The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius $R_{1}$ to another of radius $R_{2} (R_{2} > R_{1}) is$

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$2 GmM (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

$GmM (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

$\frac{1}{2} GmM (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

$GmM (\frac{1}{{R_{1}}^{2}} - \frac{1}{{R_{2}}^{2}})$

answer is D.

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Detailed Solution

Total energy of the orbiting satellite of mass m having orbital radius r is

E = $- \frac{GMm}{2 r}$ where M is the mass of the planet.

Additional kinetic energy required to transfer the satellite from a circular orbit of radius $R_{1} to another radius R_{2} is$

= $E_{2} - E_{1}$

= $- \frac{GMm}{2 R_{E}} - (- \frac{GMm}{2 R_{1}}) = - \frac{GMm}{2 R_{2}} + \frac{GMm}{2 R_{1}}$

= $\frac{GMm}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

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