The minimum energy required to launch a satellite of mass $m$ from the surface of earth of mass $M$ and…
The minimum energy required to launch a satellite of mass $m$ from the surface of earth of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ from the surface of the earth is:
$\frac{2 G m M}{3 R}$
$\frac{G m M}{2 R}$
$\frac{G m M}{3 R}$
$\frac{5 G m M}{6 R}$
Solution
Apply energy conservation,
$\begin{gathered}
U_i+K_i=U_f+K_f \\
\Rightarrow \quad-\frac{G M m}{R}+K_i=-\frac{G M m}{3 R}+\frac{1}{2} m v^2 \\
\Rightarrow \quad-\frac{G M m}{R}+K_i=-\frac{G M m}{3 R}+\frac{1}{2} \times m \times \frac{G M}{3 R} \\
\Rightarrow \quad K_i=-\frac{1}{6} \frac{G M m}{R}+\frac{G M m}{R} \\
K_i=\frac{5}{6} \frac{G M m}{R}
\end{gathered}$