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kinetic energy (Quantity)

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.



"kinetic energy" is owned by pbruin.

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See Also: temperature

Also defines:  translational kinetic energy, rotational kinetic energy

Cross-references: Taylor series, classical mechanics, speed of light, speed, object, special relativity, moment of inertia, velocity, mass, rigid body, mechanics, Lagrangian, temperature, centre of mass, work, system, motion, energy
There are 18 references to this object.

This is version 1 of kinetic energy, born on 2005-08-31.
Object id is 95, canonical name is KineticEnergy.
Accessed 5473 times total.

Classification:
Physics Classification: 03.30.+p (Special relativity)
 45.05.+x (General theory of classical mechanics of discrete systems)
 45.40.-f (Dynamics and kinematics of rigid bodies)
 45.50.Dd (General motion)
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