Successive Bounces of Ball Paradox

Question

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?

Answer 1

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).

Answer 2

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 $D$.