Discussion
The height at which the weight of a body becomes one third of its weight on the surface of earth is (R is the radius of earth)
Answer: B
The height is 0.732 R.
Gravity at height h is \(g_h = \frac{g}{(1 + h/R)^{2}}\). The weight is one third when this ratio is \(\frac{1}{3}\).
\[ \begin{aligned} \left(1 + \frac{h}{R}\right)^{2} &= 3 \\ \frac{h}{R} &= \sqrt{3} - 1 = 0.732 \end{aligned} \]
\(\frac{R}{3}\) uses the approximation \(g_h \approx g(1 - \frac{2h}{R})\), which only holds for small heights. 0.414R (\(\sqrt{2} - 1\)) is the height at which the weight is halved.
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