How To Actually Solve The Bouncing Ball Puzzle?

I recently came across the bouncing ball puzzle whilst researching literature on the notion of ‘limits’. I found it quite fun to solve. So, I thought you might like it too.

The puzzle statement is quite simple. I have with me an ideally elastic ball. With this ball in my hand, I climb to the top of a ladder that is 4 metres high and drop this ball (from the same height) to a hard floor underneath.

The ball bounces to a height of 3 metres on its first bounce. Then, on its second bounce, it bounces to ¾ of the previous height. Likewise, with each bounce, the ball bounces ¾ of the previous bounce.

Say that this ideally elastic ball is able to somehow overcome friction, etc. So, it is able to bounce an infinite number of times (it’s just a thought experiment, really). Here is the important bit: It comes to rest after a finite period of time after bouncing an infinite number of times.

After the ball has come to rest, your challenge is to figure out how much distance the ball has covered in the air (up and down) after all of its bounces from the moment I let go of it. Do you think you can solve this puzzle?

Hint

If you are struggling and don’t know where to start, you may refer to my essay on the notion of a limit for clues. However, be warned that reading that essay will most likely make this puzzle significantly easier to solve.

Spoiler Alert

If you wish to solve this puzzle on your own, I suggest that you pause reading this essay at this point.

Beyond this section, I will be discussing the solution to this puzzle. Once you are done with your attempt, you may continue reading the essay and compare approaches.

This essay is supported by Generatebg

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Setting Up the Puzzle

The key to this puzzle is the starting point. As soon as you work out where to start, the rest of the puzzle becomes significantly easier.

Let us refer to ‘important bit’ from the introduction once again:

“The ideally elastic ball comes to rest after a finite period of time after bouncing an infinite number of times.”

This ball bounces an infinite number of times, yet it comes to rest in a finite period of time. Do you remember that I mentioned ‘limits’ in the tagline and introduction of the essay? When you combine these observations together, you should be able to imagine where to start.

Have you figured it out yet? If not, no worries. We are looking at a series here. For starters, let us ignore the first drop of 4 metres. On the first bounce, the ball bounces 3 metres up (¾ of 4 metres) and then falls 3 metres down. That would make it 6 metres in total on the first bounce (up and down).

On the second bounce, the ball bounces ¾ of 6 metres (up and down). On the third bounce, it bounces ¾ of the third bounce, and so on infinitely. So, if we say that the total distance covered by the ball (from the first bounce onward) is ‘x’, then our series looks like this:

x = 6 + ¾ *(6) + ¾ * ¾ *(6) + (¾)³ * 6 + …

With this series equation setup, we are now ready to tackle this puzzle!

How to Actually Solve the Bouncing Ball Puzzle

When you look at this series equation, it looks intimidating. How are you supposed to sum up an infinite series? Does it not lead to infinity? Well, that is where a strong understanding of the notion of limits comes in handy.

This is a convergent infinite series. In other words, it will converge on a limit. If you wish to understand the notion of a limit, I recommend that you check out my essay on this topic. But if you are in a hurry, here is a one-liner explanation: a ‘limit’ is another way of denoting the sum of the series at infinity.

Our challenge, then, is to figure out the value of the sum. But how shall we go about this. Well, it is surprisingly easy to converge on the limit for such a series. This particular series decreases by a constant proportion with each term.

There is a simple trick that we can use to compute the sum of such a series. It’s far simpler to show you the trick than explain the details. Check it out:

We know that each term in the series is ¾ of the previous term. So, let us see what happens if we multiply the entire series by its reciprocal (4/3):

So, we have ended up in a situation where the sum of all terms after 8 (the first term on the right-hand side) is equal to the original series variable ‘x’. From this point on, obtaining the solution is a straight forward process.

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The Solution to the Bouncing Ball Puzzle

We start by substituting ‘x’ for the sum of all the terms after 8 and simplify the resulting equation further using algebra:

There we go. We finally have the limit/sum of the series at infinity. It turns out to be 24 metres. But before we celebrate, there is one last pending step.

If you remember, we ignored the drop height from the ladder when we started solving. Now is the time to consider it. So, the total distance travelled by the ball in the air from the moment I let go of it is as follows:

Total distance covered by the ball = x + 4 metres = 24 + 4 = 28 metres.

And that is the solution to our puzzle!

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References and credit: Martin Gardner and Sam Loyd.

Further reading that might interest you:

• Is ‘0.99999…’ Really Equal To ‘1’?
• The Thrilling Story Of Calculus
• What Really Happens When You Invent Infinite Infinities?

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