Physics - Gravitation Question with Solution | TestHub

As we know,
Gravitational potential cnergy$=\frac{-\mathrm{GMm}}{\mathrm{r}}$
and orbital velocity,${\mathrm{v}}_0=\sqrt{\mathrm{GM}/\mathrm{R}+\mathrm{h}}$
${\mathrm{E}}_{\mathrm{f}}=\frac{1}{2}{\mathrm{mv}}_0^2-\frac{\mathrm{GMm}}{3\mathrm{R}}=\frac{1}{2} \mathrm{m}\frac{\mathrm{GM}}{3\mathrm{R}}-\frac{\mathrm{GMm}}{3\mathrm{R}}$
$=\frac{\mathrm{GMm}}{3\mathrm{R}}\left(\frac{1}{2}-1\right)=\frac{-\mathrm{GMm}}{6\mathrm{R}}$
${\mathrm{E}}_{\mathrm{i}}=\frac{-\mathrm{GMm}}{\mathrm{R}}+\mathrm{K}$
${\mathrm{E}}_{\mathrm{i}}={\mathrm{E}}_{\mathrm{f}}$
Therefore minimum required energy,
$\mathrm{K}=\frac{5\mathrm{GMm}}{6\mathrm{R}}$