# Why does the speed of a ping pong ball increase when the space in which it can bounce decreases?

I was playing table tennis the other day when I my ball fell off the table. I placed my paddle above it in order to slow it down, and then I brought the paddle to the ground so that the ball would come to a stop. A diagram of what I did is below: Why did the velocity of the ping pong ball increase so much at the end? I did not apply much force while lowering the paddle, so I didn't think it was because I applied a greater force to the ball.

• Related problem: How many bounces would occur between paddle and ball without friction or gravity? It's fascinating: youtu.be/HEfHFsfGXjs Jul 13, 2020 at 9:20
• How did you measure the velocity?
– pipe
Jul 13, 2020 at 15:45
• This is a neat question, and it's important to notice that the increased mean speed is due to reduced time with high Potential Energy in the system. It's also important to notice that this is completely different from the "pingpong ball" analogy of gas molecules getting hotter when the constraining box gets smaller(increased pressure applied to the system). Jul 14, 2020 at 12:52
• @CarlWitthoft Errr... the actual speed increase at any given height, do to work done by the paddle on the ball, is exactly the same mechanism as adiabatic heating !? (The first effect in Guy's answer.) Jul 14, 2020 at 16:53
• @CarlWitthoft I was only taking issue with your "completely"; the paddle adding kinetic energy is a factor. Jul 15, 2020 at 17:04

There are three parts to the phenomenon, two real and one illusory.

While you are lowering the bat, its relative velocity to the approaching ball increases that little bit. The ball bounces off it that bit harder, gaining twice that extra velocity relative to the floor. Repeat for several bounces and the difference might become noticeable. This is one real part.

The other arises because the ball slows as it rises and accelerates again as it falls. Lowering the bat cuts out the bit where it slows down, so even though the local speed at any given point may not increase, the average speed does increase.

The illusion is to do with the scale and period of the bouncing. As you lower the bat, the period of each bounce shortens, increasing the frequency of the bouncing. This combines with the shrinking scale to create an illusion of going faster. (Credit to user Accumulation for pointing this one out in another answer).

A similar illusion takes place when you watch a scurrying insect. Compare say a horse, a cat and an insect walking along. The big horse seems slow and lazy, the tiny insect in a mad hurry, the cat somewhere in between. But in reality the horse is going the fastest and the insect the slowest.

• You can avoid the real part by only moving the paddle when the ball is away from it. Has this ever been done experimentally to see if the illusory part alone triggers the observation? Jul 12, 2020 at 20:29
• @JohnDvorak - would be difficult to do as the frequency of rebounds increases...
– tom
Jul 13, 2020 at 0:41
• I think that you're missing another real part. When the paddle is moved lower down, the ball doesn't loose as much kinetic energy to potential energy on its way up, and therefore has a higher speed when it hits the paddle. (I now see that Accumulation hos made a post about this.) Jul 13, 2020 at 12:47
• @md2perpe Agree that aspect is missing here, so I have to -1 this answer. The ball's average speed actually does increase when you lower the paddle, even if you ignore the additional impulse given by the paddle moving downward. The period of each bounce shortens not only because the distance is shrinking, but also because the ball has a higher average speed over that distance - half the distance is covered in less than half the time. It is not an illusion that the ball's average speed is increasing! Jul 13, 2020 at 13:54
• @GuyInchbald Really? I don't see it. Jul 13, 2020 at 20:02

Guy Inchbald mentions the slight force of the paddle on the ball and that velocity relative to the separation increases. For the latter, there's a further issue that every time the ball bounces, it makes a noise, and that noise becomes more frequent as the distance shortens, which increases the perception of speed.

Also, I believe there is a third phenomenon: when a ball bounces, its speed decreases as it rises. By cutting off the high part of its bounces (which is when it is moving the slowest), you are restricting it just to the fast part of its bouncing, increasing the average speed of the ball.

• Indeed. The ball is never allowed to decelerate to 0m/s on its way up before falling back down again. The paddle sends the ball back towards the ground before the ball is able to reach a slower velocity. Jul 12, 2020 at 23:50
• And I thought I had it thought all through! Kudos. Jul 13, 2020 at 8:55
• Assuming a perfect system with no energy change due to air resistance, paddle collision, etc, in the absence of a paddle, wouldn't the ball falling back from it's peak just reach the same velocity it would have had at the same point (or height) it would have bounced off the paddle? Jul 13, 2020 at 14:33
• @anjama When the ball reaches the surface of the table, it's going the same speed regardless of the paddle position. But what matters here is the average speed, not the maximum speed. The upper half of the trajectory has a lower average speed than the bottom half of the trajectory (the speed goes to 0 at the trajectory's peak), so by cutting out the upper half, you're increasing the overall average speed. Covering the bottom half of the distance takes less than half the time. Jul 13, 2020 at 14:40
• @anjama I think the point is that as the paddle approaches the floor it restricts the systems maximum gravitational potential energy. Since all of the energy is restricted to kinetic energy, the system exhibits continuous high velocity in the ball Jul 13, 2020 at 14:40

# Because Ping-Pong balls are elastic

And in elastic collisions, kinetic energy is conserved. When your ball is allowed to bounce freely, it reaches a speed of zero at the top of its arc. At that point, all of its kinetic energy has been converted to gravitational potential energy. The opposite is true at the ground, where all its gravitational energy has been converted into kinetic energy. For a perfectly elastic collision, what you are doing when you lower the paddle is truncating the bouncing ball's arc (with a corresponding increase in frequency). In an ideal case, where you add no extra energy to the ball, the ball's speed at each height does not increase, but the ball's average speed increases drastically, because the ball is moving so much faster in the free arc when it's closer to the ground.

You will note that this principle works just as well in reverse: Bounce a ping-pong ball off the ground, then catch it with your paddle. Even though the ball should bounce up back off your paddle about as high it seems to be bouncing a lot slower because it is now spending all its time in the slower part of the arc. This reverse application demonstrates some of the limits to this logic, however-- in practice the air resistance to a slowly-moving ping-pong ball means you will probably see a notable decrease in height on each bounce.

• This is a less technical answer, but it does a great job showing the mechanics at play! Jul 14, 2020 at 17:50