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Assume a Ball is bouncing and is travelling in the horizontal direction at constant horizontal velocity v.

Also assume that after each successive bounce, the ball is is in the air for half the time of the previous bounce.

We therefore conclude that each bounce covers half the distance of the previous bounce.

If the initial bounce is 1m, how far will the ball travel?

Method 1: Each bounce is half the previous bounce, giving an infinite geometric series, evaluating to a travel distance of 2m.

Method 2: The ball is travelling at a constant velocity so the ball will go on forever.

What is the explanation behind this apparant paradox?

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How many times will the ball have bounced after twice the duration of the first bounce? What happens after this time? – Lord_Farin Oct 18 '12 at 14:36
That's not consistent with your question. You specified that the ball is is in the air for half the time of the previous bounce; in your comment, it appears that you let the ball bounce for time $t/2$ twice, instead of once $t/2$ and once $t/4$. – Lord_Farin Oct 18 '12 at 14:43
To start with, you only want to count the horizontal distance covered whilst the ball is in the air (you haven't explicitly said that, though). Secondly, the horizontal distance covered whilst the ball is in the air is bounded above by 2m and you can get arbitrarily close to this by measuring the distance covered over a longer time. This is just a rewording of this: – Sam Jones Oct 18 '12 at 14:48
I think the problem is that the ball can't keep moving in the air beyond 2 seconds. Assume the velocity is 1m/s. After 1 second it has travelled 1m. Now the next bounce takes 0.5 seconds and it travels a further 0.5m, right? So after 2 seconds it has travelled 2m. But now the bounces are so small that it's rolling. Yes, the horizontal speed is still 1m/s but it's no longer covering that distance whilst in the air. So the total horizontal distance covered whilst in the air is 2m. – Sam Jones Oct 18 '12 at 15:17
If you're worried about that problem, this one might give you bad dreams: There are point-like billiard balls of equal mass on the $x$ axis, one at every coordinate $2^{-n}$ for all $n\in\mathbb N_0$. You give the first one a push to make it move in the negative $x$ direction with unit velocity. It comes to a halt as it bumps into the next one and sets it going. After one time unit, all the balls have been bumped and come to a halt again -- but where did the energy and momentum go that you'd imparted to the first ball?! – joriki Oct 18 '12 at 16:30
up vote 0 down vote accepted

I think the "paradox" arises from the fact that both 1 and 2 are correct, in the following sense.

The total horizontal distance covered whilst the ball is in the air is 2m. But after 2 seconds (assuming a horizontal speed of 1m/s) 2m has been covered whilst the ball is in the air and from now on the ball is rolling on the ground. The total horizontal distance covered is unbounded but to get the unbounded distance you have to (also) consider the distance travelled whilst the ball is on the ground.

I have made some assumptions here, like the (vertical) height of each bounce is strictly less than the height of the bounce before it but this usually happens in real life (on earth, anyway).

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You could add the assumption that the height of each bounce is constant and it is only the frequency of each bounce which increases. I'm not sure what could be said about the ball's position after 2 seconds in that case, I'm not even sure that the question would make sense. I think the main point to this question and answer is that you have to think about what assumptions or restrictions you're putting on the problem and this problem appears difficult because not all the assumptions have been stated. – Sam Jones Oct 18 '12 at 16:13
Thanks Sam, your answer appears to have resolved the apparent paradox. – Mew Oct 18 '12 at 21:19
Could it be resolved if the balls height is assumed to be fixed for all bounces? – Mew Oct 19 '12 at 2:50
I think we'd need to figure out what assumptions we would be using, we obviously wouldn't be modelling real life if we allow the height of each bounce to be fixed so what are we doing, what assumptions do we have and what reasoning is valid. What axioms are we using? It's possible that those two assumptions (constant horizontal velocity and constant height for each bounce) would lead to a contradiction and that no meaningful answer could be obtained, if this were the case then it wouldn't be possible to resolve the paradox as in an inconsistent system every statement is true (or provable). – Sam Jones Oct 19 '12 at 9:33

This is a paraphrasing of Zeno's paradoxes.

To explain a bit further, In say the ball travels at velocity $v$, then in $t$ time, it has traveled a distance $x \cdot t$. However, each bounce, it travels half the distance, so let total distance be $D$, however, then total time is $D/v$ which is a contradiction, you said it goes on for ever, not a fixed amount of time. Yes, however, in that fixed amount of time, it travels a distance $D$.

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