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# Kinetic energy

Kinetic energy (also called vis viva, or living force) is energy possessed by a body by virtue of its motion. The kinetic energy of a body is equal to the amount of work needed to establish its velocity and rotation, starting from rest.

 Contents

## Equations

### Definition

$E_k = \int \mathbf{v} \cdot \mathrm{d}\mathbf{p}$

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).

### 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:

$E_k = E_t + E_r \,\!$

where:

Ek is the total kinetic energy
Et is the translational kinetic energy
Er is the rotational kinetic energy

For the translational kinetic energy of a body with mass m, whose centre of mass is moving in a straight line with linear velocity v, we can use the Newtonian approximation:

$E_t = \begin{matrix} \frac{1}{2} \end{matrix} mv^2$
Etranslation is the translational kinetic energy
m is mass of the body
v is linear velocity of the centre of mass body

Thus, for a speed of 10 m/s the kinetic energy is 50 J/kg, for a speed of 100 m/s it is 5 kJ/kg, etc.

If a body is rotating, its rotational kinetic energy or angular kinetic energy is calculated from:

$E_r = \begin{matrix} \frac{1}{2} \end{matrix} I \omega^2$,

where:

Er is the rotational energy or angular kinetic energy
I is the body's moment of inertia
ω is the body's angular velocity.

### Relativistic mechanics

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

$E_k = m c^2 (\gamma - 1) = \gamma m c^2 - m c^2 \;\!$
$\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$
$E_k = \gamma m c^2 - m c^2 = \left( \frac{1}{\sqrt{1- v^2/c^2 }} - 1 \right) m c^2$

where:

Ek is the kinetic energy of the body
v is the velocity of the body
m is its rest mass
c is the speed of light in a vacuum.
γmc2 is the total energy of the body
mc2 is the rest mass energy (90 petajoule/kg)

It is an edifying exercise to show that the ratio of this relativistic kinetic energy to the Newtonian kinetic energy given by (1/2)mv2 approaches 1 as v approaches 0, i.e.,

$\lim_{v\to 0}{\left( \frac{1}{\sqrt{1- v^2/c^2\ }} - 1 \right) m c^2 \over mv^2/2}=1.$

This can be done by the techniques of first-year calculus.

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.

The next term in the approximation is 0.375 mv4/c², e.g. for a speed of 10 km/s this is 0.04 J/kg, for a speed of 100 km/s it is 40 J/kg, etc.

The exact Taylor series is

$E_k = {1\over 2}mv^2 + {3\over 8}m\left({v^4\over c^2}\right) + {5\over 16}m\left({v^6\over c^4}\right) + \dots \,$

## Heat as kinetic energy

Heat is a form of energy due to the total kinetic energy of molecules and atoms of matter. The relationship between heat, temperature and kinetic energy of atoms and molecules is the subject of statistical mechanics. Heat is more akin to work in that it represents a change in internal energy. The energy that heat represents specifically refers to the energy associated with the random translational motion of atoms and molecules in some identifiable matter within a system. The conservation of heat and mechanical work form the first law of thermodynamics.

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