The velocity addition topic into the special theory of relativity is interesting one, where in addition to the two observers (two frames) included;
(1) an idea of the additional object that might be a ball, a rocket or any thing that can be thrown or fired with some velocity by the moving observer O’ (in S’ frame)
(2) Also, two spacecrafts in the space in to the same direction, while speed of both the spacecraft is given with respect to the earth observer or an observer one of the two pilots
How to understand such physical situations? For this I have consider first an example of a biker, in which he throw a ball with some speed, in his frame he know the speed of this ball. Let the speed of the ball is Ux’, relative to the rider. Speed of the bike is relative to a man who is standing across the road and this is corresponding to a moving frame that is moving with speed “v”.Now we have to found the speed of this ball with respect to the man who is in stationary frame ( or you can say at rest, across the road). Relative speed of the ball with respect to the man is Ux (in unprimed frame). Question is how to develop the relationship between the , Ux’, Ux and V??
Second Physical situation may be this one; where two space craft are flying in to the sky, suppose Alpha and Beta. Speed of the alpha and beta is relative also relative to the man who is standing on the surface of the earth. (this situation we can correlate with the first example, like spacecraft alpha is suppose bike (S’- reference frame)and spacecraft beta is that ball which is also relative to the observer on the earth O (that is in S).
In any case, we have to find the relative motion of the ball or rocket either with respect to the moving frame observer’s O’, or stationary observer’s O.
How to develop the relation between them; for it we have to use Lorentz Transformation. This is shown in last of this video.