Can You Help Solve This Generalized Work Problem with an Illustrative Image?

[Ced]

Homework Statement

A crate of mass(m) is above a ramp of angle theta and a distance(L) from a spring of constant k. The ramp and the crate have a coefficient of kinetic friction(μ)
a.) What is the crate's speed before it compresses the spring
b.) What is the maximum compression of the spring
c.) How far does the box get to its initial distance once it rebounds.

Relevant Equations

I think the relevant equations are
1. Ki + Ui + Wext = Kf + Uf
2. Elastic energy U=\frac{1}{2} k \Delta x^{2}

Here is an image for better illustration,

I only managed to solve for (a) but I'm not sure if I did it right. As for (b) and (c), I have no idea how to do it.

My answer for (a):
=> Ki + Ui + Wext = Kf + Uf
=> 0+mgh1-LμmgCosΘ = 1/2mv^2 + mgh2
=>1/2v^2 = gh1- gh2 - LμgCosΘ
=> V = √2g(h1 - h2 - LμCosΘ)

 

[Zexuo]

You can express h1 - h2 in terms of L and theta. Part b will have a term for the spring energy at maximum compression on the final side.

 

[Ced]

You can express h1 - h2 in terms of L and theta. Part b will have a term for the spring energy at maximum compression on the final side.

Thank you so much! I got the answer now. You helped me a lot!