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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

#31

Post by greenlblue »

paulofisch wrote: post #27
greenlblue wrote: post #25

Earlier quoted context omitted.

No, you are making the same mistake as some of the other people. You are summing a geometric series so you are saying the ball bounces infinitely often and the time it takes for the ball to bounce infinitely often is blah. But if it bounces infinitely often then it never stops to roll on the floor because if it did stop and roll on the floor in a finite amount of time then you wouldn't have an infinite series to sum…

The series is infinite, but the sum of the series can still be finite. These two things are not at odds.

Ya, and what are the terms in the series representing? Is it air time of each bounce? Your calculation makes it clear that it is. So you are saying you are calculating the air time for infinitely many bounces, the key word here is infinite, i.e. the ball bounces up and down, up and down infinitely often. So if the ball bounces infinitely often how can it stop and roll on the floor because infinitely many bounces means not stopping after finitely many bounces and rolling on the floor. Your calculation for the time is confounding two things, air time and bouncing. You can calculate the air time assuming infinitely many bounces but then you can't go and claim that the ball stops bouncing after t = whatever because you calculated t = whatever assuming the ball never stopped bouncing and then after the calculation went back and changed your assumption, that is a logical fallacy if I ever saw one.

Re: Simple bouncing ball puzzle, with $50 prize

#32

Post by cperciva »

greenlblue wrote: post #26
btilly wrote: post #21

Earlier quoted context omitted.

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…

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.

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.

Re: Simple bouncing ball puzzle, with $50 prize

#33

Post by cperciva »

btilly wrote: post #23
cperciva wrote: post #8

Earlier quoted context omitted.

I'm more curious to know how many people have sent in incorrect answers. My immediate guess disagreed with my mathematics.

My immediate guess agreed with my mathematics. But I have the advantage that a variant of this problem had occurred to me some 20 years ago, and I worked it out. In fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely. There may be some residual vibration that is not apparent, but it sure seems to act like…

In fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely.

Indeed, I've done that -- but my intuition told me that it was stopping due to friction, not due to the exponential decay of an infinite number of bounces. :-)

It's oddly disappointing to realize that the model actually acts more or less the same as the real world for once.

Re: Simple bouncing ball puzzle, with $50 prize

#34

Post by greenlblue »

btilly wrote: post #30
greenlblue wrote: post #26

Earlier quoted context omitted.

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.

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.

Re: Simple bouncing ball puzzle, with $50 prize

#35

Post by mkarmac »

greenlblue wrote: post #31
paulofisch wrote: post #27

Earlier quoted context omitted.

The series is infinite, but the sum of the series can still be finite. These two things are not at odds.

Ya, and what are the terms in the series representing? Is it air time of each bounce? Your calculation makes it clear that it is. So you are saying you are calculating the air time for infinitely many bounces, the key word here is infinite , i.e. the ball bounces up and down, up and down infinitely often. So if the ball bounces infinitely often how can it stop and roll on the floor because infinitely many bounces mea…

1) The ball stops bouncing in a finite amount of time.

2) The ball bounces an infinite number of times.

3) This is not a contradiction, no more than the idea that a projectile passes through an infinite number of spatial points in finite time. Please go and read about geometric series and Zeno's paradox on wikipedia, as another commenter has already suggested.

Re: Simple bouncing ball puzzle, with $50 prize

#36

Post by greenlblue »

rbabich wrote: post #28
greenlblue wrote: post #24

Earlier quoted context omitted.

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…

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.

Re: Simple bouncing ball puzzle, with $50 prize

#37

Post by greenlblue »

cperciva wrote: post #32
greenlblue wrote: post #26

Earlier quoted context omitted.

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.

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.

Re: Simple bouncing ball puzzle, with $50 prize

#39

Post by cperciva »

greenlblue wrote: post #37
cperciva wrote: post #32

Earlier quoted context omitted.

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 infini…

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".

Re: Simple bouncing ball puzzle, with $50 prize

#40

Post by cperciva »

retube wrote: post #38

The ball has horizontal energy as implied by the parabolic arc. Assuming no friction the ball's horizontal velocity will be constant, hence it will just roll away at whatever initial horizontal velocity it had.

just roll away

Technically, it will slide, since without friction it can't obtain any angular momentum.

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