Page:Relativity (1931).djvu/73

ions; for other motions the variations from the laws of classical mechanics are too small to make themselves evident in practice. We shall not consider the motion of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativity the kinetic energy of a material point of mass $m$ is no longer given by the well-known expression

$m{\frac {v^{2}}{2}}$,

but by the expression

${\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$.

This expression approaches infinity as the velocity $v$ approaches the velocity of light $c$. The velocity must therefore always remain less than $c$, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain

$mc^{2}+m{\frac {v^{2}}{2}}+{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}+\ldots$

When ${\tfrac {v^{2}}{c^{2}}}$ small compared with unity, the third of these terms is always small in comparison with the second, which last is alone considered in classical mechanics. The first term $mc2$ does not contain the velocity, and requires no consideration if we