Another physics spring problem!!!?

Question

An 8.40 kg block travels on a rough horizontal surface and collides with a spring. The speed of the block just before the collision is 4.00 m/s. As it rebounds to the left with the spring uncompressed, the block travels at 3.00 m/s. The coefficient of kinetic friction between the block and the surface is 0.360.
(a) Determine the magnitude of the loss in mechanical energy due to friction while the block is in contact with the spring.
_29.4_ J

(b) Determine the maximum distance the spring is compressed.
___ m

I have the 1st answer, but I'm stuck on the 2nd one. I'm really bad at these. Please help! Thank you!

Answer 1

Part A (just to confirm)
4.2*16=4.2*9-f
f=4.2*7
=29.4

Same answer, good start

b)
while the block is in contact with the spring there are two energy transformations that occur:

The loss due to friction, which we already computed as 2X the distance of the compression since friction will work over the compression and decompression of the spring.

The frictional force is
m*g*µ
=8.4*9.81*0.360
=29.67 N

During compression the spring will store energy as potential energy in the spring =.5*k*d^2


we know the kinetic energy at the start. When kinetic energy is zero, the spring is fully compressed

4.2*16=29.67*d+.5*k*d^2

we also know the kinetic energy at the exit of the spring, so
.5*k*d^2-29.67*d=4.2*9

We can solve for both k and d
since you asked for d,
.5*k*d^2=4.2*16-29.67*d
and
.5*k*d^2=4.2*9+29.67*d

subtract the second equation from the first
0=4.2*(16-9)-2*29.67*d

d=4.2*7/(2*29.67)
d=0.495 m

j

Answer 2

On part b), I think I have an easier approach. The distance between the spring's uncompressed state and the compressed state is d. Both when the block is coming in and being thrown out, it's d. And the energy loss due to friction will be equal going both ways. So going just one way, to full compression, it looses 1/2 of the answer for part a.

So when at full compression, the energy stored in the spring compression is the original energy minus 1/2 of the answer for part a. And spring potential energy is
(1/2)*k*d^2