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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 7R, the minimum energy required to be spent by the launching vehicle on the satellite is (Gravitational constant = G) [Manipal 2012]

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$\frac{G M m}{R}$

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

$\frac{G M m}{7 R}$

$\frac{G M m}{14 R}$

answer is B.

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Detailed Solution

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 7R, then energy of artificial satellite,

$E_{2} = - \frac{1}{2} \frac{G M m}{7 R}$

The minimum energy required,

$E = E_{2} - E_{1} = \frac{G M m}{R} - \frac{1}{2} (\frac{G M m}{7 R})$

$= \frac{14 G M m - G M m}{14 R}$

$= \frac{13 G M m}{14 R}$

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