need help on this spring problem?

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 surfaace. 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.

Thanks in advance for the help!

Answer 1

x is the distance the spring is compressed
On the way into the spring the energy equation is

1/2mv^2-umgx=1/2kx^2

When the spring is released all the energy is dissipated as friction so we have

1/2kx^2=umgx
so x=2umg/k

solve for v in the first equation and use the value for x solved above

v=sqrt(8u^2m^2g^2/(km))

Answer 2

when it encounters the spring and compresses it,by work energy theorem 1\2mv^2-Fx=1\2kx^2 ------eqn1,whereF is frictional force,after recoil it stops after covering a distance x[x is assumed],therefore we have1\2kx^2=Fx -----eqn2
sub Fx in eqn 1
kx^2=1\2mv^2
k=mv^2\2x^2 ------eqn 3
from eqn2 we know,x=2F\k,sub in eqn3,we get
v^2k^\8(mu)^2mg^2,where F=(mu)mg