Why the Moon lost its atmosphere.

Newton's escape velocity is the speed a single instantaneous kick needs to give an object so it never falls back, ignoring air resistance: v_esc = √(2GM/r), with G = 6.674 × 10⁻¹¹ N·m²/kg². Real spacecraft don't work this way — they climb under continuous engine thrust over minutes, not one instant shove — and this page ignores atmospheric drag entirely. This is also not a full model of atmospheric retention: what actually keeps a gas around is comparing escape velocity to the *statistical spread* of a gas's molecular speeds (heat means molecules constantly colliding and exchanging speed, so some are always moving much faster than the average), not a single number every molecule must clear at once. The Moon's low v_esc (2.38 km/s), combined with billions of years of solar heating, let enough of the fast tail of light gas molecules escape one by one that its atmosphere thinned to almost nothing; Earth's v_esc (11.19 km/s) is high enough that even light gases like hydrogen escape far more slowly, and heavier ones like nitrogen and oxygen barely escape at all. This page shows v_esc itself, honestly — not that fuller kinetic-theory picture.