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

$\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 $ 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

$\displaystyle 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

$\displaystyle E=\gamma(v)mc^2,
$

where $ c$ is the speed of light and $ \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 $ v=0$ ) is equal to $ 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 $ 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$ :

$\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 $ 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.



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As of this snapshot date, this entry was owned by pbruin.