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Simple bouncing ball puzzle, with $50 prize

sam.ai.ki

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Re: Simple bouncing ball puzzle, with $50 prize

#51

Post by D_Alex »

btilly wrote: post #21
pero wrote: post #10

Where's the spoiler?

Here is a spoiler. Energy = force * distance After each bounce you are left with 60% of its energy, so it comes back up 0.6 times as high as the previous bounce. Distance falling in time t is proportional to the square the time, so each bounce takes sqrt(0.6) times as long as the previous bounce did. Thus the timing of the bounces forms a geometric series. It is well known that the sum of such a geometric series is 1…

I suspect this is what the intended solution to the puzzle is... Now for a follow up: Instead of assuming no friction, assume no slippage. What happens now?

Re: Simple bouncing ball puzzle, with $50 prize

#52

Post by growt »

rbabich wrote: post #22
bravura wrote: post #12

edit: I thought this over more, here is a new answer. Spoiler: it bounces an infinite amount of time in a finite space. Actually, the horizontal distance it travels is proportionately decreased with the decreased time in the air. If the ball is now in the air for k it bounces an infinite amount of time in a finite space. --- Old, wrong answer. Spoiler: It keeps going to the right to infinity, just at a height that is…

Of the responses so far, this one is closest to being correct. Rather than "asymptotically approaching zero," however, the height of the bounce will quickly converge precisely to zero. Assume that the previous bounce (up and back down) took time t. Then the ball will stop bouncing after time t/(1-sqrt(0.6)) ~ 4.4t. After that, the ball will simply continue moving ("rolling") to the right. This follows from summing th…

no rolling without friction.

Re: Simple bouncing ball puzzle, with $50 prize

#54

Post by sswam »

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.

Re: Simple bouncing ball puzzle, with $50 prize

#55

Post by sswam »

burgerbrain wrote: post #6

I'd love to know how many people have swamped him with the correct answer so far. Any junior high physic student could answer this.

My experience has been that although the topics needed are covered in high school physics and math (limits), most people did not get it right.

Re: Simple bouncing ball puzzle, with $50 prize

#56

Post by sswam »

mfukar wrote: post #49
greenlblue wrote: post #16

Earlier quoted context omitted.

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 po…

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.

Re: Simple bouncing ball puzzle, with $50 prize

#58

Post by sswam »

greenlblue wrote: post #34
btilly wrote: post #30

Earlier quoted context omitted.

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…

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.
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