Spacecraft orbit homework problem

[kreil]

Homework Statement

Suppose a spacecraft of speed $v_c$ is in a circular orbit around the Earth at a height H from the surface of the earth. The rocket motor is fired in the opposite direction of motion to reduce the satellite's speed to $v_0$ and make the orbit elliptical. Let R denote the Earth's radius and g the gravitational acceleration on the surface of the earth.

A. Express $v_c$ in terms of R, H, and g.

B. For the spacecraft to land on the Earth surface horizontally at the perigee of the elliptic orbit, find $v_0$ in terms of R, H, and g. Ignore friction.

C. Express the eccentricity of the elliptic orbit in terms of R and H.

The Attempt at a Solution

For part A I just used the energy equation for a circular orbit, noting r=R+H, k=GMm and g=GM/R^2,

$$E = T+V=\frac{1}{2}mv_c^2 - \frac{k}{r} = -\frac{k}{2r}$$

$$v_c=\left ( \frac{gR^2}{R+H} \right )^{1/2}$$

For part B, I redid the work from part A for an ellipse, i.e.

$$E = T+V=\frac{1}{2}mv_0^2 - \frac{k}{r} = -\frac{k}{2a}$$

$$v_0 = \sqrt{2gR} \left [ R \left (\frac{1}{r}-\frac{1}{2a} \right ) \right ]^{1/2}$$

This is where I start having problems. The problem asks for v0 in terms of R, H, and g..but is it not true that when the spacecraft is landing at perigee H=0 and r=R?

The answer I ended up getting for this part came from using r=R=a(1-e) where e is the eccentricity, but I feel shaky about this.

Any thoughts on part B?

 

[zzzoak]

This is my opinion.

Point $$v_0$$ is the farthest point and
$$v_0 \ \bot \ R+H.$$

The nearest point is R.

So we can conclude that

$$2a=2R+H.$$