section_path: Limits of validity/Newtonian approximation to special relativity
section_name: Newtonian approximation to special relativity
paragraph_text: Newtonian approximation to special relativity
  
  In special relativity, the momentum of a particle is given by
  
  where m is the particle's rest mass, v its velocity, v is the modulus of v, and c is the speed of light.
  
  If v is very small compared to c, v / c is approximately zero, and so
  
  Thus the Newtonian equation p = m v is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light.
  
  For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by
  
  where f is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage.