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] ]


In ], '''kinetic energy''' (or ''vis vis'', "living force") is ] possessed by a body by virtue of its ]. In ], a body with ] ''m'', moving in a straight line with velocity ''v'', has a ''translational kinetic energy'' of '''Kinetic energy''' (also called '''vis vis''', or '''living force''') is ] possessed by a body by virtue of its ].



⚫ :<math> E_k = \begin{matrix} \frac{1}{2} \end{matrix} m v^2 </math>.
==Equations==

:<math>E_k = \int \mathbf{v} \cdot \mathrm{d}\mathbf{p}</math>

For the non-relativistic ''translational kinetic energy'' for a body with ] ''m'', moving in a straight line with velocity ''v'', we can use the ] approximation:

⚫ :<math>E_k = \begin{matrix} \frac{1}{2} \end{matrix} mv^2 </math>

*''E<sub>k</sub>'' is kinetic energy
*''m'' is mass of the body
*''v'' is velocity of the body


If a body is rotating, its ] equals If a body is rotating, its ] equals
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:<math> E_{rotation} = \begin{matrix} \frac{1}{2} \end{matrix} I \omega^2 </math>, :<math> E_{rotation} = \begin{matrix} \frac{1}{2} \end{matrix} I \omega^2 </math>,


where ''I'' is its ] and &omega; its ]. *''I'' is its ]
*&omega is ].


In ]'s ], the kinetic energy of a body is In ]'s ], (used especially for near-light velocities) the kinetic energy of a body is:

:<math>E_k = m c^2 (\gamma - 1) = \gamma m c^2 - m c^2 \;\!</math>
:<math>\gamma = \frac{1}{\sqrt{1 - (v/c)^2}} </math>


:<math>E_k = \gamma m c^2 - m c^2 :<math>E_k = \gamma m c^2 - m c^2
= \left( \frac{1}{\sqrt{1- v^2/c^2 }} - 1 \right) m c^2</math> = \left( \frac{1}{\sqrt{1- v^2/c^2 }} - 1 \right) m c^2</math>


* ''v'' is the velocity of the body
⚫ where ''m'' is its mass (i.e. ]), and ''c'' is the ] in vacuum. Relativity theory states that the kinetic energy of an object grows towards infinity as its velocity approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary.
*''m'' is its rest mass
* ''c'' is the speed of light in a vacuum.
* ''&gamma;mc<sup>2</sup>'' is the '''total energy''' of the body
* ''mc<sup>2</sup>'' is the rest mass energy.

⚫ Relativity theory states that the kinetic energy of an object grows towards infinity as its velocity approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary.


Where gravity is weak, and objects move at much slower velocities than light (e.g. in everyday phenomena on Earth), Newton's formula is an excellent approximation of relativistic kinetic energy. Where gravity is weak, and objects move at much slower velocities than light (e.g. in everyday phenomena on Earth), Newton's formula is an excellent approximation of relativistic kinetic energy.


See also: ==See also==
* ] * ]
* ] * ]

Revision as of 16:34, 20 May 2004


Kinetic energy (also called vis vis, or living force) is energy possessed by a body by virtue of its motion.


Equations

E k = ∫ v ⋅ d p {\displaystyle E_{k}=\int \mathbf {v} \cdot \mathrm {d} \mathbf {p} }

For the non-relativistic translational kinetic energy for a body with mass m, moving in a straight line with velocity v, we can use the Newtonian approximation:

E k = 1 2 m v 2 {\displaystyle E_{k}={\begin{matrix}{\frac {1}{2}}\end{matrix}}mv^{2}}
  • Ek is kinetic energy
  • m is mass of the body
  • v is velocity of the body

If a body is rotating, its rotational kinetic energy equals

E r o t a t i o n = 1 2 I ω 2 {\displaystyle E_{rotation}={\begin{matrix}{\frac {1}{2}}\end{matrix}}I\omega ^{2}} ,

In Einstein's relativistic mechanics, (used especially for near-light velocities) the kinetic energy of a body is:

E k = m c 2 ( γ − 1 ) = γ m c 2 − m c 2 {\displaystyle E_{k}=mc^{2}(\gamma -1)=\gamma mc^{2}-mc^{2}\;\!}
γ = 1 1 − ( v / c ) 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-(v/c)^{2}}}}}
E k = γ m c 2 − m c 2 = ( 1 1 − v 2 / c 2 − 1 ) m c 2 {\displaystyle E_{k}=\gamma mc^{2}-mc^{2}=\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)mc^{2}}
  • v is the velocity of the body
  • m is its rest mass
  • c is the speed of light in a vacuum.
  • γmc is the total energy of the body
  • mc is the rest mass energy.

Relativity theory states that the kinetic energy of an object grows towards infinity as its velocity approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary.

Where gravity is weak, and objects move at much slower velocities than light (e.g. in everyday phenomena on Earth), Newton's formula is an excellent approximation of relativistic kinetic energy.

See also

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