Physics - Gravitation Question with Solution | TestHub

The total mechanical energy of a satellite in a circular orbit of radius $r$ is $E=-\frac{G{M}_{E}m}{2r}$. The initial orbital radius is $r_1=R_E+R_E=2R_E$. The initial energy is $E_1=-\frac{G{M}_{E}m}{2(2R_E)}=-\frac{G{M}_{E}m}{4R_E}$. The final orbital radius is $r_2=R_E+3R_E=4R_E$. The final energy is $E_2=-\frac{G{M}_{E}m}{2(4R_E)}=-\frac{G{M}_{E}m}{8R_E}$. The energy required to move the satellite is $\Delta E=E_2-E_1=-\frac{G{M}_{E}m}{8R_E}-(-\frac{G{M}_{E}m}{4R_E})=\frac{G{M}_{E}m}{4R_E}-\frac{G{M}_{E}m}{8R_E}=\frac{2G{M}_{E}m-G{M}_{E}m}{8R_E}=\frac{G{M}_{E}m}{8R_E}$.