3.7.2.4
Escape velocity: the minimum speed to leave a gravitational field
Gravitational Fields & Orbits — AQA A-Level Physics
Key Definition
Escape velocity — The minimum speed that will allow an object to escape a gravitational field with no further energy input.
$$v_{esc} = \sqrt{\frac{2GM}{r}}$$
Deriving the escape velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The minimum speed at which an object must be launched from the surface of a body to escape its gravitational field entirely (reach infinity with zero kinetic energyThe capacity to do work. Measured in joules (J).The energyThe capacity to do work. Measured in joules (J). an object possesses due to its motion.).
- At escape velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The minimum speed at which an object must be launched from the surface of a body to escape its gravitational field entirely (reach infinity with zero kinetic energyThe capacity to do work. Measured in joules (J).The energy an object possesses due to its motion.)., all KE converts to GPE. Set KE equal to the magnitude of GPE:
$$\frac{1}{2}mv^2 = \frac{GMm}{r}$$
- Cancel m, multiply by 2, and take the square root:
$$v = \sqrt{\frac{2GM}{r}}$$
- $v$: escape velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The minimum speed at which an object must be launched from the surface of a body to escape its gravitational field entirely (reach infinity with zero kinetic energyThe energy an object possesses due to its motion.). (m s⁻¹)
- $G$: Newton's gravitational constant
- $M$: mass of the body being escaped from (kg)
- $r$: distance from the centre of mass M (m)
- Escape velocity does not depend on the mass of the escaping object. A tennis ball and a rocket have the same escape velocity from Earth.
- This equation is not on the data sheet. You must be able to derive it.
- Rockets do not need to reach escape velocity to enter orbit. They receive continuous energy from fuel. Escape velocity refers to escaping the gravitational fieldA region of space in which a mass experiences a gravitational force. entirely.
Worked Example
Calculate the escape velocity at the Moon's surface. DensityMass per unit volume of a material. Measured in kg m⁻³. of the Moon = 3340 kg m⁻³. Mass of Moon = 7.35 × 10²² kg.
Show Solution
1
Find the Moon's radius from densityMass per unit volume of a material. Measured in kg m⁻³.
$$\rho = \frac{M}{V} = \frac{3M}{4\pi r^3}$$
$$r = \sqrt[3]{\frac{3M}{4\pi\rho}} = \sqrt[3]{\frac{3 \times 7.35 \times 10^{22}}{4\pi \times 3340}}$$
$$r = 1.74 \times 10^6 \text{ m}$$2
Substitute into escape velocity
$$v = \sqrt{\frac{2GM}{r}} = \sqrt{\frac{2 \times (6.67 \times 10^{-11}) \times (7.35 \times 10^{22})}{1.74 \times 10^6}}$$
3
Evaluate
$$v = 2370 \text{ m s}^{-1} = 2.37 \text{ km s}^{-1}$$
Answer
$v = 2.37$ km s⁻¹
Examiner Tips and Tricks
- When defining escape velocity, avoid saying 'escape the planet's gravity'.
- Say 'escape the gravitational fieldA region of space in which a mass experiences a gravitational force.'.
- Gravity has infinite range, so technically you never escape it, but at escape velocity you reach infinity with zero KE.