Need Some Serious Help With A Physics Problem?

Question

A 10.7 kg weather rocket generates a thrust of 210 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 400 N/m, is anchored to the ground.

1) Initially, before the engine is ignited, the rocket sits at rest on top of the spring. How much is the spring compressed?

2) After the engine is ignited, what is the rocket's speed when the spring has stretched 31.0 cm?

3) For comparison, what would be the rocket's speed after traveling this distance if it weren't attached to the spring?

**Note: I've already solved #1 so all i need to figure out are #2 and #3.

The answers to #2 and #3 are 3.20 m/s and 3.72 m/s. I just need the work to show and prove that.

Answer 1

2.

PEi(spring) = 0.5(400)(0.26215^2) = 13.7445245

PEf(spring) = 0.5(400)(0.31^2) = 19.22

PEi(gravity) = 0

PEf(gravity) = 10.7(9.8)(0.57215) = 59.995649

W = 210(.57215) = 120.1515

ΔPE(spring) + ΔKE + ΔPE(gravity) = W

(19.2 -13.7) + 0.5(10.7)(v²) + (60 - 0) = 120

5.5 + 5.35v² + 60 = 120

5.35v² = 54.5

v² = 10.2

v = 3.2 m/s

3.

if it's not attached to the spring then PEf(spring) = 0

ΔPE(spring) + ΔKE + ΔPE(gravity) = W

(0 - 13.7) + 0.5(10.7)v² + 60 = 120

5.35v² = 73.7

v² = 13.78

v = 3.72 m/s