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 ½mv². In Relativity Theory the precise formula for total energy is:

E = mc²/(1-(v/c)²)^1/2
 

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

1 + (1/2)(v/c)² + (3/8)(v/c)⁴ + (5/16)(v/c)² + ....
 

Thus the series expansion for energy is:

E = mc² + (1/2)mv² + (3/8)mv⁴/c² + ...
 

The kinetic energy is E-mc² and thus while (1/2)mv² is the first order approximation of kinetic energy it is not the precise value. The total kinetic energy E-mc² is larger than (1/2)mv² by a factor of

1 + (3/4)(v/c)² + (5/8)(v/c)⁴ + (7/16)(v/c)⁶ + ..
 

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.