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Comment by greenlblue | Hacker News Reader

Same mistake as everyone else. Geometric series means infinitely many bounces and infinitely many bounces means it never stops bouncing. Everyone is making the same logical fallacy.

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I refer you to Zeno's paradox for an example of how a geometric series can allow an infinite number of things to happen in a finite time.

In this case the time taken forms a geometric series, and the total time taken is the sum of that geometric series. Which means that, for the same mathematical reasons that let Achilles catch the tortoise, it stops in finite time.

I am familiar with Zeno's paradox and all other things Zeno and infinite series summing to finite things but there is fallacy here that nobody seems to get. Yes the time taken is indeed a geometric series but if the time taken is a geometric series, an infinite one at that, then that means there are infinitely many bounces, no? So if there are infinitely many bounces how can you claim the ball stops bouncing? You are confounding two things, air time which is indeed finite because it forms a geometric series, and the number of bounces. You can not have an infinite geometric series to calculate the time and only have finitely many bounces.

The ball bounces infinitely many times, in a finite period of time, and then stops! :) You can model this completely, i.e. describe height as a function of time. You can know exactly where the ball is at any particular time, and it follows a continuous curve. So it does make sense as a model, although it is perhaps a bit mind-bending.

You don't give up, do you? An infinite series can have a finite sum. Yes, the ball bounces infinitely often, but it does that in a finite amount of time.

I'm not disagreeing with you. Yes, an infinite series can have a finite sum, no argument there but what I am arguing with is that people are saying the ball stops bouncing after time t = whatever. Everyone is confounding two things here, air time and bouncing. Yes, the air time is finite but the bouncing isn't. So you can't say it stops bouncing after time t = whatever if you calculate t = whatever by assuming infinitely many bounces which is what everyone is doing because they are summing a geometric series where the terms of the series represent the air time of each bounce.

There is a time t such that after time t the height of the ball's bounces is zero.

Personally, I call that "not bouncing".

Ok, now we are in agreement because my definition of "stops bouncing" was finitely many bounces.

Bzzt Still wrong.

If you have infinite bounces, and I ask you "Which number bounce in the series happens at exactly time X?", there is a time for X for which you will not be able to give an answer.

This is because there is a limit for the latest time at which bounces happen.

Yes, even with infinite bounces.

Whatever. I got it sorted out so go bother someone else.

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