Escape Velocity (AQA A Level Physics): Revision Note

Escape velocity

• To escape a gravitational field, a mass must travel at the escape velocity

• This is dependent on the mass and radius of the object creating the gravitational field, such as a planet, a moon or a black hole

• Escape velocity is defined as:

  The minimum speed that will allow an object to escape a gravitational field with no further energy input
  

• It is the same for all masses in the same gravitational field ie. the escape velocity of a rocket is the same as a tennis ball on Earth

• An object reaches escape velocity when all its kinetic energy has been transferred to gravitational potential energy

• This is calculated by equating the equations:

$\frac{1}{2}m{v}^2=\frac{GMm}{r}$

• Where:

  • $m$ = mass of the object in the gravitational field (kg)

  • $v$ = escape velocity of the object (m s^-1)

  • $G$ = Newton's gravitational constant

  • $M$ = mass of the object to be escaped from (ie. a planet) (kg)

  • $r$ = distance from the centre of mass $M$ (m)

• Since mass $m$ is the same on both sides of the equations, it can cancel on both sides of the equation:

$\frac{1}{2}{v}^2=\frac{GM}{r}$

• Multiplying both sides by 2 and taking the square root gives the equation for escape velocity, $v$:

$v=\sqrt{\frac{2GM}{r}}$

• This equation is not given on the datasheet — be sure to memorise how to derive it

• Rockets launched from the Earth's surface do not need to achieve escape velocity to reach their orbit around the Earth

• This is because:

  • they are continuously given energy through fuel and thrust to help them move

  • less energy is needed to achieve orbit than to escape from Earth's gravitational field

• The escape velocity is not the velocity needed to escape the planet but to escape the planet's gravitational field altogether

  • This could be quite a large distance away from the planet