Simple arithmetic addition is commonly used as an adequate combination law for velocities. For example, somebody walking at 5 km/h on a belt rolling at 3 km/h is moving at the velocity of 8 km/h with respect to the airport hall. However, since the speed of light is a limit that cannot be overcome by any signal, this simple law must break down when the velocities are close to the speed of light. For example, if a spaceship cruising at a speed close to that of light with respect to Earth emits a light pulse, the velocity of that pulse with respect to Earth would have to be close to twice the speed of light, which is not allowed by Special Relativity.
The exact combination law for velocities must reduce, in very good approximation, to standard arithmetic addition when all velocities involved are much smaller than the speed of light. On the other hand, it must be such that velocities superior to that of light never arise and that the velocity of light is a universal constant.
Here is the link to the text “Concepts of Special Relativity” in PDF format. Velocity combination is analyzed in Chapter 7. As mentioned above, the approach known as k-calculus is adopted. One of the advantages is that the combination law of Bondi's k factors is essentially obvious and, once it is stated, the results obtained in Chapter 4 on the relationship between k factor and relative velocity can be used to recast it in the form of a combination law for velocities.
A numerical example of the combination law for collinear velocities can be found in Appendix A.5.
In order to understand clearly the concepts and the nomenclature used, it is recommended to begin by reading the first three chapters of the text.
An animation software written in the Java language is available. Velocity combination is illustrated by the sixth animation proposed.
If you are interested in another particular topic of Special Relativity, here is a list of the topics covered in the text and in the software. Clicking on an item in this list, will open a page which briefly introduces that topic and indicates the parts of the text and of the software in which it is treated.