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Comment by paulofisch | Hacker News Reader

This assumes that each bounce takes the same time to complete instead of also tending to zero.

If you expect that each bounce also takes a smaller amount of time to complete, a series of infinite bounces takes a finite amount of time and therefore a finite distance.

At the point in time and distance where the bouncing ends, the forward motion of the ball (unimpeded by friction) continues in a slide along the ground.

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Ah but it takes infinite amount of time for the ball to stop and start rolling on the floor and infinite amount of time means the ball bounces infinitely often so the ball never stops to bounce and never rolls along the ground. It is easy to calculate how long the ball stays in the air on each bounce a formula from high school physics tells you that the potential energy of an object is m x g x h so if you know the potential energy then you know how high the ball is and if you know how high the ball is then you know how long it will take for it to fall to the ground but no matter how long you wait the ball will have some potential energy left so it will be bouncing no matter how long you wait.

In addition to the fallacy that the ball will keep bouncing for an infinite amount of time, I'd like to add that the ball will never roll even when it stops bouncing, because of the frictionless environment, it will slide instead.

Unless of course it was rolling in the first place and I missed/misread it.

This is true but I was just looking for the 'infinite number of bounces in a finite time' part; after that it would slide or roll without bouncing, or sit still if there had been no horizontal component initially.

The bounces are an infinite series, each with a time attached to them tending to zero.

Time itself is not a series, this is where the fallacy creeps in.

A infinite series can have a finite sum.

True but then in what order are you calculating things. In order to have an infinite series for the air time to sum you must assume there are infinitely many jumps so if there are infinitely many jumps then the ball never rolls on the floor by definition. But then if you say you sum the infinite series of time intervals and the ball stops at that time then you don't have infinitely many bounces because if there were infinitely many bounces the ball would not roll on the floor. So you are missing something somewhere.

http://en.wikipedia.org/wiki/Geometric_series

See also: http://en.wikipedia.org/wiki/Zenos_paradox (Hint: It's not really a paradox.)

This doesn't address the fact you are making a logical fallacy. You can't make a calculation assuming the ball bounces infinitely often and then after the calculation go back and say the ball stops bouncing because it invalidates your original calculation.

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