Question

A satellite is revolving around a planet of mass $M$ in an elliptical orbit of semi major axis $a$. The orbital velocity of the satellite at a distance $r$ from the focus will be

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

Answer 1

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

Total energy of any satellite in elliptical orbit with semi-major axis a is: $E=\frac{-GMm}{2a}$ (Where m is the mass of satellite)
and it remains constant with time.
Therefore by using law of conservation of energy, in this situation we have:
$E=-\frac{GMm}{2a}=Potentialenergy+Kineticenergy$
$\Rightarrow -\frac{GMm}{2a}=-\frac{GMm}{r}+\frac{1}{2}m{v}^2$
where $m$ is mass of satellite, and $v$ is its orbital velocity at that point
Solving for $v$, we get: $v=\sqrt{GM(\frac{2}{r}-\frac{1}{a})}$