Q: The minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R is

Sol: From the law of conservation of energy

$\large -\frac{G M m}{R} + K.E = -\frac{G M m}{2 \times 3R} $

$\large K.E = \frac{G M m}{R}(1-\frac{1}{6})$

$\large = \frac{5}{6}(\frac{G M m}{R})$