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A satellite of mass $m$ is in circular orbit of radius $3 R_E$ about earth (mass of earth $M_{E}$, radius of earth $R_{E}$). How much additional energy is required to transfer the satellite to an orbit of radius $9 R_{E}$?
$\frac{{G{M_E}m}}{{18{R_E}}}$
$\;\frac{{3G{M_E}m}}{{2{R_E}}}$
$\;\frac{{G{M_E}m}}{{9{R_E}}}$
$\;\frac{{G{M_E}m}}{{3{R_E}}}$
Solution
Initial total energy of the satellite is
$E_{i}-\frac{G M_{E} m}{6 R_{E}}$
Final total energy of the satellite is
$E_{f}=-\frac{G M_{E} m}{18 R_{E}}$
The change in the total energy is
$\Delta E=E_{f}-E_{i}$
$\Delta E=-\frac{G M_{E} m}{18 R_{E}}-\left(-\frac{G M_{E} m}{6 R_{E}}\right)$
$=-\frac{G M_{E} m}{18 R_{E}}+\frac{G M_{E} m}{6 R_{E}}=\frac{G M_{E} m}{9 R_{E}}$
Thus, the energy required to transfer the satellite to the desired orbit $=\frac{G M_{E} m}{9 R_{E}}$
