The gravitational PE is U = -GMm/r (negative, zero at infinity).

Common mistake: Using mgh for large heights. This works ONLY for h << R where g is approximately constant.

Correct approach: Use energy conservation with U = -GMm/r: \frac{1}{2}$$mv_1^2 - GMm/$r_1$ = \frac{1}{2}$$mv_2^2 - GMm/$r_2$

Examples:

• Escape from surface: \frac{1}{2}$$mv_e^2 = $\frac{GMm}{R}$ -> $v_e$ = sqrt(2gR)
• Max height from speed v: h = v^2$$\frac{R}{2gR - v^2}
• Speed at height h with initial speed v: $v_h^2$ = $v^2$ - 2$gR^2$(1/R - $\frac{1}{R+h}$)

Work done against gravity (surface to height h): W = GMm(1/R - $\frac{1}{R+h}$) = $\frac{mgRh}{R+h}$ For h << R: W ≈ mgh (standard formula) For h = R: W = mgR/2 (NOT mgR)