# PlanetPhysics/Kinetic Energy

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Kinetic energy is energy associated to motion. The kinetic energy of a mechanical system is the work required to bring the system from its rest' state to a moving' state. When exactly a system is considered to be `at rest' depends on the context: a stone is usually considered to be at rest when its centre of mass is fixed, but in situations where, for example, the stone undergoes a change in temperature the movement of the individual particles will play a role in the energetic description of the stone.
Kinetic energy is commonly denoted by various symbols, such as ${\displaystyle E_{\mathrm {k} }}$, ${\displaystyle E_{\mathrm {kin} }}$, ${\displaystyle K}$, or ${\displaystyle T}$ (the latter is the convention in Lagrangian mechanics). The SI unit of kinetic energy, like that of all sorts of energy, is the joule (J), which is the same as ${\displaystyle \mathrm {kg\;m^{2}/s^{2}} }$ in SI base units.
Energy associated to motion in a straight line is called translational kinetic energy . For a particle or rigid body with mass ${\displaystyle m}$ and velocity ${\displaystyle \mathbf {v} }$, the translational kinetic energy is ${\displaystyle E_{\mathrm {trans} }={\frac {1}{2}}mv^{2}={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} .}$ Kinetic energy associated to rotation of a rigid body is called rotational kinetic energy . It depends on the moment of inertia ${\displaystyle I}$ of the body with respect to the axis of rotation. When the body rotates around that axis at an angular velocity ${\displaystyle \omega }$, the rotational kinetic energy is ${\displaystyle E_{\mathrm {rot} }={\frac {1}{2}}I\omega ^{2}.}$
In special relativity, the total energy of an object of mass ${\displaystyle m}$ moving in a straight line with speed ${\displaystyle v}$ is ${\displaystyle E=\gamma (v)mc^{2},}$ where ${\displaystyle c}$ is the speed of light and ${\displaystyle \gamma (v)}$ is the Lorentz factor: ${\displaystyle \gamma (v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}$ In particular, the rest energy of this object (obtained by setting ${\displaystyle v=0}$) is equal to ${\displaystyle mc^{2}}$. The kinetic energy is therefore ${\displaystyle E_{\mathrm {kin} }=\gamma (v)mc^{2}-mc^{2}=(\gamma (v)-1)mc^{2}.}$ For values of ${\displaystyle v}$ much smaller than ${\displaystyle c}$, this expression becomes approximately equal to ${\displaystyle {\frac {1}{2}}mv^{2}}$, the kinetic energy from classical mechanics. This can be checked by expanding ${\displaystyle \gamma (v)}$ in a Taylor series around ${\displaystyle v=0}$: ${\displaystyle \gamma (v)=1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}+{\frac {3}{8}}{\frac {v^{4}}{c^{4}}}+{\frac {5}{16}}{\frac {v^{6}}{c^{6}}}+\cdots }$ Substituting this into the expression for the kinetic energy gives the following expansion: ${\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}+{\frac {3}{8}}mv^{4}/c^{2}+{\frac {5}{16}}mv^{6}/c^{4}+\cdots }$ When ${\displaystyle v}$ approaches the speed of light, the factor ${\displaystyle \gamma (v)}$ goes to infinity. This is one way of seeing why objects with positive mass can never reach a speed ${\displaystyle c}$: an infinite amount of work would be required to accelerate the object to this speed.