Question

Derive the Newtonian addition of velocities formula
vCA = vCB + vBA
for the case shown below where the 3 objects (A, B, and C) do not pass
through the same event. Note that, without loss of generality, we may
take the worldlines of A and B to intersect at t = 0.

Answer 1

Look at the figure:

At time t, the separation between B and A (from B's viewpoint) was $V_{BA}t.$ The separation between C and A was $V_{CA}t$. And from the diagram we see that the separation between C and B is $V_{CB}t$. Hence:

$$V_{CA}t=V_{CB}t+V_{BA}t,$$

cancel t:

$$V_{CA}=V_{CB}+V_{BA},$$

because their relative velocities are not affected by the time when these objects' trajectories intersected.