Derivation of the day: escape velocity

The escape velocity is the velocity a small body needs to escape from the surface of a large body.

M = mass of large body
R = radius of large body
m = mass of small body
vesc = escape velocity of small body
r = orbital distance of small body

The kinetic energy is K = 1/2 m v^2
The potential energy is U = - GMm/r

If we equation the kinetic energy with the magnitude of the potential energy at the surface of the large body:
1/2 m vesc^2 = G M m/ R
vesc = sqrt(2GM/R)

Drop constants of order unity:

vesc = sqrt(GM/R)

Equivalently, we can say the particle can escape if it has a total energy greater than 0:
K + U > 0
1/2 m vesc^2 - G M m/ R > 0
vesc > sqrt(GM/R)