# B How many times will a ball bounce before coming to a rest

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1. Aug 21, 2016

### Biscuit

I always enjoy physics questions that aren't super complicated and give me something to think about for a day or two. One that has recently come to my head is how many times will a ball bounce before coming to a rest. what formulas should I use to find this out.

2. Aug 21, 2016

### phinds

Have you done any research at all or are you looking to be spoon-fed the answer? This is not one of those forums where you just ask a question and get an answer, you are expected to have made some effort on your own. A simple forum search here would have been a good start

3. Aug 21, 2016

### Staff: Mentor

It depends

A pretty good model of this problem is to say that when the ball it hits the floor with a certain amount of kinetic energy, it compresses and rebounds with almost the same energy for the next bounce. That's "almost the same" because it loses a bit of energy to internal friction with every bounce (otherwise, it would bounce forever). So say the ball looses 5% of its energy on each bounce.... Drop it from a height of 100 centimeters, and its first bounce will rebound to 95 centimeters.... and the bounce after that to 95% of that, which is 90.25 centimeters.... and so on.

That can't be completely right though, because it suggests that although the bounces get smaller and smaller the ball never loses all its energy - every bounce takes out 5% but leaves 95% so the energy never gets to zero. You'll have to cut off the calculation at some point when the energy remaining is so small that you can treat it as if it were zero.

You should be able to work out the necessary formulas for yourself from here - that's a pretty broad hint.

4. Aug 21, 2016

### Krylov

The example of the bouncing ball becomes considerably more complicated (and interesting) when one imagines that the surface (ex. a table) on which the ball bounces exhibits periodic motion itself. The impact map that describes this system is discussed at some length in e.g. Section 2.4 of a recent printing of Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by Guggenheimer and Holmes. Depending on the amplitude and the frequency of the table's motion, this map admits a so-called "Smale horseshoe", an invariant set that is a signature of chaotic dynamics.

Of course, by equating the amplitude of the table oscillations to zero, one recovers the impact map that describes the situation from the OP.

Last edited: Aug 21, 2016
5. Aug 21, 2016

### Biscuit

ya my problem is i measured the first drop to come back at 68 centimeters, but If i just continue at that rate its linear decay. So I'm trying to figure out a model that shows exponential (what I predict it would be). My guess is I need a formula that shows the decrease in kinetic energy and I assume that when kinetic energy is 0 the ball is no longer bouncing. Is that correct? If so, what formula would this be called

6. Aug 21, 2016

### Biscuit

Tried doing my own research, but I'm not that well educated on physics so some of the stuff I find confusing. I tried to experiment on my own but the results were disappointing because I just get stuck.

7. Aug 21, 2016

### Biscuit

Darn, I wish I had that book. I would love to try and read/understand that section looks interesting do you think you could help me out without being too complicated about it? I dropped it at 1 meter and it comes up to 68 centimeters. I also found the density of the ball but I just can figure out how to put everything together.

8. Aug 21, 2016

### PeroK

If you apply the simple model, then you find it never stops bouncing: it always bounces a percentage of the previous bounce and that is never 0. You must, therefore, enhance this model with some additional factors that tell you when there is too little KE for the ball to leave the surface and when effectively the bouncing has stopped.

9. Aug 21, 2016

### rcgldr

With the simple model, the ball bounces an infinite number of times, but only for a finite amount of time (frequency of bouncing approaches infinity). If you consider compression or deformation of the ball, at some point the bounce is less than the compression or deformation, and the bottom of the ball ceases to leave the surface that the ball was bouncing on, while the center of mass continues to oscillate a few more cycles.

10. Aug 21, 2016

### jbriggs444

In the simple model where kinetic energy decays exponentially, velocity (and therefore time until the next bounce) also decays exponentially. Add up the times of all the bounces and it is finite. The ball stops bouncing in finite time. However this does not mean that the number of bounces must be finite.

Edit: rcgldr was quicker

11. Aug 21, 2016

Staff Emeritus
If you tell us the minimum height a ball has to be above the ground before it bounces, we will tell you how many times it bounces when dropped from height h. If your answer is "there is no such minimum height" our answer will be "there will be an infinite number of bounces".

12. Aug 21, 2016

### Biscuit

That question is a part of the problem. I have determine what a bounce is and when its no longer bouncing. The best answer I can come up with is when the ball has 0 kinetic energy. Lets use a height of 2.5 meters.

13. Aug 21, 2016

### Biscuit

All I have now is f(x)=.68x, and I have no idea how to include something such as KE or how to make it exponential.

14. Aug 21, 2016

### jbriggs444

You have that each bounce is 0.68 times as high as the previous. That's not f(x) = 0.68x. That's f(x) = 0.68 f(x-1).

15. Apr 20, 2017

### new2physics

What would happen if you dropped the ball and immediately put a surface above it at, say 1m, so that the ball bounced off the top surface and back down and then up again, etc... how long would it keep bouncing between the two surfaces?

16. Apr 20, 2017

### Staff: Mentor

As this thread dates back more than 8 months, you will probably get no answer anymore.
If you're really interested in the question, please open a new one, including a precise description of the question. Preferably in the homework section and accompanied by data.