How would you tell how much the spring could be compressed?

Question

Hello, here is the problem: A.12kg block is held in place against the spring by a 35N horizontal external force. the external force is removed, and the block is projected with a velocity (v1) of 1.2m/s when it separates from the spring. The block descends a ramp and has a velocity (v1) of 1.4m/s at the bottom of the ramp. the ramp so far is a frictionless surface. after a small straight section, the block enters a rough section with a coefficient of kinetic friction of 0.26. The block eventually comes to a stop. How many centimeters was the spring initially compressed?

Answer 1

Energy is conserved therefore the energy that the block has initially is equal to the energy of the block as it leaves the spring. As it leaves the spring, the energy is kinetic energy as calculated by 1/2mv².

E_K = 1/2mv²

E_K = 1/2(12)(1.2)² = 8,64 joules

That means that the energy the spring had when it was compressed was also 8,64 joules

When it was compressed, the spring had spring potential energy as calculated by 12kx² (where "k" is the spring constant and "x" is the amount the spring is compressed. so:

E_SP = 1/2kx²

So 8.64 = 1/2kx²

We are also told that the spring was being held by a 34 N force. We know that F = kx for a spring so:

34 = kx which means k = 34/x

Plugging in "34/x" into the other equation, we get:

8.64 = 1/2(34/x)x²

8.64 = 17x

x = 0.51 meters