PlanetPhysics/Kinetic Energy

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 $E_{\mathrm {k} }$, $E_{\mathrm {kin} }$, $K$, or $T$ (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 $\mathrm {kg\;m^{2}/s^{2}}$ 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 $m$ and velocity $\mathbf {v}$, the translational kinetic energy is $E_{\mathrm {trans} }={\frac {1}{2}}mv^{2}={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} .$ Kinetic energy associated to rotation of a rigid body is called rotational kinetic energy . It depends on the moment of inertia $I$ of the body with respect to the axis of rotation. When the body rotates around that axis at an angular velocity $\omega$, the rotational kinetic energy is $E_{\mathrm {rot} }={\frac {1}{2}}I\omega ^{2}.$

In special relativity, the total energy of an object of mass $m$ moving in a straight line with speed $v$ is $E=\gamma (v)mc^{2},$ where $c$ is the speed of light and $\gamma (v)$ is the Lorentz factor: $\gamma (v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.$ In particular, the rest energy of this object (obtained by setting $v=0$) is equal to $mc^{2}$. The kinetic energy is therefore $E_{\mathrm {kin} }=\gamma (v)mc^{2}-mc^{2}=(\gamma (v)-1)mc^{2}.$ For values of $v$ much smaller than $c$, this expression becomes approximately equal to ${\frac {1}{2}}mv^{2}$, the kinetic energy from classical mechanics. This can be checked by expanding $\gamma (v)$ in a Taylor series around $v=0$: $\gamma (v)=1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}+{\frac {3}{8}}{\frac {v^{4}}{c^{4}}}+{\frac {5}{16}}{\frac {v^{6}}{c^{6}}}+\cdots$ Substituting this into the expression for the kinetic energy gives the following expansion: $E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}+{\frac {3}{8}}mv^{4}/c^{2}+{\frac {5}{16}}mv^{6}/c^{4}+\cdots$ When $v$ approaches the speed of light, the factor $\gamma (v)$ goes to infinity. This is one way of seeing why objects with positive mass can never reach a speed $c$: an infinite amount of work would be required to accelerate the object to this speed.