Escape Velocity

Escape velocity v_esc is the MINIMUM speed needed to escape a gravitational field, going to infinity with zero residual KE.

Derivation: ½mv_esc² = GMm/R ⇒ v_esc = √(2GM/R) = √(2gR).

Earth: v_esc ≈ 11.2 km/s. Moon: ~2.4 km/s. Sun (at its surface): ~617 km/s.

Relation to orbital velocity: v_esc = √2 · v_orb. So escape = ~1.41× orbital.

Independent of the direction of launch (assumes no atmosphere, no rotation).

Escape velocity does NOT depend on the escaping body's mass. Same v for an atom or a rocket.

If launch speed > v_esc: body follows a hyperbolic trajectory. v = v_esc: parabola. v < v_esc: ellipse (bound).

Above v_esc, body has POSITIVE total energy ⇒ unbound, leaves to infinity.