Wikipedia:Reference desk/Archives/Mathematics/2009 September 22

= September 22 =

Bouncing balls
Hi. Imagine a ball propelled upwards from the origin with unit speed, with gravity of magnitude 2 units acting on it. The origin is a hard surface, and a coefficient of restitution between it and the ball of e. On any bounce, the time taken to reach the ground again satisfies:

$$-u = u + at$$

$$2u = 2t$$

$$t = u$$

And after n bounces, the initial speed has been multiplied by e n times:

$$u_n = {e^n}u_1 = e^n$$ Hence: $$t_n = e^n$$

Now, let us consider the total time it spends bouncing. A coefficient of restitution is some number, $$0 \le e \le 1$$. Assuming it is less than 1, then our series of times $$t_0, t_1, t_2, ...$$ are a geometric progression, whose sum tends to a limit:

$$\sum^\infty_{n=0}t_n = \frac{1}{1 - e}$$

In other words, the total time that the ball spends bouncing is finite, even though there are an infinite number of bounces. But how can it ever stop bouncing? The ball's velocity never reaches zero. After each bounce, it is only multiplied by some number, and so merely tends towards zero. 79.78.99.44 (talk) 16:05, 22 September 2009 (UTC)


 * See Zeno's paradoxes. — Emil J. 16:10, 22 September 2009 (UTC)
 * Moreover, see geometric sequence.  Dr Dec  ( Talk )    18:45, 22 September 2009 (UTC)
 * will you please pinpoint what in geometric sequence is of use for the above question, because the article is quite long. Thx! --84.220.118.29 (talk) 19:27, 22 September 2009 (UTC)
 * Not sure I see what the problem is. The ball only covers a finite distance in its infinite number of bounces. The theory is consistent, although at some point this idealised model obviously ceases to be an accurate representation of reality. Gandalf61 (talk) 16:16, 22 September 2009 (UTC)


 * You seem to miss several idelizations here. First, the 'bounce' is not an event but rather a process: it involves interaction between a flexible ball's surface and a hard surface below it. It consists of two phases, namely compressing the ball material (say, rubber) and relaxing it. So it takes some time and space (vertical movement from the moment the ball touches the table until they separate). So an answer to your question could be: 'bouncing stops (or becomes not observable) when the jump height becomes less than the rubber grain size'. --CiaPan (talk) 19:53, 22 September 2009 (UTC)


 * It takes an infinite number of bounces for the speed to reach 0, but as you showed those infinite bounces take only a finite amount of time to elapse in total. So considering the speed over time, it reaches 0 in a finite amount of time.  As Emil J pointed out, it's Zeno's paradox.  Rckrone (talk) 21:58, 22 September 2009 (UTC)


 * ... and, in reality, all but the first dozen (or maybe score) of "bounces" are just increasingly rapid oscillations within the rubber of the ball.   D b f i r s   13:52, 24 September 2009 (UTC)