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 planet will be given by $V^2=GM[\frac{2}{r}-\frac{1}{a}]$.

Answer 1

In elliptical orbit , Total energy

$E=\frac{-GMm}{2a}=$ constant

At position r, if V is orbital speed of satellite the

$KE=\frac{1}{2}m{v}^2,PE=\frac{GMm}{r}$

$\therefore \frac{1}{2}m{v}^2-\frac{GHm}{r}=\frac{-GMm}{2a}$

$\Rightarrow V^2=GM(\frac{2}{r}-\frac{1}{a})$