Question

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 is

Difficult

Solution

$E_{1} = ∆ U = (\frac{m g h}{1 + h / R})$
E₂ = Energy of satellite-Energy of sateilite on surface of earth
$= - \frac{G M m}{2 (R + h)} + \frac{G M m}{R}$
$= m g R [1 - \frac{1}{2 (1 + h / R)}] (∵ G M = g R^{2})$
$\Rightarrow E_{2} = \frac{m g R (\frac{2 h}{R} + 1)}{2 (1 + \frac{h}{R})}$
$∴ \frac{E_{1}}{E_{2}} = \frac{m g h}{1 + \frac{h}{R}} \times \frac{2 (1 + \frac{h}{R})}{m g R} = \frac{2 h}{R} (\begin{array}{l} ∵ h < < R \\ ∴ 1 + \frac{2 h}{R} \approx 1 \end{array})$

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