Escape Velocity
Escape velocity is the minimum speed an unpowered object needs at a given distance from a massive body to coast to infinity, ending with zero kinetic and zero gravitational potential energy.
Derivation from energy conservation
At distance r from a body of mass M, the gravitational potential energy of a small mass m is U = −G·M·m/r, and its kinetic energy is K = ½·m·v². For the object to just barely escape, the total mechanical energy must be zero: ½·m·v² − G·M·m/r = 0. Solving for v gives the escape speed, which is independent of the escaping object's mass.
Numerical values
From the surface of the Earth (r ≈ 6 371 km, μ ≈ 3.986 × 10¹⁴ m³·s⁻²), v_esc ≈ 11.19 km/s, or about 40 270 km/h. From the surface of the Moon it is only 2.38 km/s, which is why the Apollo lunar module needed comparatively little propellant to leave the Moon. Escape speed decreases as √(1/r), so a satellite already in low Earth orbit needs a smaller boost to reach interplanetary space.
Relationship to orbital energy
Escape velocity is exactly √2 times the speed of a circular orbit at the same radius. In terms of conic sections, a trajectory at exactly v_esc is a parabola (eccentricity 1); below that it is an ellipse (bound), and above that a hyperbola (unbound). In the simulator, raising the spacecraft's speed past the local escape value will turn a closed orbit into an open one.
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