Physics - Gravitation Question with Solution | TestHub

The formula for the energy required to raise a satellite to height $\mathrm{h}$ is ${\mathrm{E}}_1=\Delta \mathrm{U}=\frac{\mathrm{mgh}}{1+\frac{\mathrm{h}}{\mathrm{R}}}=\frac{\mathrm{mghR}}{\mathrm{R}+\mathrm{h}}$ The formula for the energy required to set the satellite in orbit is ${\mathrm{E}}_2=\frac{-\mathrm{GMm}}{2(\mathrm{R}+\mathrm{h})}+\frac{\mathrm{GMm}}{\mathrm{R}}$ $=\mathrm{mgR}\left[1-\frac{1}{2\left(1+\frac{\mathrm{h}}{\mathrm{R}}\right)}\right]\left(\because \mathrm{GM}={\mathrm{gR}}^2\right)$ $\begin{array}{rl}\therefore E_2& =\frac{mgR\left(\frac{2h}{R}+1\right)}{2\left(1+\frac{h}{R}\right)}\\ \therefore \frac{E_1}{E_2}& =\frac{mgh}{1+\frac{h}{R}}\times \frac{2\left(1+\frac{h}{R}\right)}{mgR}\\ & =\frac{2h}{R}\left(\because h\ll R\Rightarrow 1+\frac{2h}{R}\approx 0\right)\end{array}$