Comparing Trampoline Jumps on Earth vs. the Moon

[JamieLT]

Hi!
I'm trying to help my 5th grader with her science fair project - she's comparing how a trampoline would work on Earth vs. the moon.

The trick is in the force of the jump I think - what the jumper herself puts into it. The larger the downward force she exerts on the mat the higher she goes. On the moon, the downward force you can get is less because gravity is less – you wouldn't be able to push down on it as much right? If you do just the simple energy balance

½ k x^2 = mgh
PE at the bottom = PE at the top.

you get a smaller x, smaller x^2 - and since x goes by the square, and g is not squared, seems like it would be harder to jump higher on the moon? (even though without the tramp, just jumping on the surface of the moon, you could go higher)

F = kx = mg at rest
not at rest F = kx + leg force ...
leg force is constant, just the mg that is not.

LOL, this should be simple, but for some reason I think I'm missing something.

For the experiment we're just going to drop different weighted balls onto the tramp and measure the height and deflections... then we'll have a jumper with and without weights strapped on to measure the different heights (same legs lifting different amounts of weight)... we also thought about jumping in a pool (astronauts train in pools) but the drag would mess it up. Any other way to test it out? I want to get the math right too though.

Air resistance - the moon doesn't have much air, but is it a negligible effect?

 

[xAxis]

The trick is in the force of the jump I think - what the jumper herself puts into it. The larger the downward force she exerts on the mat the higher she goes. On the moon, the downward force you can get is less because gravity is less – you wouldn't be able to push down on it as much right?

Don't forget there are two downward forces. One is the gravity, the other is the jumpers push downward as she rebounds. I think her force doesn't depend on gravity.
The other important thing to think about is that gravity acts all the time. So concentrate on what happens while the body has just rebounded upwards. Imagine that from that moment there is 6 times stronger force on Earth pulling it back.
Also, you can neglect friction of the air on earth

 

[Devils]

Acceleration due to gravity on the moon is 1.6 m/s^2, on Earth 9.8m/s^2 ...


Equivalence of gravitational & inertial mass ...

 

[JamieLT]

Don't forget there are two downward forces. One is the gravity, the other is the jumpers push downward as she rebounds. I think her force doesn't depend on gravity.
The other important thing to think about is that gravity acts all the time. So concentrate on what happens while the body has just rebounded upwards. Imagine that from that moment there is 6 times stronger force on Earth pulling it back.
Also, you can neglect friction of the air on earth

Taking out what your legs do - if you just drop balls of different mass from some initial height, it shouldn't matter if you are on the Earth or on the moon, idealistically, the ball will just bounce right back up to the same height it started at (without energy loss from heat/air resistance etc.) right? sort of like a pendulum.

Now just look at the force of your legs - if you were standing under a ceiling, where you could wedge your hands against the ceiling and then push down with your legs, to deflect the trampoline, then the deflection would be a matter of your leg muscles... but you don't have a ceiling... so it's not just your leg muscles, it's how much your weight can push it down, right?

If you have any leg muscles at all, and could add a little force each time you hit the bottom, and kept jumping higher and higher each bounce, until you reach the maximum deflection that the tramp could take before ripping through the material, then you could get to the same maximum deflection on the Earth and on the moon eventually? (material property of the elastic) and then you would go higher on the moon... but would you be able to get to the same maximum deflection on the moon? or would it be harder - take more leg muscles - to compress the elastic? harder to compress it, because gravity isn't helping you out as much? Would there be some inflection point - if your leg muscles are stronger than ________ then you could jump higher on the moon, but if you are not stronger then ____ you couldn't jump higher?

 

[Drakkith]

Hmm, I'll take a swing at this.

It seems to me that you will simply be able to reach a higher altitude on your jump than you would on Earth. Imagine just jumping straight off the ground. On the Moon, since the acceleration due to gravity is much less, you will reach a much greater height before coming down. Now, what does a trampoline do? When you land on it you stretch the springs out, which will then apply a force on you as they try to return to their initial state, pulling you upwards. They are returning some of your kinetic energy from falling back into you. This allows you to jump to a much higher height than you would otherwise since the ground normally doesn't do this.

The key here is to understand that your jump propels you at the same velocity on both the Earth and the Moon. Its just like taking a slingshot and shooting a rock into the sky. When you, or the rock, return to the ground, you will have the same amount of kinetic energy as you did upon launch. (Ignoring air resistance) The only difference is the increased time it takes to return to the ground on the Moon due to its lower acceleration.

Remember, the trampoline does nothing but convert part of your downward kinetic energy into potential energy that is used to accelerate you back up. This is the reason the trampoline works in the first place.

 

[xAxis]

Taking out what your legs do - if you just drop balls of different mass from some initial height, it shouldn't matter if you are on the Earth or on the moon, idealistically, the ball will just bounce right back up to the same height it started at (without energy loss from heat/air resistance etc.) right? sort of like a pendulum.

Exactly.

Now just look at the force of your legs - if you were standing under a ceiling, where you could wedge your hands against the ceiling and then push down with your legs, to deflect the trampoline, then the deflection would be a matter of your leg muscles... but you don't have a ceiling... so it's not just your leg muscles, it's how much your weight can push it down, right?

"Deflecting" the trampoline depends mainly on your mass and speed. I don't think your muscles can contribute much. They are important after the trampoline has reached its lowest deflection, on the way up. That's the moment when you shoud start pushing (jumping). My point is that the force of your jump doesn't depend on gravity. It would be the same on Earth and on the moon. But because the gravity on the moon is 6 times weaker, it means you would jump much higher.

@Drakkith:
I don't think that your jump propels you at the same velocity on both the Earth and the Moon. I think your initial velocity would be bigger on the moon.

 

[Drakkith]

@Drakkith:
I don't think that your jump propels you at the same velocity on both the Earth and the Moon. I think your initial velocity would be bigger on the moon.

Yeah, probably. You are working against less gravity after all.