The ratio of energy required to raise a satellite to a height ' \(h\) ' above the earth's surface to that required to put it into the orbit at the same height is ( \(\mathrm{R}=\) radius of earth)

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
\(\mathrm{E}_2=\frac{-\mathrm{GMm}}{2(\mathrm{R}+\mathrm{h})}+\frac{\mathrm{GMm}}{\mathrm{R}}\)
\(=\operatorname{mgR}\left[1-\frac{1}{2\left(1+\frac{\mathrm{h}}{\mathrm{R}}\right)}\right]\left(\because \mathrm{GM}=\mathrm{gR}^2\right)\)
\(\begin{aligned} \therefore \quad E_2 & =\frac{m g R\left(\frac{2 h}{R}+1\right)}{2\left(1+\frac{h}{R}\right)} \\ \therefore \quad \frac{E_1}{E_2} & =\frac{m g h}{1+\frac{h}{R}} \times \frac{2\left(1+\frac{h}{R}\right)}{m g R} \\ & =\frac{2 h}{R} \quad\left(\because h \ll R \Rightarrow 1+\frac{2 h}{R} \approx 0\right)\end{aligned}\)