Escape Speed (DP IB Physics: HL): Revision Note

Escape Speed

• To escape a gravitational field, a mass must travel at, or above, the minimum escape speed

  • 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 speed 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

  • For example, the escape speed of a rocket is the same as a tennis ball on Earth

• The escape speed of an object is the speed at which all its kinetic energy has been transferred to gravitational potential energy

• This is calculated by equating the equations:

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

• Where:

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

  • $v_{esc}$ = escape velocity of the object (m s⁻¹)

  • G = Newton's Gravitational Constant

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

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

• Since mass m is the same on both sides of the equation, it can be cancelled on both sides of the equation:

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

• Multiplying both sides by 2 and taking the square root gives the equation for escape velocity $v_{esc}$:

$v_{esc}=\sqrt{\frac{2GM}{r}}$

For an object to leave the Earth's gravitational field, it will have to travel at a speed greater than the Earth's escape velocity, v

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