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

Question

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

Professor PC Thomas Classes · Answer

(a) : Energy of satellite on surface of earth, \ $ E_i=-\frac{G m M}{R} $ The energy at distance $(2 R+R=3 R)$, \ $ E_f=\frac{-G M m}{2(3 R)}=\frac{-G M m}{6 R} $ \ Energy required to launch the satellite, \ $ \begin{aligned} E=E_f-E_i & =\frac{-G M m}{6 R}-\left(\frac{-G M m}{R}ight) \ & =\frac{5 G M m}{6 R} \end{aligned} $