Gravitaional potential energy

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Question

A satellite is moving around the earth in a circular orbit. Radius of the orbit is r and kinetic energy is K. Another satellite is moving around the earth in circular orbit of radius 2r. Then total energy of the second satellite is

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Solution

$V = \sqrt{\frac{G M}{r}} a n d K = \frac{1}{2} m V^{2} = \frac{G M m}{2 r}$
For the second satellite, $V' = \sqrt{\frac{G M}{2 r}} = \frac{V}{\sqrt{2}}$
$∴ K' = \frac{1}{2} m {(V')}^{2} = \frac{1}{4} m V^{2} = \frac{K}{2}$
$∴ E' = - K' = - \frac{K}{2}$

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