A launching vehicle carrying an artificial satellite of mass $m$ is set for launch on the surface of the earth of mass $M$ and radius $R$ . If the satellite is intended to move in a circular orbit of radius $7 R$ , the minimum energy required to be spent by the launching vehicle on the satellite is :
(Gravitational constant $= G$ )

\frac{G M m}{R}

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- \frac{13 G M m}{14 R}

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\frac{G M m}{7 R}

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\frac{G M m}{14 R}

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Solution

The correct option is B $- \frac{13 G M m}{14 R}$
The energy of artificial satellite at the surface of the earth
$E_{1} = \frac{G M m}{R}$
When the satellite is intended to move in a circular orbit of radius $7 R$ , then energy of artificial satellite
$E_{2} = - \frac{1}{2} \frac{G M m}{7 R}$
The minimum energy required
$E = E_{1} - E_{2}$
$= - \frac{G M n}{R} + \frac{1}{2} (\frac{G M m}{7 R})$
$= \frac{- 14 G M m + G M m}{14 R}$
$= - \frac{13 G M m}{14 R}$

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