Question
Knowing that the acceleration due to gravity on the Earth surface is g and the radius of the Earth is R, a small body of mass m falls on the Earth from a height h = R/5 above the Earth's surface. During the freefall, the potential energy of the falling body decreases by:
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Solution
[Key Concept: For heights comparable to R, use U = −GMm/r with GM = gR2 instead of mgh.]
$$\begin{matrix} & U_{I} = - \frac{{GM}_{e}\text{ }m}{R + h} \\ & U_{F} = - \frac{{GM}_{e}\text{ }m}{R} \\ & DU = U_{I} - U_{F} \\ & \Delta U = {GM}_{e}\text{ }m\left\lbrack \frac{1}{R} - \frac{1}{R + h} \right\rbrack \\ & \Delta U = {GM}_{e}\text{ }m\left\lbrack \frac{1}{R} - \frac{5}{6R} \right\rbrack \\ & \Delta U = {GM}_{e}\text{ }m\left\lbrack \frac{1}{6R} \right\rbrack = \frac{GMem}{R} \times \frac{1}{6} \\ & \ = m\left\lbrack \frac{{GM}_{e}}{R^{2}} \right\rbrack \times \frac{R}{6} \\ & \Delta U = \frac{{mgR}^{2}}{6} = \frac{5mgh}{6} \end{matrix}$$
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