simulations - Trampoline physics - what is the right time to crouch and to jump to maximize your height?

Please bear in mind that I'm neither a physics guy nor particularly a math guy. So I will probably need a bit more hand-holding than just a complex equation.

For a game, I'm trying to calculate a reasonable simulation of a human body on a trampoline. It seems to me that there are two phases to consider once the body has contacted the trampoline: the deceleration of the body once it has contacted the surface, which gets stored as potential energy in the trampoline, and then the combination of the release of that potential energy once the body has reached the nadir plus any extra force that body might apply (assuming he has bent his knees prior to that point).

I'm interested in understanding the interaction between the trampoline's storage and release of energy and the body's own contribution (addition) to that energy with its own kinetic energy.

CLARIFICATION: In short, I want to know when the best time to crouch is (pull your knees up) and when the best time to release (jump) is. Anyone who's been on a trampoline knows you can get better height by crouching and jumping at the right time - you can also bring yourself to a complete stop by crouching and jumping at different times. I can't seem to find anything documented though about what these times actually are. Since I couldn't find the information from the sports / athletics side, I thought I'd come over here to the physics side and see if anyone knew.

I'm surely not being very clear, which only further reveals my own shortcomings in this matter. Any guidance would be appreciated. I understand that this forum typically operates at a much higher level than what I'm looking for, but I thought I'd ask here, since this is where the great minds seem to gather.

EDIT I can very easily do a "good enough" simulation of a mass "bouncing"off the trampoline by taking its incoming velocity and simply inverting it and multiplying by an arbitrary restitution amount to simulate the springiness of the material. Nothing more complicated really need from the perspective of a game.

The part that's killing me is getting the body to continue moving past the plane of the trampoline while decelerating to 0 and then accelerate again up to final "release velocity". The part that baffles me is the part that happens between the time the body touches the material to the time it leaves contact, taking into account any additional force added by the person through his legs (I've decided not to model any additional lift sure to arm motion etc, to keep both the simulation and the user controls thereof straightforward).

SECOND EDIT I've read through the answers here and they're very good! And surprisingly, I seem to be following about 75% of it.

The challenge I'm having is that I'm principally interested in Velocity, rather than Energy, even though it's obviously energy that is behind all of this. Every 30th of a second (every "step") I need to know the velocity of the body so that I can plot its new position. It's really easy to plot the position of the body once it has left "contact": starting at some arbitrary position and velocity, v = v + g, y = y + v. See, it's a very different way of representing the same facts when you're dealing with visual graphics than physics equations.

So now y = the plane of the trampoline. v = v * c, where 'c' is a factor of how far the springs will compress, less than 1. y = y + v until v approaches 0 at which point we start the "bounce back". Here I suppose v = v * (1/c) * s, where 's' represents whatever factor of 'loss' we want to simulate so that we don't get 100% bounce back. Once y > the plane of the trampoline, we can then forget the more complicated stuff and go back to letting gravity take over.

Obviously this whole bit ignores mass, which is unfortunate, but I have to keep it simple for now. The "mass" factor gets rolled into the numbers for 's' and 'c' that I fudge until it 'feels' right.

Okay, now that you guys have stopped shuddering in utter horror at this wishy-washy approach, the bit I'm trying to figure out is the part about the human body bending his legs and then performing the jumping motion to add his force to the rebound of the trampoline.

It seems to me, having spent a little time on a trampoline, that you can jump at the right time and at the wrong time.

So, from a physics perspective, when is the optimal time to crouch? When is the optimal time to jump / release?

Answer

I will make the following approximating assumptions:

1. The surface of a trampoline has zero mass density. This implies standing or crouching will not instantaneously affect the position of your center of gravity.


2. The upward restoring force is a monotonic function of vertical displacement.


3. Humans can straighten their legs in zero time. This is just a way to avoid extra optimization details.

If these hold, then the best time to crouch is before landing, and the best time to straighten your legs is when your speed is zero (i.e., when your height is at a minimum). This behavior optimizes three quantities:

1. The kinetic energy you gain before reaching the trampoline is maximized. Because you are in a crouching position, your center of gravity falls further before the restoring force starts slowing you down.


2. The height of your center of gravity when your speed reaches zero is minimized. This is because for a fixed position of your center of gravity, the restoring force is minimized when you are in a crouching position.



3. The work done by the springs in releasing you is maximized. If you set $h_0$ to be the displacement of your center of gravity where you are stationary, the energy imparted is the integral $\int_{h_0}^0 F ds$ of the restoring force with respect to displacement. For any fixed position of your center of gravity, the restoring force is maximized when you are in a standing position.

If you watch experienced gymnasts gaining altitude, you will see that they straighten their legs roughly when they reach the bottom, and they let the trampoline carry them up.