Velocity Addition Transformations (HL) | HL IB Physics Revision Notes 2025

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)

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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: $u = u' + v$
- Or, $u' = u - v$

If v and u' are close to the speed of light, we have to use Lorentz velocity addition transformations instead
These equations are:

$u' = \frac{u - v}{1 - \frac{u v}{c^{2}}}$

$u = \frac{u' + v}{1 + \frac{u' v}{c^{2}}}$

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.

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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

$u = \frac{u' + v}{1 + \frac{u' v}{c^{2}}}$

$u = \frac{- 0.82 c + 0.60 c}{1 + \frac{(- 0.82 c) (0.60 c)}{c^{2}}} = \frac{(- 0.82 + 0.60) c}{1 + (- 0.82) (0.60)}$

$u = - 0.43 c$

Exam 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.

Velocity Addition Transformations (HL) (HL IB Physics)

Revision Note