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Addition of Velocities

Consider two frame S and S’, where S’ is moving towards positive x-axis relative to S with velocity v. Suppose a particle moves with velocity component u x, u Y‘ u z relative

 

According to Lorentz’s transformation

X = x vt /√1-v2 /c2

Differentiate equations (1) and (2), we get

D x = dx –vdt /√ -v2 /c2

As ,                                          u x  = dx dt

Since                                                    y =y

And                                                     dy =dy

 

From equations (4) and (6), we get

 

Similarly,                                  z = z

and                                            dz  =  dz

 

From equations, (4) and (8), we have

The inverse velocity transformation can be obtained by replacing v with (-v) and  interchanging primed and unprimed co-ordinates . thus the inverse velocities transformations are:

Note.

 

1. If particle is moving in frame S’ with velocity of light.

U x  =c

From               

which means relativistic combination of velocities is consistent

 

2. If u x = c = v, then from equation

Thus addition of velocity of light to velocity of light gives velocity of light.