rest at height h, and the same amount of time to rebound from the ground to rest at height h. Therefore, the period of time between the n’th bounce and its following bounce, is

$\large{t_n = 2\sqrt{2h_n/g} = 2\sqrt{2\times 0.9^n/g} = 2\sqrt{2/g}\sqrt{0.9}^n}$

The total time that the ball spends bouncing, starting from its first bounce, is therefore

$\large{T = \sum_{n=1}^{\infty}t_n = 2\sqrt{2/g}\sum_{n=1}^{\infty}\sqrt{0.9}^n = 2\sqrt{2/g}(\frac{1}{1-\sqrt{0.9}}-1) \approx 16.7}$

Since every bounce is followed by another bounce (of 90% its height) there is never a last bounce, yet after 16.7 seconds there is no more bouncing whatsoever!