When does the ball come to rest?

[Gopal Mailpalli]

Homework Statement

A ball is dropped vertically from a height H on to a plane surface and permitted to bounce repeatedly along a vertical line. After every bounce, its kinetic energy becomes a quarter of its kinetic energy before the bounce. The ball will come to rest after time?

Homework Equations

Kinetic Energy = $\frac{1}{2}mv^{2}$
Velocity = $\sqrt{2gRh}$ , where R is the fraction of height ball reaches after every bounce

The Attempt at a Solution

I could recognize the geometrical series and attempted the solution which gave me a wrong time.

 

[BvU]

I could recognize the geometrical series and attempted the solution which gave me a wrong time

Did you ? And how did you interpret the sum of this series to get the answer ?

 

[Gopal Mailpalli]

Did you ? And how did you interpret the sum of this series to get the answer ?

Excuse me! Some where I did a mistake, i assumed the the initial Kinetic energy be K and next consecutive terms are (1/4)K, (1/4)^2 K ...
This represents a Geometric Progression, this is how I approached the problem.

 

[BvU]

Yes, that's what it says in the problem statement. So what series did you get and how did you work the sum of that series around to the answer you gave ?

There is a simple mistake that's easy to make here, but if I ask specifically I give away the answer, which is a bit against PF culture.

 

[Gopal Mailpalli]

Yes, that's what it says in the problem statement. So what series did you get and how did you work the sum of that series around to the answer you gave ?

There is a simple mistake that's easy to make here, but if I ask specifically I give away the answer, which is a bit against PF culture.

I found the coefficient of restitution which is 0.5, found out the height using the respective COR - Height relation and formed a geometric progression. I calculated the total height and went further to solve the time it takes using basic laws of motion.

 

[PeroK]

I found the coefficient of restitution which is 0.5, found out the height using the respective COR - Height relation and formed a geometric progression. I calculated the total height and went further to solve the time it takes using basic laws of motion.

You have to post your working, equations and all. We can't spot a mistake from a description of what you did.

 

[BvU]

I don't understand what you did. Can you post your working ?
Why bother to find a total height ? The exercise asks for a time.
What total height did you find ? Something like $4h\over 3$ ?

 

[Gopal Mailpalli]

I don't understand what you did. Can you post your working ?
Why bother to find a total height ? The exercise asks for a time.
What total height did you find ? Something like $4h\over 3$ ?

I mean sum of the geometric progression, which i got 4h/3.

 

[BvU]

Nice, but useless: the total time has nothing to do with the time needed to fall 4h/3 (why not ?).

 

[Gopal Mailpalli]

Nice, but useless: the total time has nothing to do with the time needed to fall 4h/3 (why not ?).

Then how do I approach the problem? I don't have an idea.

 

[BvU]

You will also get a geometric series if you calculate the sum of the times the ball needs to get back to the ground after each bounce.

 

[Gopal Mailpalli]

You will also get a geometric series if you calculate the sum of the times the ball needs to get back to the ground after each bounce.

Still, i couldn't solve the problem.

 

[haruspex]

Still, i couldn't solve the problem.

It's a bit easier to see what is going on if you look at the time between bounces.
Between first and second bounce, how long is it in the air?
What about between second and third bounces?
What is the pattern?