Escape Velocity

Escape velocity is the minimum velocity required to escape the gravitational field of a body from its surface.

v_{\text{esc}} = \sqrt{\frac{2GM}{r}}

An object has ‘escaped’ from a gravitational field when its total orbital energy is equal to zero. This is because the object must have enough kinetic energy to continue moving away until its gravitational potential energy is equal to zero. Otherwise, the object will return from whence it came.

Begin the derivation by equating the total orbital energy to zero.

K+U=0

Next, substitute the relevant formulae for both the kinetic (K) and potential (U) elements of the orbital energy, as seen below. The large M refers to the mass of the larger body, while the smaller m refers to the escaping object.

\frac{1}{2}mv_{\text{esc}}^2 - \frac{GMm}{r}=0

Finally, cancel m and re-arrange the equation. Take the square root of both sides to arrive at the final equation for escape velocity.

v_{\text{esc}}^2 = \frac{2GM}{r} \space \space \rightarrow \space \space v_{\text{esc}} = \sqrt{\frac{2GM}{r}}