Calculating Total Distance Traveled by an Elastic Ball with Infinite Bounces

[danni7070]

Homework Statement

When dropped, an elastic ball bounces back up to a height three-quarters of that from which it fell. If the ball is dropped from a height 2 m and allowed to bounce up and down indefinitely, what is the total distance it travels before coming to rest?

Homework Equations

I think I have to use Partial sums of geometric series.

If r is not equal to 1 then


$$S_n = a + ar + ar^2 + ... ar^n-1 = a(1-r^n)/1-r$$

The Attempt at a Solution

It's really easy to understand the question, but setting it up mathematecally is other story.

I tried to do 2 + 3/2 + 9/8 + 27/32 + 81/128 + ... + 3n/4n where a_1 = 2 and a_2 = 3/2
but this is not leading me to an final answer.

Trying to use the equation above saying a = 2 and r = 3/4

I get the final answer 8 m But the answer is 14 m.

What is the correct setup?

 

[otg]

If you use only 3/2+9/8+27/32+...+3n/4n where a_1=3/2, you get half the distance, save 2m [since it bounces up AND down the same distance every time except the first drop]
S_n=8 according to your calculations, so the new sum would be S_n-2

Thus you get the total distance as 2+2(S_n-2) = 2+12=14

 

[danni7070]

duh?

Thank you for this eye opening reply.

Of course!