Question

A satellite is revolving around a planet of mass M in an elliptic orbit of semi-major axis $a$. The orbital velocity of the satellite when it is at a distance $r$ from planet will be given by

• A. $\sqrt{GM(\frac{2}{r}-\frac{1}{a})}$
• B. $\sqrt{GM(\frac{2}{r}+\frac{1}{a})}$
• C. $\sqrt{GM(\frac{4}{r}-\frac{1}{a})}$
• D. $\sqrt{GM(\frac{4}{r}+\frac{1}{a})}$

Answer 1

The correct option is A $\sqrt{GM(\frac{2}{r}-\frac{1}{a})}$

Total energy of a satellite in elliptical orbit is:

$E=\frac{-G{M}_m}{2a}$ ($m\to$ mass of satellite)

Now,

$K.E+P.E=E$

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

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