Show that the variation of gravity with height can be accounted for approximately by the following potential function
V = mgz(1+z/re)
in which re is the radius of the Earth. find the force given by the above potential function.
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)x2/2
V = (-Gm/R)(1-z/R + (z/r)2
V = -GM/R + GMz/R2 - GMz2/R3
but the force is given by
F = GM/R2 = mg
replace GM/R2 with mg to get
V = -mgR + mgz - mgz2/R
= -mgR + mgz(1-z/R)
I'm not sure why I have an extra -mgR term.