stylistic:style_transfer, stylistic:tone, structural:scholar_edition, structural:expansion
Relativistic kinetic energy

If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy. In relativity, the total energy is given by the energy-momentum relation:

E 2 = (p c) 2 + (m 0 c 2) 2 {\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,}

Here we use the relativistic expression for linear momentum: p = m γ v {\displaystyle p=m\gamma v}, where γ = 1 / 1 − v 2 / c 2 {\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}.with m {\displaystyle m} being an object's (rest) mass, v {\displaystyle v} speed, and c the speed of light in vacuum.Then kinetic energy is the total relativistic energy minus the rest energy: E K = E − m 0 c 2 = (p c) 2 + (m 0 c 2) 2 − m 0 c 2 {\displaystyle E_{K}=E-m_{0}c^{2}={\sqrt {(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}}}-m_{0}c^{2}}

At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy.