Q.Prove that at points near the surface of the Earth, the gravitational potential energy of the object is $U=mgh$.

Step 1. For a mass $m$ at height $h$ above Earth's surface, its distance from Earth's centre is $r=R_e+h$, so its general potential energy is

Step 2. Since $h\ll R_e$ near the surface, apply the binomial approximation $(1+x)^{-1}\approx1-x$ for small $x=h/R_e$:

Step 3. Using $GM_e/R_e^2=g$ (the surface value), the second term becomes $mgh$, so

$U\approx-mgR_e+mgh.$ …

Binomially expand the general potential energy formula for small h/Re an …

• Keeping the constant -mgRe term instead of recognising it can be dropped (or re-zeroed) since only differences in U matter. …