The Zeno's paradox and one of its variations

physics baws

Hi,

I was learning about collisions, and I stumbled upon this materials, which is interesting because a guy who wrote it gave this interesting example. He was talking about "somewhat inelastic collisions", as he calls them, and he gave an example of a ball bouncing of the floor. Here are the snapshots of that part of lecture

http://i.imgur.com/OVkS2.png

http://i.imgur.com/g6DTC.png

This part, when he posed the question
intrigued me the most. It reminded me of Zeno's paradox, which I never quite understood.

The way I'm seeing it, as far as the thing about distance goes, mathematically speaking, the ball should bounce less and less, so the total distance traveled converges to some final value (but mind you, it should never reach that value, only gradually become closer and closer to it).

Now, as far as the thing about time goes, I am completely baffled by it, I cannot make myself understand it. Again, mathematically speaking, the formula for total time converges to some value. That should mean that the total time will become closer and closer to that value but it should never reach it. This is the logic I picked while learning integrals and derivatives and the limit process behind them. In an example he gave, that time was 1.36 seconds.

But we all know that those 1.36 seconds must and will pass, so we will reach this total time value. And when it does, the ball will come to rest, which essentially means that it did reached the total distance value. So clearly, my thought process is wrong. How is it possible then for the ball to bounce infinitely many times?

Maybe someone can shed some light on my troubled mind?

Much obliged.

Simon Bridge

The ball bouncing is a dissipative system with a lot of things going on at the same time. The model used (in the lecture for eg) to describe it's behavior while the bounces are large does not apply for very small bounces ... consider what happens if, say, the amount of energy the ball can have or lose is quantized - so there is a minimum [itex]\Delta[/itex]E at each bounce.

On each bounce the ball loses an integer multiple of [itex]\Delta[/itex]E with the classical behavior acting as an envelope determining the multiple. At some point the ball will have only a single quanta left, lose it, and come to a standstill.

the_emi_guy

I have attached a paper for you reading pleasure. Let's assume we are talking about the idealized mathematical situation (no quantization of energy, no loss). Time does not "approach" 1.36 seconds. Time flows normally and the number of bounces approaches infinity at 1.36 seconds. The ball cannot still be bouncing at > 1.36 seconds because it has already bounced an infinite number of times at 1.36 seconds.

Attached Files

bouncing_ball.pdf (473.6 KB, 21 views)

Neandethal00

Anytime I see such never ending actions of something, I have one suspicions.
1) The mathematical model is not exact. It lacks something.

It may be in the 'e'.

the_emi_guy

The OP is not trying to model a real, physical, bouncing ball which would obviously include *lots* of complexity. He is trying to understand the time limit which occurs even in the mathematically idealized case.

Simon Bridge

Is there a time limit in the mathematically idealized case? I thought the problem was that there is a time-limit observed even though the idealized case suggests that the bounces continue forever (if very very small).

Though - in that case we could point out that there is a smallest bounce that can be seen and when the ball's bounces are smaller than that it will appear to have stopped bouncing. IRL bounces would get small enough to get washed out by thermal vibrations - everything is jiggling all the time.

So - how do you know the ideal ball has stopped bouncing?

Neandethal00

First of all e=.5 is very low for the example.

The results OP got t=1.36 sec, and L=1.67 m, are for infinite bounces.

Actual time and length when we will see the ball comes to a rest is lower than 1.36 sec and 1.67 m.

A.T.

Yes. See the paper that emi_guy has posted. Equation 3.
No, the idealized case, where the ball loses the same fraction of KE in each bounce, suggests an infinite number of bounces in a finite time.

sophiecentaur

There is no problem if you remember that the mathematical model of a situation like this is convergent. If the model isn't then it can't be an appropriate model.
The great thing about Maths is that it takes care of the difficulties that you can get into when you try to appreciate things only in an arm waving way. The arm waving will often let you down because waving arms don't always follow the rules properly and can lead to wrong conclusions.

azizlwl

The dichotomy paradox
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
—Aristotle, Physics VI:9, 239b10

In Zeno's paradox, the distance/time divisions are not constant.
Thus the rabbit will never catch up with the tortoise.

In calculus the division can be small as we can but the division are constant.
Thus there will be limit.

Simon Bridge

There are physical models which have the features being examined in a less trivial way - I'm talking about Feynman sums in QED and the self-energy of, say, an electron. There is nothing in the math to suggest that the sum should converge ... it would be a nice confirmation of the model if it did, but we cannot tell - maybe it doesn't. We know it must do, for a correct theory, because electrons have a finite mass.

But, since we know the mass of an electron already we can cheat - and replace the troublesome sum with the empirical mass and be done with it. The hope is that the model is close enough to being right for it all to come out in the wash and so far it seems to.

It's a bit like that (not much - just a bit) with the various sums above too - we just take it for granted that it's the way to do things because the methods are 1000s of years old. Like when we set up the division of the intervals in calculus (they don't have to be equal BTW - you just have to makes sure you don't count bits more than once, it's just that equal size intervals makes the math easier) you just make sure the size of each interval multiplied by the number of intervals ends up as the total length you need ... the condition under which this works is called "convergence".

Mind you - I am aware reading through the previous that I have an unconventional viewpoint on this.

A.T.

We take it for granted that dividing a finite interval into sub-intervals doesn't change the length of the original interval. That kind of follows from the definition of division.

sophiecentaur

I think he was making the distinction between 'equal' intervals and 'non-equal' intervals. Most of the time we just make the assumption that we are dealing with all the same step sizes but it is by no means necessary - as anyone involved with data compression of pictures and sound will instantly catch onto.
Interestingly, our mental awareness of objects and processes uses far from regular 'step sizes'. Then along comes Excel, and we're all into equal increments. Brains are far more efficient - missing out the bits that don't really count and concentrate on the important bits - then building a good spatial or temporal model of the experience.

Simon Bridge

Yeah - we take it for granted that some "sensible" way of dividing time or distance up into infinitesimals is being used ... a technique that was invented and developed to address the kinds of problems Xeno etc were talking about.

Notice that in order to divide the interval, you need some trick to figure out how long the interval is first ... so you are sort-of starting with the answer.

I had to repeatedly analyse wave-functions by computer for my thesis ... and I used an unequal step size - smaller steps where the wavefunction oscillated more - for that. Saves a lot of time.

You just have to be careful in how you do the dividing.

Of course, from the POV of the brain designer, and model is "good" if it increases your chance of getting laid. Which explains a lot of the weird systems out there but does not explain the sex lives of math geeks.

sophiecentaur

Problem is that ten times nothing is still nothing!!

the_emi_guy

Folks, I think we have strayed significantly from the OP's question.
Let me offer up a simple thought experiment that illustrates the "infinite bounces in finite time" concept.
Suppose I invent a ball that bounces in a special way. Specifically, each bounce lasts exactly 1/10 of the time of the previous bounce. First bounce time is 3/10 seconds.
Let's add up the times:
3/10 + 3/100 + 3/1000 + 3/10000 + ...
Expressing this in decimal:
0.33333...
We all recognize this as *exactly* 1/3. This ball will not be bouncing after 1/3 of a second no matter how many times we let it bounce.