Physics - Gravitation Question with Solution | TestHub

We know

${\text{V}}_0=\sqrt{\frac{\text{GM}}{\text{r}}}\text{\&}\text{g}=\frac{\text{GM}}{{\text{R}}^2}$

From energy conservation

${\text{U}}_{\text{i}}+{\text{K}}_{\text{i}}={\text{U}}_{\text{f}}+{\text{K}}_{\text{f}}$

$-\frac{\text{GM}\text{m}}{\text{R}}+{\text{K}}_{\text{i}}=-\frac{\text{GM}\text{m}}{2\text{R}}+\frac{1}{2}{\text{mv}}_0^2$

${\text{K}}_{\text{i}}=\frac{\text{GM}\text{m}}{2\text{R}}+\frac{1}{2}\text{m}{\left(\sqrt{\frac{\text{GM}}{2\text{R}}}\right)}^2\Rightarrow {\text{K}}_{\text{i}}=\frac{3\text{GM}\text{m}}{4\text{R}}\Rightarrow {\text{K}}_{\text{i}}=\frac{3}{4}\text{mgR}$