Relativistic speed
Relativistic speed refers to speed at which relativistic effects become significant to the desired accuracy of measurement of the phenomenon being observed. Relativistic effects are those discrepancies between values calculated by models considering and not considering relativity. Related words are velocity, rapidity, and celerity which is proper velocity. Speed is a scalar, being the magnitude of the velocity vector which in relativity is the four-velocity and in three-dimension Euclidean space a three-velocity. Speed is empirically measured as average speed, although current devices in common use can estimate speed over very small intervals and closely approximate instantaneous speed. Non-relativistic discrepancies include cosine error which occurs in speed detection devices when only one scalar component of the three-velocity is measured and the Doppler effect which may affect observations of wavelength and frequency.Relativistic effects are highly non-linear and for everyday purposes are insignificant because the Newtonian model closely approximates the relativity model. In special relativity the Lorentz factor is a measure of time dilation, length contraction and relativistic mass increase of a moving object.
Velocity-addition formula
The proof as given is highly formal. There are other more involved proofs that may be more enlightening, such as the one below.
Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the - and -axes, it is convenient to use the same units for space and time, otherwise a unit conversion factor appears throughout relativistic formulae, being the speed of light. In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to . A velocity is then expressed as fraction of the speed of light.