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The mass of the Moon is $$\frac{1}{81}$$ times the mass of Earth while the radius of the Moon is $$\frac{1}{3.7}$$ times the radius (R) of the Earth. At what height h above the surface of Earth, an object will have the same weight as it weighs on the surface of the Moon?
On the Moon, $$g_m = \frac{G(M/81)}{(R/3.7)^2} = g \times \frac{3.7^2}{81}$$. At height h above Earth, $$g_h = g\frac{R^2}{(R + h)^2}$$. Equating, $$\frac{R + h}{R} = \frac{9}{3.7} \approx 2.43$$, so $$h \approx 1.43R$$.
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