The minimum energy required to launch a satellite of mass $m$ from the surface of earth of mass $M$

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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 $2R$ from the surface of the earth is:

Accepted Answer

Sol. [Key Concept: Energy to launch a satellite (binding energy)] Apply energy conservation,$$U_{i} + K_{i} = U_{f} + K_{f}$$$$\begin{matrix} & \ \Rightarrow \ - \frac{GMm}{R} + K_{i} = - \frac{GMm}{3R} + \frac{1}{2}mv^{2} \ & \ \Rightarrow \ - \frac{GMm}{R} + K_{i} = - \frac{GMm}{3R} + \frac{1}{2} 	imes m 	imes \frac{GM}{3R} \ & \ \Rightarrow \ K_{i} = - \frac{1}{6}\frac{GMm}{R} + \frac{GMm}{R} \end{matrix}$$$$K_{i} = \frac{5}{6}\frac{GMm}{R}$$

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