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The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius $R_1$ to another of radius $R_2$ ($R_2 > R_1$) is
A
$GmM\left(\frac{1}{R_1^2} - \frac{1}{R_2^2}\right)$
B
$GmM\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
C
$2GmM\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
D
$\frac{1}{2}GmM\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
Detailed Solution
Total energy of a satellite in a circular orbit of radius R: $E = KE + PE = \frac{GMm}{2R} - \frac{GMm}{R} = -\frac{GMm}{2R}$
Energy in the first orbit: $E_1 = -\frac{GMm}{2R_1}$; energy in the second orbit: $E_2 = -\frac{GMm}{2R_2}$
Energy to be supplied: $E_1 + \Delta E = E_2$
$\Delta E = E_2 - E_1 = -\frac{GMm}{2R_2} + \frac{GMm}{2R_1}$
$\Delta E = \frac{1}{2}GmM\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
Energy in the first orbit: $E_1 = -\frac{GMm}{2R_1}$; energy in the second orbit: $E_2 = -\frac{GMm}{2R_2}$
Energy to be supplied: $E_1 + \Delta E = E_2$
$\Delta E = E_2 - E_1 = -\frac{GMm}{2R_2} + \frac{GMm}{2R_1}$
$\Delta E = \frac{1}{2}GmM\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$