Figure $3$: Bumper problem

Consider the following system: a cart with a spring$-$based shock absorber is released from rest and

starts rolling down a hill angled at $=20,$ towards a big rock a distance $d=1\mathrm{.}5m$ away. We can

model this system as a mass with a spring and damper as shown in Fig. $3.$

When we built the cart, we used a spring with stated spring constant $k=90,000\frac{N}{m}$ and measured

the mass of the entire cart $+$ bumper as $70$ kg $.$ We did not have a direct measure of damping but

observed a damped natural frequency of $5$ Hz while the cart bumper is in contact with a surface.

Let's define time $t=0$ and initial cart position $)=(0$ right when the cart impacts the rock.

Direction of positive motion is indicated in the figure. While the bumper remains in contact with the

rock:

$($a$)$ What is $c,$ numerically? You should be able to find this knowing $m,k,$ and $f_d$ as stated in the

problem above. Report your answer in standard units Ns$/$m$.$

$($b$)$ Draw a free body diagram of the cart. Indicate positive direction of motion in your diagram.

$($c$)$ Find the equation of motion of the cart$-$bumper system when in contact with the rock.

$($d$)$ What is the initial velocity of the cart, $)=(0,$ when it hits the rock?

$($e$)$ Find and plot the solutions for cart position and velocity from time $t=0$ until the cart breaks

contact with the rock. Indicate this time and the maximum distance the cart travels on your

plots.