PlanetPhysics/Kinetic Energy

Kinetic energy is [[../CosmologicalConstant/|energy]] associated to [[../CosmologicalConstant/|motion]]. The kinetic energy of a mechanical [[../GenericityInOpenSystems/|system]] is the [[../Work/|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 [[../CenterOfGravity/|centre of mass]] is fixed, but in situations where, for example, the stone undergoes a change in [[../BoltzmannConstant/|temperature]] the movement of the individual [[../Particle/|particles]] will play a role in the energetic description of the stone.

Kinetic energy is commonly denoted by various symbols, such as $E_{\mathrm{k}}$, $E_{\mathrm{k}\mathrm{i}\mathrm{n}}$, $K$, or $T$ (the latter is the convention in [[../LagrangesEquations/|Lagrangian]] [[../Mechanics/|mechanics]]). The SI unit of kinetic energy, like that of all sorts of energy, is the joule (J), which is the same as $\mathrm{k}\mathrm{g}{\mathrm{m}}^{\mathrm{2}}/{\mathrm{s}}^{\mathrm{2}}$ in SI base units.

Energy associated to motion in a straight line is called translational kinetic energy . For a particle or [[../RigidBody/|rigid body]] with [[../Mass/|mass]] $m$ and [[../Velocity/|velocity]] $𝐯$, the translational kinetic energy is $E_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}=\frac{1}{2}m{v}^2=\frac{1}{2}m𝐯\cdot 𝐯.$ Kinetic energy associated to rotation of a rigid body is called rotational kinetic energy . It depends on the [[../MomentOfInertia/|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 $E_{\mathrm{r}\mathrm{o}\mathrm{t}}=\frac{1}{2}I{\omega }^2.$

In [[../SR/|special relativity]], the total energy of an [[../TrivialGroupoid/|object]] of mass $m$ moving in a straight line with [[../Velocity/|speed]] $v$ is $E=\gamma (v)m{c}^2,$ where $c$ is the [[../CosmologicalConstant/|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 $m{c}^2$. The kinetic energy is therefore $E_{\mathrm{k}\mathrm{i}\mathrm{n}}=\gamma (v)m{c}^2-m{c}^2=(\gamma (v)-1)m{c}^2.$ For values of $v$ much smaller than $c$, this expression becomes approximately equal to $\frac{1}{2}m{v}^2$, the kinetic energy from [[../MathematicalFoundationsOfQuantumTheories/|classical mechanics]]. This can be checked by expanding $\gamma (v)$ in a [[../TaylorFormula/|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: $E_{\mathrm{k}\mathrm{i}\mathrm{n}}=\frac{1}{2}m{v}^2+\frac{3}{8}m{v}^4/c^2+\frac{5}{16}m{v}^6/c^4+\cdots$ 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.