If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Infinite geometric series word problem: bouncing ball

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.3 (EK)
,
LIM‑7.A.4 (EK)

## Video transcript

let's say that we have a ball that we drop from a height of 10 meters and every time it bounces it goes half as high as the previous bounce so for example you drop it from 10 meters the next time it's peak height is going to be at 5 meters so the next time around on the next bounce let me draw R injure and the next bounce the ball is going to go 5 meters this distance right over here is going to go it's going to be 5 meters and then the bounce after that is going to be half as high this is going to go two and a half meters is two and a half meters and it's just going to keep doing that and it's just going to keep doing that so it's going to go two and a half meters right over here and what I want to think about in this video is what is the total vertical distance that the ball travels so let's think about that a little bit so it's first going to travel its first going to travel 10 meters straight down so it's going to travel 10 meters just like that and then it's going to travel half of 10 meters twice it's going to go it's going to go up five meters up half of 10 meters and then down half of 10 meters so then it's so it's going to go it's going to go two times let me put it this way too so it's going to go each of these is going to be 10 meters 10 meters I don't have to write the unit's here let me take the unit's out of the way so let me write the clear so the first bounce once again it goes straight down 10 meters then on the next bounce it's going to go up ten times one half and then it's going to go down ten times one half ten times one half notice we just care about the total vertical distance we don't care about the direction so it's going to go up ten times one half up five meters and then it's going to go down five meters this it's going to travel a total vertical distance of 10 meters five up and five down now what about on this jump are on this bounce I should say well here it's going to go half as far as it went there so it's going to go it's going to go times 1/2 squared up and then 10 times 1/2 squared 10 times 1/2 squared down and I think you see a pattern here this looks an awful lot like a geometric series an infinite geometric series it's going to just keep on going like that forever and ever so let's try to X me let's try to clean this up a little bit so it looks a little bit more like a traditional geometric series so if we were to simplify this a little bit we could rewrite this as 10 plus plus 20 20 to the tie or 20 times 1/2 to the what first power plus 10 1/2 times 1/2 squared plus 10 times 1/2 squared is going to be 20 times 1/2 times 1/2 squared and we'll just keep on going on and on so this would have this would be a little bit clearer if this were a 20 right over here but we could do that we could rewrite negative 10 oh sorry we could write 10 as negative 10 plus 20 and then we have plus all of this stuff right over here plus all of this so I'm just copy and paste that copy and paste so plus plus all of this right over here and we can even write this first we can even write this 20 right over here is 20 times 1/2 to the 0 power 20 times 1/2 to the 0 power plus all of this so now it very clearly looks like an infinite geometric series we can write our entire sum and maybe I'll write it up here since I don't want to lose the diagram we could write it as negative 10 that's that negative 10 right over here plus the sum from k is equal to zero to infinity of 20 20 times our common ratio our common ratio to the K power so what's this going to be what's this going to turn out to be well we've already derived in multiple videos already here that the sum of an infinite geometric series so the sum from k equals 0 to infinity of a times R to the K is equal to a over 1 minus R so we just apply that right over here this business right over here is going to be equal to 20 over 1 minus 1/2 which is the same thing as 20 over 1/2 which is the same thing as 20 times 2 or 40 so how what's the total vertical distance that our ball travels it's going to be negative 10 plus 40 which is equal to 30 meters our total vertical distance that the ball travels is 30 meters
AP® is a registered trademark of the College Board, which has not reviewed this resource.