Let there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, they are at the same position that is Observers O and O’ coincides. After that S’ frame starts moving with a uniform velocity v along x axis.

Suppose a particle P is place in frame S’ and it is moving.

The velocity component of particle P from observer O’ in frame S’ will be:

u’_{x} = dx’/dt’ (1a)

u’_{y} = dy’/dt’ (1b)

u’_{z} = dz’/dt’ (1c)

The velocity component of particle P from observer O in frame S will be:

u_{x} = dx/dt (2a)

u_{y} = dy/dt (2b)

u_{z} = dz/dt (2c)

From Lorentz transformation equations:

x’ = (x – vt)/(√1 – v^{2}/c^{2}) (3a)

y’ = y (3b)

z’ = z (3c)

t’ = (t – xv/c^{2})/(√1 – v^{2}/c^{2}) (3d)

Differentiate equations (3)

dx’ = (dx – vdt)/(√1 – v^{2}/c^{2}) (4a)

dy’ =d y (4b)

dz’ = dz (4c)

dt’ = (dt – dxv/c^{2})/(√1 – v^{2}/c^{2}) (4d)

Now substitute equations (4a) and (4d) in equation (1a), we get

u’_{x} = (dx – vdt)/ (dt – dxv/c^{2})

Divide numerator and denominator of R.H.S with dt, we get

u’_{x} = (dx/dt – v)/ (1 – dx/dt(v/c^{2}))

Put equation (2a) in above equation,

u’_{x} = (u_{x} – v)/ (1 – u_{x}(v/c^{2})) (5)

Similarly by putting equations (4b) and 4d) in equation (1b) and then dividing the numerator and denominator of R.H.S with dt, and then putting equation (2b), we get

u’_{y} = u_{y}(√1 – v^{2}/c^{2})/ (1 – u_{x}(v/c^{2})) (6)

Similarly by putting equations (4c) and 4d) in equation (1c) and then dividing the numerator and denominator of R.H.S with dt, and then putting equation (2c), we get

u’_{z} = u_{z}(√1 – v^{2}/c^{2})/ (1 – u_{x}(v/c^{2})) (7)

Equations (5, 6 and 7) represent the addition of velocity relations as observed by observers O’ from frame S’.

From the observer O in frame S, the relations (5, 6 and 7) will become:

u_{x} = (u’_{x} + v)/ (1 + u’_{x}(v/c^{2})) (8)

u_{y} = u’_{y}(√1 – v^{2}/c^{2})/ (1 + u’_{x}(v/c^{2})) (9)

u_{z} = u’_{z}(√1 – v^{2}/c^{2})/ (1 + u’_{x}(v/c^{2})) (10)

Generally equation (8) is written as

u = (u’ + v)/ (1 + u’(v/c^{2})) (11)

**Special case**: If v <<< c, then v/c^{2} will get neglected and the equation 11 will become

u = u’ + v

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