A running track with some compliance $("$springiness$")$ can increase runners' speed by increasing stride length $($the distance between consecutive footprints of the same foot$)$ and stride frequency $($the number of strides per second$).$ The track can influence the runner's dynamics only when one of the feet is in contact with the ground, so we can model the runner as a mass$-$spring$-$damper system in series with a spring representing the track:

In this model, a passive spring and damper $("$dashpot$")$ represent the mechanical properties of the muscles, and the spring under the foot represents the compliance of the track. Note that this model is useful only when the track spring is at or shorter than its length prior to landing.

The dynamics of this system are governed by two second$-$order ordinary differential equations, which can be derived from the schematic shown here:

$m{x}_r^{}+b_m(x_r^{}-x_t^{})+k_m(x_r-x_t)=o$

$b_m(x_r^{}-x_t^{})+k_m(x_r-x_t)-k_t{x}_t=o$

where $m$ is the mass of the runner's body, $x_r(t)$ is its displacement from equilibrium, $x_t(t)$ is the displacement of the track under the foot, $b_m$ is the damping coefficient, and $k_m$ and $k_t$ are the stiffness of the muscle and track springs, respectively.

$($a$)[4$ marks$]$ Derive the equations of motion shown above. Assume the system is initially in equilibrium$-$i$.$e$.,$ the springs are initially compressed by the amount that is necessary to balance gravity. With this assumption, we can omit gravity $(mg)$ from our free$-$body

diagrams and measure the displacements of the runner and track with respect to their positions at equilibrium. The small piece of track under the foot is modelled as a massless body. You may wish to assign a mass to this piece of track $($e$.$g$.,m_t)$ when deriving the equations of motion and then substitute $m_t=0$ at the end.