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  4. A satellite of mass m is in a circular orbit of radius 2RE about the earth. The energy required to transfer it to a circular orbit of radius 4RE is (where ME and RE is the mass and radius of the earth respectively)

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Q. A satellite of mass $m$ is in a circular orbit of radius $2R_E$ about the earth. The energy required to transfer it to a circular orbit of radius $4R_E$ is (where $M_E$ and $R_E$ is the mass and radius of the earth respectively)

Gravitation

Solution:

Initial total energy of the satellite is
$E_{i}= -\frac{GM_{E}m}{4R_{E}}$
Final total energy of the satellite is
$E_{f}= -\frac{GM_{E}m}{8R_{E}}$
The change in the total energy is
$\Delta E = E_{f} - E_{i}$
$ \Delta E = -\frac{GM_{E}m}{8R_{E}} -\left(- \frac{GM_{E}m}{4R_{E}}\right)$
$ = - \frac{GM_{E}m}{8R_{E}} + \frac{GM_{E}m}{4R_{E}} = \frac{GM_{E}m}{8R_{E}}$
Thus, the energy required to transfer the satellite to
the desired orbit $= \frac{GM_E m}{8R_E}$