The relative velocity is the velocity of an object or observer in the rest frame of another object or observer. If it is constant
v_AB=〖-v〗_AB
In the Newtonian limit where the Galilean transformation is approximately:
r=r-vt
t'=t
This holds true and is identical with the vector difference between the velocities A and B as evaluated in terms of a single coordinate system:
v_AB= v_B- v_A
Due to Einstein’s Special Relativity, this does not hold true in general. The expression “the velocity of A relative to B” is shorthand for “the velocity of A in the coordinate system where B is always at rest.
To obtain vAB from vA and vB in an arbitrary reerence frame, the Lorentz transformation is necessary. If vA and vB are collinear, the formula is:
v_AB= (v_B- v_A)/(1- (v_A v_B)/c^2 )
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