- #1

DeldotB

- 117

- 8

## Homework Statement

Hello!

A ball is dropped and falls to the floor (no horizontal velocity). It hits the floor and bounces with inelastic collisions. The velocity after each bounce is [itex] \mu [/itex] times the velocity of the previous bounce (here [itex]\mu [/itex] is the constant of restitution). The velocity of the first bounce is just [itex] v_0[/itex]. Find the time it takes for the ball to stop bouncing.

## Homework Equations

Newtons Laws

## The Attempt at a Solution

Well:

I know this will turn into a convergent geometric series. I am just trying to find what that series will look like.

using the formula [itex]h=x_0+v_0t+1/2at^2 [/itex] its easy to see that the time it takes for the ball to reach the ground is:

[itex] h=1/2gt^2[/itex] so [itex] t=\sqrt{2h/g}[/itex].

Using energy I also have: [itex] mgh=1/2mv_0^2[/itex] so [itex]gh=1/2v_0^2 [/itex]

Time for the next bounce: well, the ball now has an upward velocity of [itex] \mu v_0[/itex] and the height of the first bounce is [itex] h'=\mu v_0t-1/2gt^2[/itex].

I realize this is a simple problem but for some reason I'm not seeing it. If I solve this equation for time, (using quadratic formula) the resulting series for the times [itex]t=t_1+t_2+... [/itex] isn't geometric and actually quite complicated. Is my approach right?