Derivation

Using energy conservation (set E_final = 0 at ∞):

Initial state (at surface): KE = ½mv_$e^{2}$, PE = −GMm/R, Total = ½mv_$e^{2}$ − GMm/R

Final state (at ∞, just barely): KE = 0, PE = 0, Total = 0

Energy conservation: ½mv_$e^{2}$ − GMm/R = 0

v_e = $\sqrt{2GM/R}$ = $\sqrt{2gR}$

Numerical Value for Earth

v_e = √(2 × 9.8 × $6.4 \times 10^{6}) = √(\1.2544 \times 10^{8}$) = 11,200 m/s = 11.2 km/s

Critical Independence Properties

v_e is independent of:

1. Mass of projected body (m cancels in derivation)
2. Angle of projection (energy approach has no direction dependence)
3. Shape of escape trajectory (only energy at surface matters)

v_e depends on:

1. Mass M of the planet (v_e ∝ √M)
2. Radius R of the planet (v_e ∝ 1/√R)

Scaling Rule for Other Planets

v_e(planet) / v_e(Earth) = $\sqrt{M_planet/M_Earth × R_Earth/R_planet}$ = $\sqrt{M'/M × R/R'}$

For same-density planets: $v_e \propto R$ (linear with radius)

Comparison with Other Planets

Why the Moon Has No Thick Atmosphere

Moon's escape velocity (2.4 km/s) is comparable to thermal velocities of gas molecules, so lighter gases escape over geological time.