## Bounce, part 1

When I was a kid, my dad sat my sisters and I down one day to inquire why the ceiling in our garage was peppered with blurry brown splotches. I speculated that perhaps it was camouflage – the ceiling needed to blend into the herd. He didn’t buy it, and I accused him of not paying much attention to what the signs at the zoo had said about the ungulates when we visited two weeks before.

Juvenile creativity aside, the reason for the disfigured ceiling was that someone at school had told me about that thing with the basketball and the tennis ball. You know, the one where you hold a tennis ball directly over a basketball, drop them both straight down, and the tennis ball goes flying away crazily. You have seen that, right? It looks like this:

That experiment is why the ceiling was messed up: we had dirty tennis balls.

At that time, I didn’t know why the experiment should work. Why does the tennis ball bounce higher than it was dropped from? Not even a super-bouncer ball does that. Evidently it has something do with the basketball being so much larger than the tennis ball. So, what if I got something even bigger than a basketball? If I had a ball ten times as big as the basketball, and dropped the tennis ball on top of that, then how high would it blast up? Could I send something up to outer space this way? What if I use three balls – say a ping pong ball on top the tennis ball? What if I use four, or five, or infinity? What if I use two basketballs rather than a basketball and tennis ball? Can I still make one shoot up super-high then? What if I reverse the order, dropping the basketball on top the tennis ball? Will it bounce higher or lower than it was dropped from? Will it go higher than it would if the tennis ball wasn’t even there? Is there a way to make the basketball and tennis ball trick work even better? Does it matter that they’re ball-shaped, rather than bouncy cubes, for example?

The remarkable thing is that although I didn’t understand my experiment at the time, I could have, in principle, if I’d had someone around to explain it. That’s what I’m going to try to do now – answer the questions from the last paragraph with a series of thought experiments, rather than with forces, potential and kinetic energy, and a bevy of equations. It’ll take a few posts to do it, but the fundamental ideas are simple: that a ball bounces off a wall as fast as it came in, that you won’t get carsick if the ride is smooth enough, and that gravity pulls evenly.

Let’s begin by thinking about bouncing in the simplest context: bounce the basketball all by itself. When we do this in real life, it turns out that the basketball bounces back up, but not quite all the way. If we drop it from waist height, it might rise back to thigh-level, then bounce again, going to knee level, and so on.

Why lower than the original height? Why can’t it bounce higher? If I pump it up more, it bounces better. So if I pump it up high enough, could it bounce up higher than it began? The answer is no. You might get it to bounce better by pumping it up, but each bit of pumping would give you less and less benefit, so that you’d never quite get the ball to come all the way back up.

Suppose the ball bounced a little higher than it was dropped from. Then if it bounced a little higher on the next bounce, and a little higher the bounce after that, it would bounce to arbitrarily great heights, all by itself. Eventually it’d go to the moon. This is absurd, so the basketball can’t keep bouncing higher. But that doesn’t mean it can’t bounce higher for some particular height. Why can’t it, for example, bounce from 1 meter to 1.1 meters, then to .8 meters, then to .9 meters, then to .6 meters, etc. in some strange up-and-down pattern? Or why not bounce higher than dropped for all heights under one meter, and lower than dropped for all heights above one meter? Or higher than dropped on the first bounce, and lower than dropped on all subsequent bounces?

All of these scenarios are contrived, intuitively wrong, and even silly. Intuition can fail though, as it did for me when seeing the tennis ball bounce away so high. So it’s interesting that what intuition says about the single bouncing basketball can be derived from a simple principle.

Here is the first bit of physics: We assume that perpetual motion is impossible. By this, I mean that it is impossible to build a cuckoo clock that keeps on going forever without drawing in energy from the outside (for example, a solar-powered cuckoo clock with a battery to store energy for the night might run indefinitely, but if we block out the sun with a large wall of Coppertone bottles, the clock will wind down quickly and halt.)

There’s nothing special about my example of a cuckoo clock. If you could build something else, like a train, that ran forever, you could use its motion to power a cuckoo clock. And if you could build a cuckoo clock that ran forever, you could make a million copies of it and use their combined motion to power a train. So the cuckoo clock is just a convenient example for a general idea that you can’t build any machine that goes on forever.

One way to make a cuckoo clock go is to let a basketball fall, and use this to drive the clock. What you do is tie a string to the basketball, then tie the other side of the string to the gears of the clock. As the clock ticks, the basketball pulls on the gears, forcefully turning them a little bit. The gears deliver a tiny push to the pendulum each swing to keep it going. This way, the clock can run for a long time. Eventually it wears down though, because the basketball falls all the way to the floor. Then there’s nothing pulling the gears around, so the gears can’t power the pendulum, and friction slows everything to a stop. (For more on how a pendulum clock like a cuckoo clock works, see the How Stuff Works article. The important point for this argument is that you can power the clock by carefully and slowly lowering the basketball.)

