Mechanics, acceptable method?

[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.

 

[Nickie]

I would suggest the following method to solve this problem:

First, we need to determine the initial velocity of the ball before it hits the ground for the first time. This can be done using the conservation of energy equation, where the potential energy at the top of the building is equal to the kinetic energy just before the first bounce. We can then use this initial velocity to calculate the maximum height the ball will reach after the first bounce.

Next, we can use the geometric series equation xn = arn-1 to find the number of bounces needed for the ball to reach a height of 2.44m above the ground. We know the initial height and the common ratio (0.9) and we can solve for the number of terms (bounces) needed.

Alternatively, we can also use the equation for the sum of a geometric series Sn = a(1-rn)/(1-r) to directly calculate the number of bounces needed to reach a certain height.

Both of these methods are more efficient and systematic than guessing and checking. They also take into account the final height we need the ball to reach, making the solution more accurate.