53

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: picture

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.

Qmechanic
  • 201,751
Nico Damascus
  • 701
  • 5
    Related problem: How many bounces would occur between paddle and ball without friction or gravity? It's fascinating: https://youtu.be/HEfHFsfGXjs – Peter - Reinstate Monica Jul 13 '20 at 09:20
  • 17
    How did you measure the velocity? – pipe Jul 13 '20 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). – Carl Witthoft Jul 14 '20 at 12:52
  • 2
    @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.) – Peter - Reinstate Monica Jul 14 '20 at 16:53
  • @Peter-ReinstateMonica but in the case in question, that's a very small contributor and is not the source of increased average speed . Adiabatic gases do not under go delta Hamiltonian whereas the bouncing real-world pingpong ball has massive transfers between kinetic and potential energy. – Carl Witthoft Jul 14 '20 at 17:47
  • @CarlWitthoft Could you please clarify what you mean by "Adiabatic gases do not under go delta Hamiltonian"? – theorist Jul 15 '20 at 06:34
  • @theorist - they don't 'exchange' kinetic for potential energy in any significant way. – Carl Witthoft Jul 15 '20 at 11:10
  • 1
    @CarlWitthoft I was only taking issue with your "completely"; the paddle adding kinetic energy is a factor. – Peter - Reinstate Monica Jul 15 '20 at 17:04

3 Answers3

107

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.

Guy Inchbald
  • 7,372
  • 9
    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? – John Dvorak Jul 12 '20 at 20:29
  • 7
    @JohnDvorak - would be difficult to do as the frequency of rebounds increases... – tom Jul 13 '20 at 00:41
  • @John Dvorak - Instead of using a bat, project the ball into a narrow vertical v-shape between two planes. – chasly - supports Monica Jul 13 '20 at 12:34
  • 15
    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.) – md2perpe Jul 13 '20 at 12:47
  • 1
    The audible bouncing getting faster would tend to reinforce the impression too - another sense confirming the increase in speed, but only sampling at the bounces – Chris H Jul 13 '20 at 13:09
  • 8
    @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! – Nuclear Hoagie Jul 13 '20 at 13:54
  • 1
    Thanks all for pointing out the missing third effect, I have now added it. – Guy Inchbald Jul 13 '20 at 17:32
  • 3
    @GuyInchbald Really? I don't see it. – Brilliand Jul 13 '20 at 20:02
  • 1
    Sorry, I must have failed to click the button. Should be up now.. – Guy Inchbald Jul 14 '20 at 09:14
  • Another way to avoid the moving paddle effect is to move the paddle in from the side:
    1. Drop the ball from a height of say 76 cm (ITTF regulation)
    2. With the paddle at say 1/2 height, 38 cm, but not in the path of the ball,

    slide the paddle horizontally, not changing its height above the floor, into the path of the ball.

    You will see the effects the ball not reaching 0 velocity at the apex, until the energy has decayed.

    Imagine a vertical. rod with a series of "paddles" closely stacked. Since the change in height is small, it would easier to swing each into the path of the ball.

     – Andrew Dennison Jul 14 '20 at 13:31
  • I'm very skeptical that the "real" contributions would be significant compared with the illusory contribution, especially when considering that there's also a "real" contribution in the opposite direction: collisions between the ball and paddle are not perfectly elastic, so the ball will actually lose some speed on each collision and this effect will accelerate as collisions become more frequent. – Robin Saunders Jul 15 '20 at 14:40
  • In particular, the first "real" contribution from the answer can be made negligible by lowering the paddle slowly enough (or only lowering it in between bounces), as others have noted. This will also cause the second "real" contribution to be more gradual, while the "negative" contribution from imperfect elasticity will remain constant. Alternatively: drop the ball onto a hard surface and note that the bounces get "faster" (more frequent) even though the ball's average or maximum speed cannot be increasing. – Robin Saunders Jul 15 '20 at 15:00
  • The answer states that "Repeat for several bounces and the difference might become noticeable." That is to say, it might equally remain insignificant/negligible; the skepticism is already built into the answer. – Guy Inchbald Jul 15 '20 at 15:12
69

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.

Acccumulation
  • 8,745
  • 1
    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. – DKNguyen Jul 12 '20 at 23:50
  • And I thought I had it thought all through! Kudos. – Peter - Reinstate Monica Jul 13 '20 at 08:55
  • 2
    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? – anjama Jul 13 '20 at 14:33
  • 2
    @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. – Nuclear Hoagie Jul 13 '20 at 14:40
  • 2
    @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 – Steve Cox Jul 13 '20 at 14:40
  • This is genius. I was thinking the "cutting off the high part" was basically just statistical fluff, until I realized: you're doing this continuously, stripping more and more of that 'average' away. Even without any momentum added from the paddle, if you manage to lower the paddle down as far as possible, the ball is going to be continuously moving at the maximum speed it would have only temporarily had with the large bounces. – Kevin Jul 13 '20 at 18:00
  • Hm, the point is certainly valid, but I´m not sure this isn´t overcompensated by the loss of energy while getting deformed (semi-elastically) at the paddle and table. Perhaps not, but that would also have to be taken into account. – Karl Jul 13 '20 at 20:55
4

Because Ping-Pong balls are elastic

And in elastic collisions, kinetic energy is conserved.

enter image description here

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.

Please stop being evil
  • 141
  • 6