Question

If a satellite is revolving around a planet of mass M in an elliptical orbit of semi-major axis a. Show that the orbital speed of the satellite when it is at a distance r from the focus will be given by
$v^2=GM[\frac{2}{r}-\frac{1}{a}]$

Answer 1

Total mechanical energy of a satellite in an elliptical I orbit of semi major axis a is $-\frac{GMm}{2a}$.
$E=K+U$
$\therefore$ $-\frac{GMm}{2a}=\frac{1}{2}m{v}^2-\frac{GMm}{r}$
or $v^2=GM[\frac{2}{r}-\frac{1}{a}]$
Hence proved.