Physics

Physics

Problem

For small heights hh (with h≪Rh \ll R), which of the following expressions correctly gives the acceleration due to gravity ghg_h at height hh above Earth’s surface? (Here gg is the surface value and RR is Earth’s radius.) a) gh=g(1−hR)g_h = g\left(1 - \dfrac{h}{R}\right) b) gh=g(1−h2R)g_h = g\left(1 - \dfrac{h}{2R}\right) c) gh=g(1−2hR)g_h = g\left(1 - 2\dfrac{h}{R}\right) d) gh=g(1−3h2R)g_h = g\left(1 - \dfrac{3h}{2R}\right)
Intuitive explanation

The key is recognizing that gravity follows an inverse-square law, so it weakens as you move away from Earth's center. For small heights where h≪Rh \ll R, you can use a Taylor expansion to approximate ghg_h without doing the full calculation—this reveals how gravity decreases linearly with height to first order. The trick is expanding 1(R+h)2\frac{1}{(R+h)^2} and keeping only the leading correction term.

Intuitive explanation

The key is recognizing that gravity follows an inverse-square law, so it weakens as you move away from Earth's center. For small heights where h≪Rh \ll R, you can use a Taylor expansion to approximate ghg_h without doing the full calculation—this reveals how gravity decreases linearly with height to first order. The trick is expanding 1(R+h)2\frac{1}{(R+h)^2} and keeping only the leading correction term.

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