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Relativistic Mechanics

The mechanics of particle motion is often treated as a closed field in which all
problems have been properly formulated and reduced to a system of equations which are,
in principle, mathematically solvable. But this refers to Newtonian mechanics. When
relativistic considerations are taken into account the precise
equations and formulas
of Newtonian mechanics turnout to be only first order approximations of the correct
quantities. For example, the kinetic energy of a particle is taken to be ½mv^{2}.
In Relativity Theory the precise formula for total energy is:

E = mc^{2}/(1-(v/c)^{2})^{1/2}

The Maclaurin series expansion of (1-(v/c)^{2})^{-1/2} is:

E = mc^{2} + (1/2)mv^{2} + (3/8)mv^{4}/c^{2} + ...

The kinetic energy is E-mc^{2} and thus while (1/2)mv^{2} is the
first order approximation of kinetic energy it is not the precise value. The total kinetic
energy E-mc^{2} is larger than (1/2)mv^{2} by a factor of

When v/c=0.1 this factor is equal to about 1.00813. Thus even at a speed of 18,600 miles
per second the Newtonian formula for kinetic energy is in error only by less than one percent.
But for v/c=0.5 the error is approximately 20 percent and at v/c=0.8 the errror is more than
50 percent.