Relativistic addition of velocities

Imagine you are a stationary observer on Earth and see a spaceship travelling at 0.9c. The spaceship fires a rocket at 0.3c relative to it. According to classical physics, one would say that you would observe the rocket to be travelling at 1.2c (0.9c + 0.3c). But we learned in the previous section that nothing can travel faster than the speed of light, so classical physics is wrong here. This is where relativistic addition of velocities come into play. The 2 equations which are used to calculate relativistic velocities are:

Calculating relativistic velocities

To figure out which equation to use when solving relativistic velocity questions, we first need to define the variables. There are only 3 variables used in the above equations; u, u’, and v. Consider you are an observer in an inertial frame of reference and you see another observer (ie. another frame of reference) travelling at speed v. Therefore, the variable v can be thought of as the velocity of the moving frame of reference as observed by you. Now imagine there is a travelling object which is visible through your frame of reference and the moving frame of reference. The velocity of the object that you measure in your inertial frame of reference is considered to be the variable u and the velocity of the object that is measured through the moving frame of reference is considered to be the variable u’.

Remember the earlier example of a spaceship firing a rocket. The speed of the spaceship is the variable v. The speed of the rocket that you, a stationary observer on Earth will measure, is the variable u. The speed of the rocket that the spaceship, which is a moving frame of reference, will measure is the variable u’. In some question it won’t be obvious what the moving frame of reference is and what the moving object is, but once the variables are determined through the use of the definitions, the last step is to simply substitute them into the equation.

An important point to note when using the relativistic velocity equations is that the variables are velocities, not speed, implying that directionality is important. To incorporate directionality in calculations, we assign positive and negative to opposite direction. For example consider the same spaceship firing a rocket example from above. If both the spaceship and rocket are travelling to the right, then for the sake of calculations we can assign right to be positive and so the velocity values of the rocket and spaceship will be positive. If the spaceship is travelling to the left and the rocket is travelling to the right, then (again) for the sake of calculations we can assign right to be positive and left to be negative, so the velocity of the spaceship will be a negative value and the velocity of the rocket will be a positive value. Don’t be surprised if you get a negative answer because it simply represents the direction of the velocity.