How Many Bounces Until the Ball Reaches 2.44m?

[BOAS]

Homework Statement

A ball is dropped from rest from the top of a 6.10m building, falls straight down, collides inelastically with the ground and bounces back. The ball loses 10% of it's K.E every time it hits the ground. How many bounces can happen and the ball still reach a height of 2.44m above the ground.

Homework Equations

mgh = 1/2 mv²

x_n = ar^n-1 (finding a term in a geometric series)

The Attempt at a Solution

I can come to an answer easily enough using the two equations stated above and the ideas of gravitational P.E being converted to K.E (so mass cancels out) and constructing a geometric series that uses r = 0.9 to accommodate the energy loss.

I have to essentially just guess terms until I'm in the right region and then increase or decrease my term until I reach the right answer. This seems somewhat messy to me, using trial and error.

Is there a cleaner method, that will use the information of the final height I need it to reach?

I'm just curious,

thanks!

 

[Enigman]

$x_n=ar^{n-1}$
$r^{n-1}=x_n/a$
$(n-1)logr=log x_n-loga$

 

[BOAS]

$x_n=ar^{n-1}$
$r^{n-1}=x_n/a$
$(n-1)logr=log x_n-loga$

Ahh, thank you!

 

[Chestermiller]

$$x_n=x_0r^n$$ But, it's much easier just to do 1 bounce at a time. It couldn't take more than about 10 bounces with r = .9. Or, do it using: every two bounces is 0.81, every 3 bounces is 0.729.