Escape Velocity

Escape velocity is the minimum initial speed needed to escape a planet's gravity without additional propulsion.

Consider an object of mass $m$ at the surface of a planet with mass $M$ and radius $R$. The total mechanical energy at launch is

$$E = \frac{1}{2}mv^2 - \frac{GMm}{R}$$

The first term is kinetic energy. The second is gravitational potential energy, which is negative because we define zero potential at infinity.

To just barely escape, the object must reach infinity with zero speed. At infinity, both kinetic and potential energy vanish. By conservation of energy:

$$\frac{1}{2}mv^2 - \frac{GMm}{R} = 0$$

Rearranging:

$$\frac{1}{2}mv^2 = \frac{GMm}{R}$$

The mass $m$ cancels, showing escape velocity is independent of the object's mass:

$$\frac{1}{2}v^2 = \frac{GM}{R}$$

Solving for $v$:

$$v_{\text{esc}} = \sqrt{\frac{2GM}{R}}$$

Escape velocity depends only on the planet's mass $M$ and radius $R$. A faster launch leaves residual speed at infinity. A slower launch means the object falls back.