Find the Value of k: Ball's Momentum and Height

[Robin04]

Homework Statement

We release a ball from a height h and it bounces for a time t. What is the value of k (the quotient of the ball's momentum before and after collision with the ground)?

Homework Equations

The Attempt at a Solution

I'm kind of lost here. :/
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[PeroK]

How many times do you think it bounces before it comes to rest?

 

[Robin04]

How many times do you think it bounces before it comes to rest?

Well, according to my equations it never comes to rest because k is a quotient and we have to multiply the speed infinite times for it to reach zero. Somehow t has to define the end of the movement, but I don't see how I could do that.

 

[PeroK]

Well, according to my equations it never comes to rest because k is a quotient and we have to multiply the speed infinite times for it to reach zero. Somehow t has to define the end of the movement, but I don't see how I could do that.

With the simple mathematical model it bounces an "infinite" number of times, but as each bounce takes less time than the last, that doesn't mean the bouncing lasts an infinite time.

 

[Robin04]

With the simple mathematical model it bounces an "infinite" number of times, but as each bounce takes less time than the last, that doesn't mean the bouncing lasts an infinite time.

So, if I understand it well, the t(n) function that I wrote for the total time of the movement has to have a limit in n->infinity

 

[PeroK]

So, if I understand it well, the t(n) function that I wrote for the total time of the movement has to have a limit in n->infinity

Yes, mathematically, take the limit as $n \rightarrow \infty$.

If you are practically minded, in reality the ball bounces a finite number of times, so the mathematical limit gives an approximation of reality!

 

[Robin04]

Yes, mathematically, take the limit as $n \rightarrow \infty$.

If you are practically minded, in reality the ball bounces a finite number of times, so the mathematical limit gives an approximation of reality!

I found a solution. I think I'll do an experiment to check if I got it right. Thank you very much! :)

 

[PeroK]

I found a solution. I think I'll do an experiment to check if I got it right. Thank you very much! :)

You can always check the two extreme cases:

As $t \rightarrow \infty$ your formula should have $k \rightarrow 1$. And, if $t = t_0$ then you should get $k =0$.

 

[Robin04]

You can always check the two extreme cases:

As $t \rightarrow \infty$ your formula should have $k \rightarrow 1$. And, if $t = t_0$ then you should get $k =0$.

Yes, my solution gives that :)

 

[Robin04]

By the way what other ways are there to describe this problem mathematically?

 

[PeroK]

By the way what other ways are there to describe this problem mathematically?

The next thing to do would be to investigate the coefficient $k$. Is it really a constant or does it depend on the velocity of impact?

That said, you probably still end up with an infinite sum.

Or, you could consider the deformation of the ball. At some point it is no longer bouncing and the motion has reduced to internal damped oscillations.