91Ó°ÊÓ

The relative momentum p of a material particle of mass m and velocity v is

$\mathit{p}\mathbf{=}\frac{\mathbf{m}\mathbf{v}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{v}}^{\mathbf{2}}\mathbf{/}{\mathbf{c}}^{\mathbf{2}}}}$

We see that even when the speed v of the particle approaches the upper limit speed c, the momentum p takes on arbitrarily high values with no upper limit (becomes infinite); Figure 37.20 shows the graph of the magnitude of the momentum of a particle with rest mass m as a function of speed v. Also, from Section 37.8 we have known the total energy of a particle is the sum of its kinetic energy (related to its momentum) and the rest energy (related to its rest mass). Equation (37.39) gives us the famous energy-momentum formula,$E^2=(m{c}^2{)}^2+(pc{)}^2$

from which we can see the energy E of a particle of rest mass m and momentum p can assume arbitrarily high values without an upper limit, since the momentum p has no upper limit as we have shown above.

Thus, the energy and momentum of a particle have no upper limits