Q: If a satellite is revolving around a planet of mass M in an elliptic orbit of semi major axis a, then show that the orbital speed of the satellite when it is at a distance r from the focus will be given by $v = \sqrt{GM (\frac{2}{r}-\frac{1}{a})}$

Sol: Total energy of the system is E = -GMm/2a

Which is conserved. So, KE + PE = -GMm/2a

At position ‘r’, orbital speed of the satellite is v .

Then, KE = (1/2)mv^{2} and PE = (-GMm)/r

$\large -\frac{G M m}{r} + \frac{1}{2} m v^2 = -\frac{G M m}{2a}$

$\large v = \sqrt{GM (\frac{2}{r}-\frac{1}{a})}$