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A $400 \;kg$ satellite is in a circular orbit of radius $2 R_{E}$ about the Earth. How much energy is required to transfer it to a circular orbit of radius $4 R_{E} ?$ What are the changes in the kinetic and potential energles?
Solution
Answer Initially,
$E_{t}=-\frac{G M_{E} m}{4 R_{E}}$
While finally
$E_{f}=-\frac{G M_{E} m}{8 R_{E}}$
The change in the total energy $1 s$
$\Delta E=E_{f}-E_{t}$
$=\frac{G M_{E} m}{8 R_{E}}=\left(\frac{G M_{E}}{R_{E}^{2}}\right) \frac{m R_{E}}{8}$
$\Delta E=\frac{g m R_{E}}{8}=\frac{9.81 \times 400 \times 6.37 \times 10^{6}}{8}$$=3.13 \times 10^{9} J$
The kinetic energy is reduced and it mimics
$\Delta E,$ namely, $\Delta K=K_{f}-K_{t}=-3.13 \times 10^{9} J$
The change in potential energy is twice the change in the total energy, namely
$\Delta V=V_{f}-V_{t}=-6.25 \times 10^{9} J$
