A 5.0 kg box slides down a 5-m-high frictionless hill, starting from rest, across a 2-m-long horizontal surface, then hits a horizontal spring with spring constant 500 N/m. The other end of the spring is frictionless, but the 2.0-m-long horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.25.
(a) What is the speed of the box just before reaching the rough surface?
(b) What is the speed of the box just before hitting the spring?
(c) How far is the spring compressed?
(d) Including the first crossing, how many complete trips will the box make across the rough surface before coming to rest?
W_nc = delta-E
The Attempt at a Solution
So I actually solved (a)-(c) using non-conservative work and delta energy. My solutions for (a)-(c), if that would help, were
(a) 9.9 m/s
(b) 9.39 m/s
(c) 0.959 m
However I honestly haven't a clue as to how to approach (d) besides the "obvious" long way by going through the velocities over and over again until it reaches zero. Any elegant way here?
Thanks in advance!