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Mathematics for Physics

172 Kinetic Energy

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

Kinetic energy as excess over rest energy. Kinetic energy is the excess of total energy over rest energy. Rest energy is the energy equivalent of rest mass when the particle is at rest. Total energy is the full energy of a free particle. This principle is used to compute the energy of motion at high speed.

The relativistic kinetic energy is

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

where

  • \(K\) is the kinetic energy.
  • \(\gamma\) 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.

Unbounded growth at the speed limit. For a massive particle, kinetic energy grows without bound as the speed approaches the speed of light. This principle is used to show that reaching the speed of light would require unbounded kinetic energy.

Newtonian correspondence. When the speed is much smaller than the speed of light, the relativistic kinetic energy reduces to the Newtonian kinetic energy. This principle is used to recover everyday mechanics at low speed.

The Newtonian kinetic energy is

\[ K = \dfrac{1}{2}mv^{2} \]

where

  • \(v\) is the speed.

Conversion of kinetic energy into rest mass. In an inelastic collision, lost kinetic energy appears as an increase in rest mass of the composite body. An inelastic collision is a process that conserves total energy but not kinetic energy. This principle is used to compute the larger rest mass of a hotter or more tightly bound composite.

Note: Also called \(K\). Also called \(T\).

172.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(K=(\gamma-1)mc^{2}=E-E_{0}\); Newtonian \(K=\dfrac{1}{2}mv^{2}\); inelastic mass increase.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(E_{\mathrm{kin}}=E-mc^{2}\); unbounded \(K\) as \(v\to c\); inelastic collisions.
  3. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(\mathrm{K.E.}=(\gamma-1)mc^{2}\); Newtonian limit.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — Newtonian correspondence; inelastic conversion into rest mass.
  5. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — Newtonian kinetic energy \(\dfrac{1}{2}mv^{2}\).