Each time a ball bounces, it loses e.g. 40% of its vertical kinetic energy. Can you describe what happens to the ball?

(Clarification for the pedantic: the ground is flat, and each part of the ball’s path is a parabolic arc. Don’t consider friction, atoms, relativity, quantum, etc! Just describe qualitatively what happens to the ball. There’s no need for calculations or numbers, although they might help support your reasoning. You will likely need some knowledge of physics and math.)

I invented this little puzzle and I know the answer. I’ve asked people to solve it, and no one has ever given me the right answer. So, just for fun, I’m offering a $50 prize for the first person who can answer it correctly before Monday 15th November, and I will be pleased to meet you! I can send you the prize by PayPal, or in person if you live in Melbourne. It’s a nice puzzle; the answer is surprising and interesting.

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You say it looses 40% of its vertical energy, but you do not say about the horizontal component of its velocity. If that is unaffected (you do say ignore friction) then it’s just going to keep bouncing to the right at constant horizontal speed, each time with a lower amplitude (as it’s only got 60% of the vertical energy of the last bounce). I think (if my memory serves me right) the line described by the top of the bounce becomes asymptotic.

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

In pure mathematics from the Geometric series web page:

S-2/5*S=1

Therefore S=5/3, so the total distance covered by bouncing will be 5/3 of the distance of the first bounce and the ball will bounce forever in infinitely small values.

In practice there is a minimum amount of energy required for a ball to bounce and it will roll when the bounce doesn’t have enough energy, rather than the height/length of each bounce being a multiple of the previous bounce, it will be a multiple minus some fixed value.

My entry might be disqualified because I learned all this a couple of decades ago but forgot enough that I had to look up Wikipedia. 😉

Congratulations to everyone who figured it out, and especially to Glenn from Alaska who was the first to email me with the correct answer, and wins $50. In his words:

“The ball makes an infinite number of progressively smaller bounces in a finite amount of time, and then proceeds to slide (roll?) along the ground at a constant speed.”

I was just looking for “infinite number of bounces in a finite amount of time/distance”.

I think it’s a nice puzzle, because it illustrates Zeno’s ‘paradox’, with a simple model of an everyday occurrence. The answer makes sense, but it is not obvious unless you understand limits. I thought of this puzzle while playing with a pool cue, you can really hear/feel them bouncing faster and faster.