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.