Orbits: Firing satellite to Moon

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Homework Statement


Printed exam question 11 on orbital mechanics: effective potential for a satellite, sketching orbits after rocket firing, Jules Verne's 1865 Moon projectile, and final speed relative to the lunar surface.

Homework Equations


The Attempt at a Solution


Handwritten notes: effective potential U = J^2/(2mr^2) - GMm/r, energy E equation, turning point formulas, then sketches of U against r for cases (a) E and J increase and (b) E increases with J unchanged, with orbit sketches beside each.

Handwritten derivation of orbital energy E = (1/2)m(dr/dt)^2 + J^2/(2mr^2) - GMm/r, then r_max = 6OR with dr/dt = 0 there, leading to v = sqrt(26M/R); sketch of Earth E with moon and dashed orbit, and note that the moon must be approaching at 90 degrees.


Parts I'm unsure of:
Sketching of graphs in (a) and (b). I know increasing its energy will shift the entire graph upwards. But does it shift it rightwards if the angular momentum is increased?

Last part, they said to ignore the gravitational effects of the moon. But clearly the satellite does not go into an elliptical orbit, as without the moon it's just going to go to a maximum distance, then come back in a straight line. By this analogy, the satellite is just going to be at rest at rmax. Thus the relative velocity is going to be simply the velocity of the moon.

Handwritten working: Vs = 0 (in Earth's frame), then Vrel = Vm = sqrt(GM_E / r) = sqrt(GM_E / 6αR) = 1020 m s^-1.
 
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