Gravitational Potential Energy Near the Surface of the Earth

The formula $U=mgh$, used constantly for objects near the Earth's surface, is not an independent rule -- it is simply the general result $U(r)=-GM_em/r$ approximated for the special case where the height $h$ above the surface is tiny compared with the Earth's radius $R_e$.

Derivation. With $r=R_e+h$, the general potential energy is

Since $h\ll R_e$, a first-order binomial expansion $(1+x)^{-1}\approx 1-x$ gives

Using $GM_em/R_e^2 = mgR_e/R_e = mg$ (from $GM_e/R_e^2=g$), this becomes

The first term, $-mgR_e$, does not depend on $h$ at all -- it is a fixed constant offset for the chosen reference (infinity). Since only differences in potential energy have physical meaning, this constant term can simply be dropped, or equivalently, we can choose the surface of the Earth ($h=0$) as our new zero reference. What remains is exactly the familiar

$U=mgh,$ …