A body of mass ' \(\mathrm{m}\) ' \(\mathrm{kg}\) starts falling from a distance 3R above earth's surface. When it reaches a distance ' \(R\) ' above the surface of the earth of radius ' \(R\) ' and Mass ' \(M\) ', then its kinetic energy is

Initial height:
\(\mathrm{h}=3 \mathrm{R}+\mathrm{R}=4 \mathrm{R}\)
The potential energy of the body initially will be:
\(\mathrm{U}_1=-\frac{1}{4} \frac{\mathrm{GMm}}{\mathrm{R}}\)
\(\therefore \quad\) At the height \(\mathrm{R}\),
\(\mathrm{h}=\mathrm{R}+\mathrm{R}=2 \mathrm{R}\)
\(\therefore \quad\) Potential energy:
\(\mathrm{U}_2=-\frac{1}{2} \frac{\mathrm{GMm}}{\mathrm{R}}\)
Gain in kinetic energy is equal to loss in potential energy.
\(
\begin{aligned}
\therefore \mathrm{KE} & =\mathrm{U}_1-\mathrm{U}_2 \\
& =-\frac{1}{4} \frac{\mathrm{GMm}}{\mathrm{R}}-\left(-\frac{1}{2} \frac{\mathrm{GMm}}{\mathrm{R}}\right)=\frac{1}{4} \frac{\mathrm{GMm}}{\mathrm{R}}
\end{aligned}
\)