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Mathematics for Physics

128 Relativistic Kinetic Energy Formula Derivation

A derivation of the relativistic kinetic energy formula that is used to obtain \(K_{\mathrm{rel}} = (\gamma - 1)mc^{2}\) from the work done by a force.

definition [d] (Relativistic Kinetic Energy Formula Derivation) From OpenStax: the relativistic kinetic energy follows from the work-energy theorem. With force

  • \(\mathbf{F} = \dfrac{d\mathbf{p}}{dt} = m\dfrac{d(\gamma\mathbf{u})}{dt}\) ,

the work that accelerates a particle from rest to final speed \(u\) is, in one dimension,

  • \(K = \displaystyle\int F\, dx = m\displaystyle\int u\,\dfrac{d}{dt}\!\left(\dfrac{u}{\sqrt{1-\left(\dfrac{u}{c}\right)^{2}}}\right) dt\) .

Integrating by parts from rest to speed \(u\) gives

  • \(K = \dfrac{mu^{2}}{\sqrt{1-\left(\dfrac{u}{c}\right)^{2}}} - m\displaystyle\int\dfrac{u}{\sqrt{1-\left(\dfrac{u}{c}\right)^{2}}}\, du\) ,
  • \(K = \dfrac{mu^{2}}{\sqrt{1-\left(\dfrac{u}{c}\right)^{2}}} + mc^{2}\sqrt{1-\left(\dfrac{u}{c}\right)^{2}}\Big|_{0}^{u}\) ,
  • \(K = \dfrac{mc^{2}}{\sqrt{1-\left(\dfrac{u}{c}\right)^{2}}} - mc^{2}\) ,

hence

  • \(K_{\mathrm{rel}} = (\gamma - 1)mc^{2}\) ,

with

  • \(\gamma = \dfrac{1}{\sqrt{1-\dfrac{u^{2}}{c^{2}}}}\) .

At low speed, the binomial approximation \(\gamma \approx 1 + \dfrac{1}{2}\left(\dfrac{u^{2}}{c^{2}}\right)\) yields

  • \(K_{\mathrm{rel}} \approx \dfrac{1}{2}mu^{2} = K_{\mathrm{class}}\) .

where

  • \(\mathbf{F}\) is the force.
  • \(F\) is the one-dimensional force component.
  • \(\mathbf{p}\) is the relativistic momentum.
  • \(t\) is time.
  • \(dt\) is the time differential.
  • \(m\) is the rest mass.
  • \(\gamma\) is the Lorentz factor.
  • \(\mathbf{u}\) is the velocity.
  • \(u\) is the particle speed.
  • \(c\) is the speed of light in vacuum.
  • \(dx\) is the displacement differential.
  • \(du\) is the speed differential.
  • \(K\) is the work integral equal to the gained kinetic energy.
  • \(K_{\mathrm{rel}}\) is the relativistic kinetic energy.
  • \(K_{\mathrm{class}}\) is the Newtonian kinetic energy.
  • \(\displaystyle\int F\, dx\) is the work done by the force.

128.1 Elementary Example

128.1.1 Simple

At \(u = 0\), one has \(\gamma = 1\), so

\[ K_{\mathrm{rel}} = (1 - 1)mc^{2} = 0 \]

where

  • \(u\) is the particle speed.
  • \(\gamma\) is the Lorentz factor.
  • \(K_{\mathrm{rel}}\) is the relativistic kinetic energy.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light in vacuum.

128.1.2 General

An electron with speed \(u = 0.990c\) has \(\gamma = 7.0888\), so

\[ K_{\mathrm{rel}} = (\gamma - 1)mc^{2} = 3.12\,\mathrm{MeV} \]

where

  • \(u\) is the electron speed.
  • \(c\) is the speed of light in vacuum.
  • \(\gamma\) is the Lorentz factor.
  • \(m\) is the electron rest mass.
  • \(K_{\mathrm{rel}}\) is the relativistic kinetic energy.