Relativistic Mechanical Quantities

A number of ordinary mechanical quantities take on a different form as the speed approaches the speed of light.

At what energies must relativistic expressions be used?

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Relativistic Momentum

The relativistic momentum is given by

which is the ordinary definition of momentum with the mass replaced by the relativistic mass.

For v = c, gamma = , and rest mass
m0 = x10^kg = me= mp
m0 = MeV/c2 = GeV/c2

the relativistic momentum is

p =x10^kg m/s = MeV/c = GeV/c
compared to the non-relativistic result p = mv = x10^kg m/s
which would then be in error by %.

In the above calculations, one of the ways of expressing mass and momentum is in terms of electron volts. It is typical in high energy physics, where relativistic quantities are encountered, to make use of the Einstein relationship to relate mass and momentum to energy. In relativistic mechanics, the quantity pc is often used in momentum discussions. It has the units of energy.

For extreme relativistic velocities wherethen

A useful application of the quantity pc is in the calculation of the velocity as a fraction of c.

and as

with the limiting case applying for the momentum of a photon.
Momentum of photonUsefulness of the quantity pc
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Momentum of Photon

For a photon, the relativistic momentum expression

approaches zero over zero, so it can't be used directly to determine the momentum of a zero rest mass particle. But the general energy expression can be put in the form

and by setting rest mass equal to zero and applying the Planck relationship, we get the momentum expression:

Wave-particle duality application

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Relativistic Energy in Terms of Momentum

The famous Einstein relationship for energy

can be blended with the relativistic momentum expression

to give an alternative expression for energy.
The combination pc shows up often in relativistic mechanics. It can be manipulated as follows:

and by adding and subtracting a term it can be put in the form:

which may be rearranged to give the expression for energy:

Note that the m with the zero subscript is the rest mass, and that m without a subscript is the effective relativistic mass.
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