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Mathematics for Physics

106 Kinetic Energy

An energy associated with motion that is used to measure how much of a particle’s total energy exceeds its rest energy.

Note: Also called \(K\). Also called \(T\). At low speed it reduces to the Newtonian kinetic energy.

definition [d] (Kinetic Energy) From Knight: a relativistic expression for the kinetic energy is

  • \(K = (\gamma - 1)mc^{2} = \gamma mc^{2} - mc^{2} = E - E_{0}\) ,

where

  • \(\gamma\) is the Lorentz factor of the particle.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light in vacuum.
  • \(E = \gamma mc^{2}\) is the total energy.
  • \(E_{0} = mc^{2}\) is the rest energy.

definition [d] (Kinetic Energy) From Griffiths: the remainder attributable to the motion is kinetic energy,

  • \(E_{\mathrm{kin}} = E - mc^{2} = mc^{2}\left(\dfrac{1}{\sqrt{1 - \dfrac{u^{2}}{c^{2}}}} - 1\right)\) .

where

  • \(E\) is the total energy.
  • \(u\) is the particle speed.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light in vacuum.

definition [d] (Kinetic Energy) From Emam: relativistically the kinetic energy is total energy minus rest energy,

  • \(\mathrm{K.E.} = E - E_{\mathrm{rest}} = m\gamma c^{2} - mc^{2} = (\gamma - 1)mc^{2}\) .

where

  • \(E = m\gamma c^{2}\) is the total energy.
  • \(E_{\mathrm{rest}} = mc^{2}\) is the rest energy.
  • \(\gamma\) is the Lorentz factor.

definition [d] (Kinetic Energy) From Knight: the Newtonian kinetic energy of a particle is

  • \(K = \dfrac{1}{2}mv^{2}\) .

where

  • \(m\) is the mass.
  • \(v\) is the speed.

106.1 Elementary Example

106.1.1 Simple

For \(m = 2\,\mathrm{kg}\) and \(v = 3\,\mathrm{m/s}\), the Newtonian kinetic energy is

\[ K = \dfrac{1}{2}(2)(3)^{2} = 9\,\mathrm{J} \]

where

  • \(K\) is the kinetic energy.

106.1.2 General

At relative speed \(v = 0.6c\), one has \(\gamma = 1.25\), so

\[ K = (1.25 - 1)mc^{2} = 0.25\, mc^{2} \]

where

  • \(K\) is the relativistic kinetic energy.
  • \(mc^{2}\) is the rest energy.