Can conservation of energy be applied to this problem?

[Sferics]

Homework Statement

http://www.aapt.org/physicsteam/2010/upload/2010_FmaSolutions.pdf
#25

Homework Equations

$$\tau$$_net = 0 $$\Rightarrow$$ L_o = L
Conservation of Mechanical Energy

The Attempt at a Solution

I started by writing two equations: one for conservation of angular momentum and one for conservation of mechanical energy for the two given points of the elliptical orbit. This gave me,

3/2v_o² = GM/(2R)

Obviously taking the square root provides the answer given. And I know that sqrt[GM/(2R)] is an equation for the speed of a particle in a circular orbit. However, something does not seem quite right here.

Is it valid to apply conservation of energy? Is mass not expelled to thrust the rocket? Moreover, by applying conservation of mechanical energy instead to the maximum distance point on the elliptical orbit and any point on the circular orbit, would then the speed not remain remain v_o due to the potential energy not changing (same radius)?

I also tried applying the conservation of angular momentum to the same two points. Following that logic, I also reached the conclusion that the speed not remained v_o, also because the radius did not change. Since this is wrong, it lead me to question whether the external torque really was zero... is it incorrect to regard the rocket thrust as an internal torque?

I guess what I am asking here is for an outline of the correct thought process involved in solving this problem. Thank you.

 

[zorro]

Is it valid to apply conservation of energy? Is mass not expelled to thrust the rocket?

Your approach is perfect.
The mass of the spaceship is given to be negligible. So don't worry about mass being expelled.