If a satellite revolving around a planet of a mass M in an elliptical orbit of semi major axis a then which of following equations are correct, when satellite is at a distance r from the focus.

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$v^{2} = G M [\frac{2}{r} - \frac{1}{a}]$

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

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

$v = G M [\frac{2}{r} - \frac{1}{a}]$

answer is A, B.

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Detailed Solution

Total mechanical energy of the system is $E = \frac{G M m}{2 a}$ which remains conserved

$K E + P E = \frac{- G M m}{2 a}$

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

Then $K E = \frac{1}{2} M v^{2}, P E = \frac{- G M m}{r}$

So, $\frac{1}{2} m v^{2} - \frac{G M m}{r} = \frac{- G M m}{2 a} o r V^{2} = G M (\frac{2}{r} - \frac{1}{a})$

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