find k, the spring constant?

Question

An object of mass m is traveling on a horizontal surface. There is a coefficient of kinetic friction (mu) between the object and the surface. The object has speed v when it reaches x=0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches x=0 on its return trip, it stops.

Find k , the spring constant.
Express k in terms of (mu),m ,g , and v.

Answer 1

The object has kinetic energy (1/2)*m*v^2 when first at x=0. The work friction does consumes all that energy as the spring compresses and pushes the object back to x=0. Let the distance the spring is compressed be x1. Then
(1/2)*m*v^2 = Ff*2*x1 = mu*m*g*2*x1

When the spring is compressed to x1, the friction has decreased the energy in the system by half and the remaining half is all stored in the spring compression because the object has come to a stop - about to start the rebound. So
Fspring*x1 = k*x1^2 = (1/4)m*v^2
If you solve the 1st equation for x1 and substitute the result into the 2nd, you should get the required result.