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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 $7 R$, the minimum energy required to be spent by the launching vehicle on the satellite is (Gravitational constant $=G$ )
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The correct answer is:
$-\frac{13 G M m}{14 R}$
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
$E=E_1-E_2$
$=-\frac{G M m}{R}+\frac{1}{2}\left(\frac{G M m}{7 R}\right)$
$=\frac{-14 G M m+G M m}{14 R}$
$=-\frac{13 G M m}{14 R}$
$E_1=-\frac{G M m}{R}$
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
$E=E_1-E_2$
$=-\frac{G M m}{R}+\frac{1}{2}\left(\frac{G M m}{7 R}\right)$
$=\frac{-14 G M m+G M m}{14 R}$
$=-\frac{13 G M m}{14 R}$
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