Problem — Sum of bounce times

Problem

A ball with coefficient of restitution $e < 1$ starts from the ground with an upward velocity $v_0$ and bounces repeatedly. Show that the sum of the times of all the bounces is finite and equal to $T_{\text{tot}} = 2\tau_0/(1-e)$, where $\tau_0 = v_0/g$. What mathematical result guarantees that the sum is finite?

Solution

Duration of the first flight. The ball starts upward with velocity $v_0$; it rises, stops, and falls back, returning to the ground with the same speed $v_0$ (symmetric motion). The time to rise is $v_0/g = \tau_0$, and the same holds for the descent, so the first flight lasts $t_0 = \frac{2 v_0}{g} = 2\tau_0$

Effect of the coefficient of restitution. At each bounce, the take-off velocity is reduced by a factor $e$: after the first impact the ball sets off again with $v_1 = e\,v_0$, after the second with $v_2 = e^2 v_0$, and in general $v_n = e^{n} v_0$. Since the duration of each flight is proportional to the take-off velocity, the $n$-th flight lasts $t_n = \frac{2 v_n}{g} = 2\tau_0\, e^{n}$

Sum of the series. The total time is the sum of all the flights: $T_{\text{tot}} = \sum_{n=0}^{\infty} 2\tau_0\, e^{n} = 2\tau_0 \left(1 + e + e^2 + e^3 + \cdots\right)$ The bracket is a geometric series with ratio $e$. Since $e < 1$, the series converges and its sum is $1/(1-e)$: $T_{\text{tot}} = 2\tau_0 \cdot \frac{1}{1-e} = \frac{2\tau_0}{1-e}$

Mathematical result that guarantees finiteness. It is the convergence of the geometric series for $|e| < 1$: although the bounces are infinite in number, their times decrease in geometric progression quickly enough for their sum to be a finite number. The ball performs infinitely many bounces in a finite total time.

$\ev{T_{\text{tot}} = \dfrac{2\tau_0}{1-e}, \qquad \tau_0 = \dfrac{v_0}{g}}$

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Topics: Experimental method Skills: Dimensional analysis