I'm looking for some clarification on what each of the terms in the relativistic velocity transformation law are.
The formula is: $s = (v+u)/(1 + uv/c^2)$
It would be really great if you could give me some kind of example to explain what u, v and s are. I know I'm supposed to say what I've tried, but I really have no idea.
Thanks
In relativity, everything is relative to a frame of reference (hence the name). In this example, the way to think about it is that we have three frames of reference to consider.
The first is our frame, the laboratory frame. The second is a frame moving at speed $v$ relative to us. The third is a frame moving at speed $u$ relative to the second frame. If you want, you can imagine that we are looking at a moving spaceship with speed $v$ and inside that moving spaceship, there is a person moving at speed $u$.
The reason that we cannot simply add $u$ and $v$ is that we could end up breaking relativity, meaning that if $v=. 75c$ and $u =. 75c$, then if we just added them, then the relative speed of $u$ with respect to us would be $1.5c$. This is not at all reasonable since nothing travels faster than the speed of light. Moreover, the addition formula is somewhat dependent upon a universal frame of reference. We know that there is no preferred frame of reference by Einstein's postulate, so the Gallilean velocity addition rule is not the truth of the matter.
We do not a priori know what speed the person is traveling at in our frame of reference. Its speed is only known relative to the spaceship. This is where the addition formula comes into play. It tells us that the speed of the person on the spaceship in our frame of reference is given by $s$ in your formula. Does this make sense?