Speed of a planet in an elliptical orbit with semi major axis 'a' about the sun of mass M at a distance 'r' from the sun is

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$\sqrt{G M (\frac{2}{r} - \frac{1}{a})}$

$\sqrt{\frac{G M r}{2 a^{2}}}$

$\sqrt{G M (\frac{2}{a} - \frac{1}{r})}$

$\sqrt{G M (\frac{1}{r} - \frac{1}{a})}$

answer is A.

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Detailed Solution

if major axis is ‘a’ then mean distance is also a and total energy is $- \frac{G M m}{2 a}$

at a distance r if the velocity is v then kinetic energy + potential energy = - $- \frac{G m M}{2 a}$

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

$v^{2} = 2 (\frac{G M}{r} - \frac{G M}{2 a}) v = \sqrt{G M (\frac{2}{r} - \frac{1}{a})}$

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