I'm looking for an equation to calculate the force of falling into bed. I mean literally if someone who weights 100 lbs just drops into bed (from a height of 2 feet), how much force is transferred to the bed?
The gravity not ever stops performing in this character. The most effective factor that reverses is the path of his motion. So the gravity nonetheless acts at the character on the maximun peak
You cannot know for certain without knowing what kind of elasticity the bed has. You need to know BOTH how stiff the bed is AND how well the bed does at restoring and rebounding the person, because when this person hits the bed and remains on the bed after impact, it is much different than when this person rebounds off the bed. A person rebounds because no energy is dissipated. A person sticks to the bed because as much energy as possible is dissipated as heat. A riverbed is still a bed (just not one on which you normally sleep), isn't it? And it is made out of very stiff rocks. Let the water run out of it without replenishing it and there you have a bed on Most bed mattresses are made spring formed and enclosed in fabric. Other mattresses may be made from foam or enhanced foam-like materials, and are much less stiff than a typical spring. The stiffer the material and geometry of the bed structure, the less time spent in contact and acceleration with the bed before coming to rest or rebounding, and the briefer the impulses. Brief impulses imply greater forces of impact than a protracted impulse delivering the same total value of impulse.
The force is F = dP/dt + W = mV/dt + W; where m = W/g = 100/32.2 = ? slugs is someone's mass and V = sqrt(2gH) = sqrt(2*32.2*2) = ? fps is someone's impact velocity (speed). The only thing we don't have is dt = ? the impact interval. How long does it take to go from V to zero when hitting the bed covers? That is bed dependent isn't it? If the mattress is hard, then dt will be short. If it's way soft, then dT dt will be longer. Anyway, we can see that F = W because V = 0 when just lying on the bed. That means the obvious, there is only someone's weight as the force transferred to the bed when not jumping on it. But the force increases by the mV/dt = m sqrt(2gH)/dt term due to impact. We can also see the impact force increases with the sqrt of the height of he jump and with the shorter impact intervals. As you probably don't know what dt is, put in several reasonable guesses (e.g., dt = .5, 1.0, 1.5 sec) and see what the results are. This is called sensitivity analysis. It's a technique used often when exact values are not known.
