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: (1/2)mv_1^2 - GMm/r_1 = (1/2)mv_2^2 - GMm/r_2

Examples:

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

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