The escape velocity of a planet is the velocity required to escape from the gravitational field of that planet.
The escape velocity, \(v\) is the minimum velocity needed by an object on the surface of the Earth to overcome the gravitational force and escape to outer space.
For a satellite of mass \(m\) at the surface of a planet, radius \(R\), the gravitational potential energy, \(U=\frac{-G M m}{R}\). Thus the energy required to escape from the gravitational field is \(\frac{+G M m}{R}\). Kinetic energy needed \(=\frac{1}{2} m v^2\) so \(\frac{1}{2} m v^2=\frac{G M m}{R}\)Escape velocity, \(v=\sqrt{\frac{2 G M}{R}}\)

Example

Given \(G=6.67 \times 10^{-11} N m ^2 kg ^{-2}\), mass of the Earth \(=5.97 \times 10^{24} kg\) and radius of the Earth \(=6.37 \times 10^6 m\).
(a) Calculate the escape velocity from the Earth.
(b) Calculate the radius of a planet with an escape velocity of 6.2 \(m s ^{-1}\). Assume the planet has the same average density as the Earth, \(5500 kg m ^{-3}\).
Solution
(a)
\[
\begin{array}{l}
M=5.97 \times 10^{24} kg , \\
R=6.37 \times 10^6 m , v=?
\end{array}
\]
Using escape velocity
\[
\begin{array}{l}
v=\sqrt{\frac{2 G M}{R}} \\
=\sqrt{\frac{\begin{array}{c}
2\left(6.67 \times 10^{-11}\right)(5.97 \times \\
\left.10^{24}\right)
\end{array}}{6.37 \times 10^6}} \\
=1.12 \times 10^4 ms ^{-1} \\
\end{array}
\]
(b) Escape velocity, \(v=\sqrt{\frac{2 G M}{R}}\)
Therefore, \(R=\frac{2 G M}{v^2}\)
\[
=\frac{2 G(4 / 3) \pi R^3 \rho}{v^2}
\]

Radius of planet, \(R\)
\[
\begin{array}{l}
=\sqrt{\frac{v^2}{2 G\left(\frac{4}{3}\right) \pi p}} \\
=\sqrt{\frac{(6.2)^2}{2\left(6.67 \times 10^{-11}\right) \times 1.33 \times}} \\
=3541 m
\end{array}
\]

Importance of escape velocity:
1. The high escape velocity of the Earth allows it to retain its atmosphere. The molecules in the atmosphere move with linear speed of \(500 m s ^{-1}\) which is lower than the escape velocity of the Earth.
2. The high escape velocity also enables commercial airplanes or fighter jets to reach high altitudes in the atmosphere without the possibility of flying into space. Airplanes can fly at a speed of \(250 m s ^{-1}\) while fighter jets can reach supersonic linear speeds of up to \(2200 m s ^{-1}\) which is lower than the escape velocity of the Earth.
3. A rocket launch requires a large quantity of fuel. The combustion of the fuel must produce sufficient thrust power to allow the rocket to reach the Earth’s escape velocity and send the spacecraft into space.

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