We have experience of such rapid
motions only in the case of electrons and 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
eq. 15: file eq15.gif
but by the expression
eq. 16: file eq16.gif
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
eq. 17: file eq17.gif
When eq. 18 is 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
mc^2 does not contain the velocity, and requires no consideration if
we are only dealing with the question as to how the energy of a
point-mass; depends on the velocity.
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