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First teaching 2023

First exams 2025

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Velocity Addition Transformations (HL) (HL IB Physics)

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Ashika

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Ashika

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Velocity Addition Transformations

  • Similar to velocity addition to Galilean transformations, the Lorentz transformation equations lead to relativistic velocity addition equations
  • These are again used when there are multiple velocities in the scenario but now some are close to the speed of light
  • Let's go back to the example of Person F in the rocket ship. They now release a missile in front of them
  • In this example:
    • u is the speed of the missile measured in frame S (by Person E)
    • u' is the speed of the missile measured in frame S' (by Person F)
    • v is the speed of frame S' (Person F)

1-5-6-lorentz-velocity-transformation-ib-2025-physics

Person F releases a missile in front of them. Both observers will view the missile travelling at different speeds

  • In Galilean velocity addition, when v << c, these were:
    • The speed of the missile as measured by Person E: u space equals space u apostrophe space plus thin space v
    • Or,  u apostrophe space equals space u space minus space v
  • If v and u' are close to the speed of light, we have to use Lorentz velocity addition transformations instead
  • These equations are:

u apostrophe space equals space fraction numerator u space minus space v over denominator 1 space minus fraction numerator space u v over denominator c squared end fraction end fraction

u space equals space fraction numerator u apostrophe space plus space v over denominator 1 space plus fraction numerator space u apostrophe v over denominator c squared end fraction end fraction

  • Where:
    • u = the velocity of an object measured from the stationary reference frame
    • u' = the velocity of an object measured from a moving reference frame
    • v = the velocity of the moving reference frame
    • c = the speed of light

Worked example

A rocket moves to the right with speed 0.60c relative to the ground.

1-5-6-lorentz-velocity-worked-example-ib-2025-physics

A probe is released from the back of the rocket at speed 0.82c relative to the rocket.

Calculate the speed of the probe relative to the ground.


Answer:

Step 1: List the known quantities

  • Speed of the rocket, v = 0.60c
  • Speed of the probe relative to the rocket, u' = 0.82c

Step 2: Analyse the situation

  • We have multiple velocities in this scenario in terms of c, so we need to use the Lorentz velocity addition equations
  • The probe is travelling in the opposite direction to the rocket, so its velocity is –0.82c
  • We want the speed relative to the ground, which is a reference frame at rest, so this is u

Step 3: Substitute values into the equation

u space equals space fraction numerator u apostrophe space plus space v over denominator 1 space plus fraction numerator space u apostrophe v over denominator c squared end fraction end fraction

u space equals space fraction numerator negative 0.82 c space plus space 0.60 c over denominator 1 space plus fraction numerator space open parentheses negative 0.82 c close parentheses open parentheses 0.60 c close parentheses over denominator c squared end fraction end fraction space equals space fraction numerator open parentheses negative 0.82 space plus space 0.60 close parentheses c over denominator 1 space plus open parentheses negative 0.82 close parentheses open parentheses 0.60 close parentheses end fraction

u space equals space minus 0.43 c

Examiner Tip

Be very careful which reference frame you are asked to calculate the velocity from, as this determines whether you find u or u'. Notice the equations are very similar, except one is with – and the other +. However, the signs will match on the numerator and denominator. 

The equation for u' is given in your data booklet.

Anytime you see the word 'relativistic' in physics such as 'relativistic speeds' it just means 'close to the speed of light'. Physics gets a bit weird at this point!

It is fine, and often encouraged, to give your final answers for relativistic velocities in terms of c. In the denominator of the velocity addition equations, the c2 will cancel out if two velocities u and v are given in terms of c.

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    Ashika

    Author: Ashika

    Expertise: Physics Project Lead

    Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.