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Watching a person on a trampoline they can get higher for about 4-5 bounces then they are losing as much energy as they are putting in. A 70 kg person bouncing 3 meters is roughly 2100 J

What are the energy loss mechanisms?

  • Spring losses.
  • Elastic losses on the matt
  • Air displacement.

I think that the largest of these is air displacement. The professional mats on olympic trampolines are nets, with only about 10% of the surface blocked. They can reach heights of 16 feet (5 m) This is close to double the energy.

A 5m circular trampoline with a 1 meter displacement, would (if a perfect cone) have a volume of about 6 cubic meters, or about 8 Kg of air. Air gets moved on both sides, so there is 16 kg of air involved.

To bring him to a stop in 1 meter is going to take more than 1 g. I think it will be square root of the distance ratio or 1.7 g. 3m corresponds to a velocity of about 8 m/s

If the landing accelerated 8 kg of air to 8m sec twice (does this on the rebound too) it would be mv2 = 512 J. This is a huge oversimplication, but I can't see it as being much larger than this, and likely smaller. Only the center of the trampoline is moving that fast, and only at the moment of impact. Moving air on both sides however doubles it.

Springs generally have a low hysteresis. Sanity check: If it takes 4 jumps to get close to maximum height, then the jumper is pumping in about 500 J/jump. 100 springs at 100g = is 10 kg of metal. 1 jump per second (too fast) would be 500W or about 5W per spring. This may be detectable, but they aren't going to glow.

The mat isn't deforming much. It will have wrinkles running from the spring attachments to the center

The final component is the jumper himself. If he starts his squat for the next jump as he touches the mat he is extracting some energy from the system, and also making his deceleration non linear.

Mind you: If he's pulling 1.7 g's, there may be an issue of him being unable to pump more energy into the system. If this is the case, then designing a trampoline to have a larger displacement and smaller forces would increase the jumping height.

This can be checked: Can a more fit athlete jump significantly higher on the same unit?

There is some sort of impedance matching happening here. Someone who is too light has a hard time getting much bounce. Two effects: Small mass doesn't deflect the mat much. Small mass also means that for a given deflection, the g forces will be larger.

So where is the energy going.

One answer says, "look at the noise" But trampolines aren't that loud. Even a watt of actual sound energy is LOUD.

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I think that the largest of these is air displacement.

I disagree. The rule of thumb is the biggest loss is where you find the most sound or heat. Nothing in this scenario is hot enough to visibly glow, so let's focus on sound. You'll hear the mat and springs, which absorb kinetic energy from the user and convert it into sound. You won't hear them tearing through the air much during a jump, although of course the sound from them landing then rebounding travels through the air.

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    $\begingroup$ For air, a useful rule is that air displacement becomes important when the mass of displaced air approaches or exceeds to mass of the moving object. $\endgroup$
    – rob ♦
    Mar 29, 2022 at 17:53
  • $\begingroup$ @rob: wow, that's kind of more than I thought it would be. Let's guess a 10 $m^{2}$ trampoline, with a 1 m vertical displacement (both probably a biiiit large, but not absurd), you get something like 12 kg of air displaced. Much less than a(n adult) person, but at least in the same order of magnitude. $\endgroup$
    – Zo the Relativist
    Mar 29, 2022 at 18:02
  • $\begingroup$ 5 m diameter (16 feet) is the large end of consumer grade tramps. Area of 20m2. One meter displacement isn't unreasonable. To do less than than that and still get 3 m vertical for the jumper would require more than 2 g's. Standing rapidly in 2 g's is hard. $\endgroup$
    – Sherwood Botsford
    Mar 30, 2022 at 14:03
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Further research on various places:

A: Primary loss is air displacement. Changing from a tight weave mat to a string or web mat can easily double the height you reach.

B: Second big loss is the kinetic energy of the mat when the jumper's feet leave the mat. The mat is still moving upward with jumper's velocity at the moment of separation. You end up losing ${kmv^2}\over2$ where k is a constant that accounts for not all the mat going up at the same speed. I think k is somewhere between 1/3 and 1/6. M is typically about 10 kg which is an appreciable fraction of a gymnast's mass.

So far I haven't gotten a handle on the energy loss to fabric hysteresis.

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