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:
- Where:
- m = mass of the object in the gravitational field (kg)
- = 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 cancel on both sides of the equation:
- Multiplying both sides by 2 and taking the square root gives the equation for escape velocity :
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 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
Worked example
Calculate the escape speed at the surface of the Moon.
- Density of the Moon = 3340 kg m−3
- Mass of the Moon = 7.35 × 1022 kg
Answer:
Step 1: List the known quantities
- Gravitational constant, G = 6.67 × 10−11 N m2 kg−2
- Density of the Moon, ρ = 3340 kg m−3
- Mass of the Moon, M = 7.35 × 1022 kg
Step 2: Rearrange the density equation for radius r
Density: and volume of a sphere:
Step 3: Calculate the radius by substituting in the values
Step 4: Substitute r into the escape speed equation
Escape speed of the Moon: = 2.37 km s−1
Examiner Tip
When writing the definition of escape velocity, avoid terms such as 'gravity' or the 'gravitational pull / attraction' of the planet. It is best to refer to its gravitational field. This equation is given on the data sheet, but make sure you know how it is derived.