When a body is projected vertically upward with velocity v (not necessarily small compared to v_e), the maximum height is NOT simply v^2/(2g).

Correct formula (using energy conservation): (1/2)mv^2 - GMm/R = -GMm/(R+h) Solving: h = v^2*R / (2gR - v^2)

For v << v_e: h ≈ v^2/(2g) (standard formula) For v = v_e: h -> infinity (escapes) For v = v_e/2: h = R/3

This is a common JEE numerical question.