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Infinite geometric series word problem: bouncing ball

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Watch Sal determine the total vertical distance a bouncing ball moves using an infinite geometric series. Created by Sal Khan.

Video transcript

Let's say that we have a ball that we dropped 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 its peak height is going to be at 5 meters. So the next time around, on the next bounce, let me draw in that same orange color. And the next bounce the ball is going to go 5 meters. This distance right over here is going to be 5 meters. And then the bounce after that is going to be half as high. So it's going to go 2 and 1/2 meters. And it's just going to keep doing that. So it's going to go 2 and 1/2 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 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 up 5 meters, up half of 10 meters, and then down half of 10 meters. Let me put it this way. So each of these is going to be 10 meters. Actually, I don't have to write the units here. Let me take the units out of the way. Let me write that clear. So the first bounce, once again, it goes straight down 10 meters. Then on the next bounce it's going to go up 10 times 1/2. And then it's going to go down 10 times 1/2. Notice we just care about the total vertical distance. We don't care about the direction. So it's going to go up 10 times 1/2, up 5 meters, and then it's going to go down 5 meters. So it's going to travel a total vertical distance of 10 meters, 5 up and 5 down. Now what about on this jump, or 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 10 times 1/2 squared up, and then 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 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 20. 20 times 1/2 to the first power, plus 10 1/2 times 1/2 squared plus 10 times 1/2 squared is going to be 20 times 1/2 squared, and we'll just keep on going on and on. So this would be a little bit clearer if this were a 20 right over here. But we could do that. We could write 10 as negative 10 plus 20, and then we have plus all of this stuff right over here. Let me just copy and paste that. So 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 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 0 to infinity of 20 times our common ratio to the k-th 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 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.