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Relativistic kinetic energy of rigid bodies

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If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy. In special relativity theory, the expression for linear momentum is modified. With m being an object's rest mass, v and v its velocity and speed, and c the speed of light in vacuum, we use the expression for linear momentum p = m γ v {\displaystyle \mathbf {p} =m\gamma \mathbf {v} } , where γ = 1 / 1 − v 2 / c 2 {\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}} . Integrating by parts yields E k = ∫ v ⋅ d p = ∫ v ⋅ d ( m γ v ) = m γ v ⋅ v − ∫ m γ v ⋅ d v = m γ v 2 − m 2 ∫ γ d ( v 2 ) {\displaystyle E_{\text{k}}=\int \mathbf {v} \cdot d\mathbf {p} =\int \mathbf {v} \cdot d(m\gamma \mathbf {v} )=m\gamma \mathbf {v} \cdot \mathbf {v} -\int m\gamma \mathbf {v} \cdot d\mathbf {v} =m\gamma v^{2}-{\frac ...
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Kinetic energy

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In physics, the kinetic energy ( KE ) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is 1 2 m v 2 {\displaystyle {\begin{smallmatrix}{\frac {1}{2}}mv^{2}\end{smallmatrix}}} . In relativistic mechanics, this is a good approximation only when v is much less than the speed of light. The standard unit of kinetic energy is the joule, while the imperial unit of kinetic energy is the foot-pound.
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Newtonian kinetic energy

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Kinetic energy of rigid bodies In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body depends on the mass of the body as well as its speed. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. In formula form: E k = 1 2 m v 2 {\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}} where m {\displaystyle m} is the mass and v {\displaystyle v} is the speed (or the velocity) of the body. In SI units, mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as E k = 1 2 ⋅ 80 kg ⋅ ( 18 m/s ) 2 = 12 , 960 J = 12.96 kJ {\displaystyle E_{\text{k}}={\frac {1}{2}}\cdot ...
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