Question

A 13kg box slides 4.0m down the frictionless ramp shown in the figure , then collides with a spring whose spring constant is 170 N/m.
What is the maximum compression of the spring?
At what compression of the spring does the box have its maximum speed?

Answer 1

The force on the box is:
F = mgsin∅
If we multiply by this with the distance it traveled, we will know the work done by the box. 
W = dmgsin∅

This work will be converted to elastic potential energy in the spring which is:
1/2 kx². Equating these and substituting values:
1/2 * 170 * x² = 4 * 13 * 9.81 * sin(30)
x = 1.73 m

The box's maximum speed will at the point right before contact with the spring, when the compression is 0.