Escape velocity

Newton's concept of escape velocity.

Newton conducted a thought experiment where a cannon would be shot from a very tall mountain on Earth.

2WRQ-orbits.png
  • He thought that the faster the cannon was fired, the greater its range would be.
  • On the diagram:
    • Black paths: This is what would happen if the speed at which it fired wasn't fast enough to escape Earth's gravity. It falls back to Earth as the radius of its path wasn't greater than Earth's curvature.
    • Red path: This is what would happen when the velocity caused the cannonball to fall at the same radius of Earth, causing a circular orbit.
    • Green path: This is what would happen if the speed was increased, causing an elliptical orbit.
    • Blue and Purple paths: Eventually, the velocity becomes so great that it escapes Earth's gravitational pull and flies off into space. These orbits are called Parabolic (Blue) or Hyperbolic (Purple) orbits.

A formula for escape velocity

  • To find a mathematical formula for escape velocity, 2 things need to be considered:
    • The Law of Conservation of energy. This is so that Kinetic Energy + GPE = 0.
    • This is calculating from the minimum amount of energy needed to escape, so that $KE_\infty = 0$
(1)
\begin{align} \therefore GPE_{at Earth} + KE_{at Earth} = GPE_{at \infty} + KE_{at \infty} \end{align}
(2)
\begin{align} \therefore -\frac{GMm}{r} + \frac{1}{2}mv_e^2 = 0 + 0 \end{align}
(3)
\begin{align} \frac{1}{2}mv_e^2 = \frac{GMm}{r} \end{align}
(4)
\begin{align} v_e^2 = \frac{2GM}{r} \end{align}
(5)
\begin{align} v_e = \sqrt{\frac{2GM}{r}} \end{align}

Where:

  • $v_e$ is the escape velocity ($ms^{-1}$)
  • $G$ is the Universal Gravitation Constant ($6.67 \times 10^{-11}$)
  • $M$ is the mass of the PLANET (kg)
  • $r$ is the radius of the planet (m)