Estimation of the limits for the contractile element force fc in Hill's lumped parameter model (below) requires the use of Hill's force-velocity equation (also below). This equation requires knowledge of the maximum isometric force f0 in the contractile element, muscle constants a and b, and velocity u1. (Note that a dot above a displacement indicates its derivative with respect to time.) This lumped parameter model also requires stiffnesses of the parallel and series elements kp and kz respectively. Suppose we know or measure force F and displacement u and hence can compute their derivatives with respect to time. Derive an expression for contractile element velocity u1 as a function of u,u,F,F,kz,kp for Hill's 3 parameter model below. This expression will allow us to see what the contractile element must be doing to produce the forces and displacements we measured. fe(u1)=b+u1f0bau1