## Bounce, part 3

Today we’ll introduce the principle of Galilean relativity and use it to continue out examination of the tennis ball/basketball experiment. I don’t want to talk about relativity much, because I’m too stupid to say anything new or interesting. Instead, I’ll just start using it.

We’ll think about the experiment as a series of steps. First, the tennis ball and basketball are dropped, one on top the other, with a tiny gap in between them. Next, the basketball hits the ground, bounces, and changes direction. Then, the tennis ball, coming down, hits the basketball, now coming up, and bounces off. The question we will try to answer is, “Once the tennis ball bounces off the basketball in our experiment, was is the maximum speed it could have going back up?” That will be enough for this post.

We begin by dropping the tennis ball and basketball from some distance above the ground. Just as we drop them, they aren't moving.

From our previous investigations, we already understand the first step of the basketball bouncing by itself. Ideally, it can bounce back up with exactly the same speed it had coming down. Let’s call that speed $v$.

The tennis ball and basketball fall together, and pick up some speed v just before reaching the ground.

In the next step, the basketball is coming up at a speed $v$, and the tennis ball coming down at speed $v$ when they collide. To understand this case, we’ll begin with a simpler one.

The basketball bounces off the ground, changing direction.

The tennis ball bounces off the moving basketball, shooting back up at an unknown speed.

If we were in an elevator moving up at speed $v$ right at the moment of the collision, we would have a different opinion on the speeds of the basketball and tennis ball. The basketball is going up the same speed we are, so from our perspective it isn’t moving at all.

We go back to just before the tennis ball bounces off the basketball, and imagine we're riding up in an elevator at the same speed as the basketball.

On the other hand, by looking at things from the point of view of the ground, we see that the distance between us on the elevator and the falling ball is shrinking at a rate $2v$. So, if we believe that we in the elevator aren’t moving, then the tennis ball must be falling towards us at speed $2v$, to keep the gap between us and the tennis ball shrinking at the same rate.

To look at things from the elevator's point of view, we add a downward velocity v to everything in the scene, including the ground.

Now the tennis ball bounces off the basketball. If the basketball is much larger than the tennis ball, it is essentially like bouncing off a brick wall, or the ground, and the tennis ball reverses direction keeping the same speed. So from your point of view in the elevator, the tennis ball is going up at speed $2v$.

The tennis ball bounces off the stationary basketball, reversing its direction and keeping the same speed, all viewed from the elevator's reference frame.

Finally, we return to the point of view of the ground. We know that the distance between you and the tennis ball is increasing at a speed $2v$. Since you’re going up at $v$, the tennis ball must be going up at $3v$. So in the ideal case, where the basketball is so much larger than the tennis ball that it isn’t deflected at all, and the tennis ball’s collisions don’t lose any energy, the tennis ball can shoot upward with three times the velocity it picks up by falling. This shows us why it can bounce higher than it came from. It bounces up going faster, and so reaches a greater height. But it also tells us that there’s still a maximum height. By making the basketball bigger and bigger, and pumping it up better, we’ll still only approach a certain limit where the tennis ball bounces back at $3v$, so we can’t launch the tennis ball into outer space this way.

To go back to the ground's reference frame, we add a speed v in the upward direction to everything, and see that the tennis ball goes up at speed 3v in the ideal case.

Before continuing, I’d like to look at where the principle of relativity came into this discussion. Most of what I’ve said, I hope, seems obviously true. It is based on the argument “if the distance between A and B is changing at a certain rate, it will change at that rate even if you begin moving”. For example, if you are playing catch, and you throw a ball away from you at 20 mph, then someone driving past in a car still thinks the difference in speeds between you and the ball is 20mph. This isn’t relativity – it’s simple kinematics. The only place we needed relativity, the idea that physics is the same in different reference frames, was in saying that in the elevator frame, the tennis ball bounces off the basketball reversing its direction with the same speed, just as it would if bouncing off a stationary basketball on Earth.

In fact, in the theory of special relativity, it’s this physics principle that holds, and not the kinematics of switching between reference frames (but that’s a different story).

In the next post, we’ll look at what today’s conclusion means in terms of how high the tennis ball goes.

### 4 Responses to “Bounce, part 3”

1. Bounce, part 4 « Arcsecond Says:

[…] Bounce, part 4 By Mark Eichenlaub Previous parts: 1 2 3 […]

2. 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 top […]

3. Rosina modipana Says:

Thank for a good,professional and scientific respond it helped me to complet my assignment but i would have apreciate it more if you were lit more specific. Keep on doing the great work

4. Bounce, part 1 « Arcsecond Says:

[…] 2 part 3 part 4 part 5 part […]