A discrepancy-free transformation of velocity is derived using the Time relativity transformation of coordinates because relativistic transformation of velocity creates a discrepancy. The relativistic transformation of velocity expresses the velocity u2 of an object q in frame 2 in terms of its velocity in frame 1. In frame 2 at time tq, the position of q is xq=u2*tq.
As q departs from the origin of frame 2, we may compute its position by using the time at the origin which is t2o. The so computed position is x2o=u2*t2o. Notice that x2o does not equal xq. So, the relativistic transformation of velocity gives 2 different positions to the object q, which we call the discrepancy of double position. The cause of this discrepancy is that the time at the origin and at the abscissa xq are not equal due to relativity of simultaneity.
Below, we derive a discrepancy-free transformation of velocity. Let us take 2 frames of reference with frame 2 moving at the velocity vo in frame 1. An object q moves at the velocity v2 in frame 2. We will derive the velocity of q in frame 1, which is v1.
The object q moves a distance during a time interval. At the start, q passes by the point x2a in frame 2 which is a fixed point of the x axis of frame 2. x2a coincides with the point x1a which is a fixed point of the x axis of frame 1, see Figure 1. At the end, q passes by the point x2b of frame 2. x2b coincides with the point x1b of frame 1, see Figure 2.
We emphasize that x2a, x2b, x1a and x1b are fixed points on their respective x axes because x2a and x2b move with frame 2 in frame 1, but x1a and x1b stay still in frame 1. x2a, x2b, x1a and x1b do not move with q. Because the point x2a moves with frame 2 at the velocity vo, it arrives at the point x’1a in frame 1 at the end of the time interval, see Figure 2.