This leads into why the basketball can’t bounce higher than we dropped it from. Say we drop the basketball from 1 meter, and it bounces back up to 1.1 meters. Then we can concoct a scheme to run a cuckoo clock forever. Start by lifting the ball up to 1.1m. Tie the string to it and use it to make the cuckoo clock run. When the ball falls to just 1 meter, take off the string and drop the ball. It bounces up to 1.1 meters again. Catch it there and re-attach the string. Let the ball fall back to 1 meter, running the clock… This creates an infinite process that’s self-contained. The system is the clock, the basketball, the floor it’s bouncing off, and the Earth, which creates gravity. Nothing outside this system is supplying any energy, and yet the cuckoo clock can run forever. If we assume that such perpetual motion is impossible, then it’s also impossible for a basketball to bounce higher than it was dropped from, no matter what the height.

In the next post, we’ll look at why the basketball doesn’t bounce back up to exactly the same height, which will lay the foundation for understanding the interaction of the basketball and tennis ball.

### 5 Responses to “Bounce, part 1”

1. Ian Says:

What, no part 2? I was hoping to use your derivation to build some insight into solving the following problem, originally inspired by a discussion that took place at one of the Dabney Pumpkin Drops, and now revived since I’m supposed to teach gas laws in the spring:

If you dropped 1000 super-bouncy-balls en masse off the roof of a tall building (say 100m high) what is the probability that, following the ensuing series of collisions at ground level, one ball shoots out fast enough to crack someone’s skull? I guess we’d have to define “fast enough to crack someone’s skull,” for this to make sense, so let’s just assume that corresponds to some kinetic energy threshold.

The simplest starting point seems to be to treat the balls like atoms of an ideal gas and assume that just before hitting the ground they all have the same kinetic energy $E\left( h\right)$ (determined by the height $h$ of the building), which is then distributed among them by numerous collisions (assumed to be completely elastic) until the distribution of ball speeds corresponds to a Maxwell-Boltzmann distribution with average kinetic energy $\left\langle E\left(h\right)\right\rangle$. Then the number of balls dropped simply influences the probability that at least one ends up above the kinetic energy threshold for cracking someone’s skull.

However, one realistic issue immediately makes the problem more complex: the collisions aren’t perfectly elastic. If I recall correctly (from Mr. Wizard, no less), the very best rubber balls return to about 90% of their initial height after bouncing off a hard surface, meaning 10% of their energy is dissipated inelastically. But a 10% loss (or even a 1% one) becomes significant as soon as you need to have many collisions take place in order to go from a narrow kinetic energy distribution to a broad Maxwell-Boltzmann-esque one. Plus I imagine the outliers at the tails of the distribution have probably undergone (on average) a greater number of collisions than the median.

Anyway, that’s as far as I’ve ever gone in my thinking. Perhaps you’ll get further.

2. Mark Eichenlaub Says:

Hey Ian,

WordPress parses $\LaTeX$ by typing “$lat.ex $”, (without the period in the middle of “latex”. I just put that there so it would display) in case you want it for future comments.

I’d take the same approach to that problem as you outlined. Another problem is that if the collisions are perfectly elastic, then no speed is sufficient to crack your skull, since cracking your skull takes energy.

I’m not sure whether the balls at the tail ends of the distribution should be expected to have had more impacts. If a ball is on the tail end, then impacts are more likely to bring it back towards the center of the distribution. Off the top of my head it’s a hard question.

When the collisions are inelastic, I suppose we could try to estimate the number of collisions necessary to come to equilibrium, and then estimate what the remaining energy would be. I might think about it a little more. You could calculate, for example, the distribution in energies after a single collision, and see how wide that is.

3. Bounce, part 4 « Arcsecond Says:

[…] part 4 By Mark Eichenlaub Previous parts: 1 2 […]

4. Bounce, part 5 « Arcsecond Says:

[…] part 5 By Mark Eichenlaub This post is a digression from the topic of the previous parts (1 2 3 4). We’ll move away from discussing how high a tennis ball should bounce when dropped on […]

5. Tennis Ball and Basketball « Kappa Alpha Theta Summer 2011 Says:

[…] I wrote this out a while ago.  You can check it out <a href=”https://arcsecond.wordpress.com/2009/11/07/bounce-part-1/“>here</a&gt;. […]