Spring and Energy: Rocket clamped to an anchored spring

[Max]

Homework Statement

A 11.6 kg weather rocket generates a thrust of 200 N . The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 580 N/m , is anchored to the ground.

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

After the engine is ignited, what is the rocket's speed when the spring has stretched 32.0 cm ?

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

Homework Equations

:[/B]

The Attempt at a Solution

:[/B]
I found 0.1962m for part A
but part B and C, I don't think I am on the right track. I don't know how to use LaTex, so hopefully you understand

for question b:
I used conservation of energy of the spring systems, saying 1.2mv^2=1/2 k* delta(x)^2
delta x I used 0.1962m of part A to solve for v by all given variables

for part c:
my answer was funky. I found net acceleration on y direction and then used integral to solve for v, but it was not working.

 

[haruspex]

conservation of energy of the spring systems, saying 1.2mv^2=1/2 k* delta(x)^2

Conservationof energy says initial total energy equals final total energy. Does that match your equation?

(You will not be able to use conservation of energy here. Consider forces and accelerations.)