Question

Using conservation of angular momentum, find the ratio of a spacecraft's speed at perigree and the speed at apogee. period=7910s, apogee=10380000m perigree=6780000m (after adding earth's radius

Answer 1

angular momentum, for a central force (spherical symmetry), is constant and is equal to
\[L = \mu r ^{2} d \theta/dt=\mu r v=constant\], where \[\mu\] is the reduced mass.
(note that the second equality above applies only to the turning points, perigee and apogee, since dr/dt = 0 there, and the velocity vector v has only a tangential component)...ok so now we can simply write:
\[\mu r _{p} v _{p}=\mu r _{a}v _{a}\rightarrow v _{p}/v _{a}=r _{a}/r _{p}=1.038\times10^{7}m/6.78\times10^{6}m \approx1.53\]