DP IB Physics: HL

Revision Notes

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First teaching 2014

Last exams 2024

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10.2.5 Escape Speed

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Escape Speed

  • 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:
1 half cross times m cross times v squared equals fraction numerator G cross times M cross times m over denominator r end fraction

  • 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:
1 half cross times v squared equals fraction numerator G cross times M over denominator r end fraction

  • Multiplying both sides by 2 and taking the square root gives the equation for escape velocity, v:
v equals square root of fraction numerator 2 cross times G cross times M over denominator r end fraction end root

  • This equation is not given on the datasheet. Be sure to memorise how to derive it

Escape Velocity Diagram, downloadable AS & A Level Physics revision notes

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 velocity at the surface of the Moon given that its density is 3340 kg m-3 and has a mass of 7.35 × 1022 kg.Newton's Gravitational Constant = 6.67 × 10-11 N m2 kg-2

Step 1: Rearrange the density equation for radius:

rho space equals space M over V  and  V space equals space 4 over 3 pi space r cubed

    

rho space equals space fraction numerator M over denominator 4 over 3 pi space r cubed end fraction space equals space fraction numerator 3 M over denominator 4 pi space r cubed end fraction

   

r space equals space cube root of fraction numerator 3 M over denominator 4 pi space rho end fraction end root

Step 2: Calculate the radius by substituting in the values:

  • M = 7.35 × 1022 kg
  • ρ = 3 340 kg m−3

r space equals space cube root of fraction numerator 3 space cross times space left parenthesis 7.35 space cross times space 10 to the power of 22 right parenthesis over denominator 4 pi space cross times space 3340 end fraction end root space equals space 1.7384 space cross times space 10 to the power of 6 space straight m

Step 3: Substitute r into the escape velocity equations:

V space equals space square root of fraction numerator 2 G M over denominator r end fraction end root space equals space square root of fraction numerator 2 space cross times space left parenthesis 6.67 cross times 10 to the power of negative 11 end exponent right parenthesis cross times left parenthesis 7.35 cross times 10 to the power of 22 right parenthesis over denominator 1.7384 space cross times space 10 to the power of 6 end fraction end root

   

V = 2.37 km s−1

Exam 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.

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