The potential energy at the start is

PE = mgh = 15.0 kg*9.81 m/s

The kinetic energy at the start is

KE = mv

So, the total energy at the start is PE+KE = 2943+750 = 3693 J

When the stone comes to rest, the work done against friction and the spring will total this amount. The work done against the spring is

W

The work done against friction is

W

This gives rise to a quadratic equation:

d

This is solved in the usual way to get

d = 47.81 m is the distance the spring will be compressed.

At that distance, the spring is exerting a force of k

= (2.0 N/m)*(47.81 m) = 95.62 N

The static friction is k

= .8*15.0 kg*9.81 m/s

Static friction is higher than the force on the stone, so it will not move again.

PE = mgh = 15.0 kg*9.81 m/s

^{2}*20.0 m = 2943 JThe kinetic energy at the start is

KE = mv

^{2}/2 = 15.0*(10.0)^{2}/2 = 750 JSo, the total energy at the start is PE+KE = 2943+750 = 3693 J

When the stone comes to rest, the work done against friction and the spring will total this amount. The work done against the spring is

W

_{spring}= 1/2*k_{s}*d^{2}= 1/2(2.0 N/m)*d^{2}= d^{2}N/mThe work done against friction is

W

_{friction}= k_{fd}mgd = .2*15.0 kg*9.81 m/s^{2}*d = d*29.43 NThis gives rise to a quadratic equation:

d

^{2}+ 29.43d = 3693, or d^{2}+ 29.43d - 3693 = 0.This is solved in the usual way to get

d = 47.81 m is the distance the spring will be compressed.

At that distance, the spring is exerting a force of k

_{s}*d= (2.0 N/m)*(47.81 m) = 95.62 N

The static friction is k

_{fs}mg= .8*15.0 kg*9.81 m/s

^{2}= 117.7 NStatic friction is higher than the force on the stone, so it will not move again.