Escape speed

Escape speed is the least launch speed that lets a body reach infinity and never return, with air resistance ignored.

Energy conservation: ½ m vᵢ² − G M_E m / (R_E + h) must be at least zero, the least total energy a body can have at infinity.

From the surface (h = 0): vₑ = √(2 G M_E / R_E) = √(2 g R_E) ≈ 11.2 km/s.

Escape speed does not depend on the mass of the body or the direction of launch; it does depend on the planet and on the launch height.

For the Moon, with its smaller g and radius, the escape speed is about 2.3 km/s, about five times smaller. Gas molecules moving faster than that leave for good, which is why the Moon has no atmosphere.

Between two spheres of masses M and 4M, radius R, centres 6R apart, the pulls cancel at 2R from the smaller one. A body fired from M only needs to reach that neutral point: v_min = (3GM/5R)^½.