The additional kinetic energy to be provided to a satellite of mass...

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\left(R_2>R_1\right)

Options

A $\mathrm{GmM}\left(\frac{1}{\mathrm{R}_1^2}-\frac{1}{\mathrm{R}_2^2}\right)$
B $\mathrm{GmM}\left(\frac{1}{\mathrm{R}_1}-\frac{1}{\mathrm{R}_2}\right)$
C $2 \mathrm{GmM}\left(\frac{1}{\mathrm{R}_1}-\frac{1}{\mathrm{R}_2}\right)$
D $\frac{1}{2} \mathrm{GmM}\left(\frac{1}{\mathrm{R}_1}-\frac{1}{\mathrm{R}_2}\right)$

Correct answer

D. $\frac{1}{2} \mathrm{GmM}\left(\frac{1}{\mathrm{R}_1}-\frac{1}{\mathrm{R}_2}\right)$

Step-by-step solution

The kinetic energy changing the orbit of satellite \begin{aligned} & \mathrm{KE}=\left(-\frac{\mathrm{GMm}}{2 \mathrm{R}_2}\right)-\left(-\frac{\mathrm{GMm}}{2 \mathrm{R}_1}\right) \\ & \mathrm{KE}=\frac{\mathrm{GMm}}{2}\left[\frac{1}{\mathrm{R}_1}-\frac{1}{\mathrm{R}_2}\right] \end{aligned}

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