A $30-$kg child jumps on a trampoline. Initially the child is moving downwards with a speed of $2\mathrm{.}5$ m$/$s right before they hit the trampoline. They are in contact with the trampoline for a total $1\mathrm{.}0$ s$.$ You can treat the trampoline as a spring with a spring constant of k $=800$ N$/$m which the child displaces by a maximum amount of $0\mathrm{.}3$ m$.$ Your job is to determine how fast the child leaves the trampoline.

A$.$ Represent this process with a sketch $(5$ points$)$ and an impulse$-$momentum bar chart $(5$ points$).$ Clearly identify your system, axes, initial and final states, and label all quantities given to you in the problem.

B$.$ Assume the trampoline stretches uniformly to $0\mathrm{.}3$ m and back to unstretched in the $1\mathrm{.}0$ s of contact. Sketch the graph of F$\_$Trampoline on child vs time during the period of contact. Explain the shape of the graph. $(5$ points$)$

C$.$ Determine the amount of impulse the trampoline imparts onto the child. Explain your work. $(5$ points$)$

D$.$ Use your representations and the information given in the problem to determine the velocity with which the child leaves the trampoline. Show your process and explain what you did. $(5$ points$)$