Relativistic Addition of Velocity, Addition of Velocities, Addition of Velocity in Relativity

RELATIVISTIC ADDITION OF VELOCITIES - ADDITION OF VELOCITIES - ADDITION OF VELOCITY IN RELATIVITY

Consider a frame of reference s’ moving with uniform velocity v relative to a stationary frame s along the positive x direction. A particle is moving along the positive direction of x axis with a velocity given by dx/dt as observed by an observer in frame s.

As observed by an observer in frame s’, he distance travelled in dx’ in time dt’- the velocity of the particle as seen by the observer in observer s’ is given by

To find a transformation between u’ and u we use Lorentz-Transformation which are as follows:

Differentiating Eq. (2) and (3) w.r.t t’, we obtain

Rearranging Eq. (4) and using Eq. (8)

Equation (9) represents the relativistic addition formula. The inverse relation will be given by

Consider two photons approaching each other such that the velocity of each photon as observed by a stationary observer is c. In order to calculate the relative velocity of one photon w.r.t  another

Thus for photon B the relative speed of photon A is equal to "c".