Questions about basic jump physics

Question

This is a problem a friend and I are working on for an undergrad reading course. Our goal at the end is to make an accurate two-dimensional model of the human jump using Processing by the end of next semester.

Suppose the human body could be modeled as a one dimensional rod of length 2L, where the length L is the distance from the ground to the human's center of mass. Over some time t, the portion of the human from h = 0 to L contracts to Lmin and then extends to an Ymax in our simulation of a jump.

I've drawn out a crude graph to model the function of Y(t) = Lmin + kt^2 where k is an arbitrary constant

The questions we have are, where does the jump begin and end? What does it mean to be in the air or on the ground? How we do we write the max height of the jump as a function of Lmin, Lmax, k, and t?

Intuitively the jump begins right before Lmax is reached in the real world, but I need a sanity check. I'm defining "in the air" as any point where the distance from the bottom of the rod to the base position is nonzero, and when "on the ground" as any point when that value is zero. As for a function of h based on L(t), I'm a bit stumped.

Answer 1

This is a project, so I don't feel comfortable answering every bit of this.

But one way to model when something leaves the ground is to consider what the value of the normal force $N$ must be at that instant. Then if you can find an expression for $N(t)$, you're set.