Kinetic energy

Kinetic energy is energy that a body has as a result of its speed.

Rigorous definitions

: E_k = int mathbf{F} cdot mathrm{d}mathbf{s} = int mathbf{v} cdot mathrm{d}mathbf{p} = (1/2)mv^2

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the words in the above equation state that the kinetic energy (Ek) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal of the body's momentum (p).

Related Topics:
Integral - Dot product - Velocity - Infinitesimal - Momentum

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In Newtonian mechanics

For non-relativistic mechanics, the total kinetic energy of a body can be considered as the sum of the body's translational kinetic energy and its rotational energy, or angular kinetic energy:

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: E_k = E_t + E_r ,

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where:

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:Ek is the total kinetic energy

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:Et is the translational kinetic energy

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:Er is the rotational energy or angular kinetic energy

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For the translational kinetic energy of a body with constant mass m, whose centre of mass is moving in a straight line with linear velocity v, as seen above is equal to

Related Topics:
Mass - Centre of mass - Linear velocity

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: E_t = egin{matrix} rac{1}{2} end{matrix} mv^2

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where:

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:m is mass of the body

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:v is linear velocity of the centre of mass body

Related Topics:
Linear velocity - Centre of mass

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Thus, for a speed of 10 m/s the kinetic energy is 50 J/kg, for a speed of 10 km/s it is 50 MJ/kg, etc.

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If a body is rotating, its rotational kinetic energy or angular kinetic energy is simply sum of kinetic energies of its moving parts, and thus is equal to:

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: E_r = egin{matrix} rac{1}{2} end{matrix} I omega^2 ,

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where:

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:I is the body's moment of inertia

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:ω is the body's angular velocity.

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In Relativistic mechanics

In Einstein's relativistic mechanics, (used especially for near-light velocities) mass no longer stays constant and accurate calculation of work to accelerate body results in the following expression for kinetic energy:

Related Topics:
Einstein - Relativistic mechanics

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:E_k = m c^2 (gamma - 1) = gamma m c^2 - m c^2 ;!

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:gamma = rac{1}{sqrt{1 - (v/c)^2}}

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:E_k = gamma m c^2 - m c^2

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