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Homework Statement
Suppose a spacecraft of speed [itex]v_c[/itex] 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 [itex]v_0[/itex] 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 [itex]v_c[/itex] 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 [itex]v_0[/itex] 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,
[tex] E = T+V=\frac{1}{2}mv_c^2 - \frac{k}{r} = -\frac{k}{2r}[/tex]
[tex]v_c=\left ( \frac{gR^2}{R+H} \right )^{1/2}[/tex]
For part B, I redid the work from part A for an ellipse, i.e.
[tex] E = T+V=\frac{1}{2}mv_0^2 - \frac{k}{r} = -\frac{k}{2a}[/tex]
[tex]v_0 = \sqrt{2gR} \left [ R \left (\frac{1}{r}-\frac{1}{2a} \right ) \right ]^{1/2}[/tex]
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?
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