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-v^{2} /c^{2}

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

D _{x} = dx –vdt /√ -v^{2} /c^{2}

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

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.