A square case rests on a track and is pushed against a horizontal spring, compressing the spring from its relaxed length. After releasing the case, the spring launches it horizontally along the track and up a ramp, as shown below. The case has enough speed to leave the ramp, and it reaches a maximum vertical height above the ground. Assume air resistance is negligible, and there is negligible friction between the case and the track.

h is the maximum height reached by the box.

k is the spring constant of the spring.

M is the mass of the box.

x is the distance the spring is compressed.

$?$ is the angle of the ramp from the horizontal axis.

a$.$ Indicate how using a softer spring $($one with a lower spring constant$)$ would affect the maximum height reached by the box, and explain why without mathematically deriving a relation for h$.$

b$.$ A physics student derives this equation for the maximum height, h equals the quantity sine of theta squared times k times x squared all divided by $2$ times mass times g$.$

Can this equation be used to accurately support your claim in $($a$)?$ Briefly explain.

c$.$ Consider the case$-$spring$-$Earth system. Create an energy bar graph to represent position A$,$ the energy of the system when the spring is fully compressed, compared to position C$,$ the energy of the system at the moment immediately before the case hits the ground. Write your name and today's date in the upper right corner. Scan or take a picture of your graph and attach it to this item.

d$.$ Suppose air resistance was not negligible. Would this affect your answer in $($c$)?$ Justify your answer.

0000000 M -X- A B