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## Homework Statement

Show that the variation of gravity with height can be accounted for approximately by the following potential function

V = mgz(1+z/r

_{e})

in which r

_{e}is the radius of the Earth. find the force given by the above potential function.

## Homework Equations

V = GM/r

## The Attempt at a Solution

Let z be a point above the surface of the Earth and R be the radius of the earth

V(z) = -GM/(R+z) = (-GM/R)(1+z/R)

^{-1}

Using the binomal expansion of (1+x)

^{n}≈ 1 +nx +n(n-1)x

^{2}/2

I get

V = (-Gm/R)(1-z/R + (z/r)

^{2}

Expanding gives

V = -GM/R + GMz/R

^{2}- GMz

^{2}/R

^{3}

but the force is given by

F = GM/R

^{2}= mg

replace GM/R

^{2}with mg to get

V = -mgR + mgz - mgz

^{2}/R

= -mgR + mgz(1-z/R)

I'm not sure why I have an extra -mgR term.