Escape speed is the minimum speed with which a mass should be projected from the Earth’s surface in order to escape Earth’s gravitation field.

$V_{escape} \, = \, \sqrt{2g R_{E}}$

, where

- V is escape speed,
- g is gravitational field strength,
- R is radius of the Earth.

### Derivation of Escape Speed From Earth

We know that:

$$\text{Total Energy at infinity} = 0$$

Hence,

$$\begin{aligned} \text{Kinetic energy } + \text{ Potential energy} &= 0 \\ \frac{1}{2} m v^{2} + \left( – \frac{\left( G M m \right)}{R} \right) &=0 \\ \frac{1}{2} m v^{2} &= \frac{ G M m}{R} \\ v^{2} &= 2 \frac{G M}{R} \end{aligned}$$

where v is the velocity of the object

m is the mass of the object

M is the mass of Earth

R is the radius of the Earth

G is the universal gravitation constant

From Gravitational Field Strength, we know that $g = G \frac{M}{R^{2}}$. Substitute this into the equation above, we will have:

$$v^{2} = 2 gR$$

In the context of this derivation, we have:

$V_{escape} \, = \, \sqrt{2g R_{E}}$