variation of g with height

Question

At small height h ≪ R above Earth, g decreases approximately by factor?

RankGuru · Accepted Answer

1. **General Case**: Acceleration at height $h$ is $g_h = \frac{GM}{(R+h)^2}$.
2. **Relation to Surface g**: $g_s = \frac{GM}{R^2}$. So, $g_h = g_s \frac{R^2}{(R+h)^2} = g_s (1 + \frac{h}{R})^{-2}$.
3. **Approximation**: For $h \ll R$, we use the binomial expansion $(1+x)^n \approx 1 + nx$. Here $x = h/R$ and $n = -2$.
4. **Result**: $g_h \approx g_s (1 - \frac{2h}{R})$.
5. **Impact**: Gravity decreases as we go higher; the decrease is roughly linear for small heights.

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variation of g with height

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