Kinetic energy

The kinetic energy (from the Greek kinesis = movement ) or kinetic energy is the energy that holds an object due to its motion. It corresponds to the work that must be expended to put the object from the rest in the instantaneous movement. It depends on the mass and the velocity of the moving body.

As a symbol for the kinetic energy is used in theoretical physics usually, more rarely (eg in physical chemistry ).

The unit of kinetic energy is joules.

The concept of kinetic energy ( even without the pre-factor 1/2) in the 18th century by Émilie du Châtelet, building on ideas of Gottfried Wilhelm Leibniz, introduced (as vis viva, living force). Up to that time, was of the opinion of Newton, the kinetic energy is proportional to the velocity.

Kinetic energy in classical mechanics

Mass point

In classical mechanics, the kinetic energy T of a particle depends on its mass ( in kg) and its state of motion. Is described (in m / s) of the mass point of the state of motion by the rate shall apply

Shuts example, a car of mass at a rate of, it consequently has a kinetic energy of.

In special coordinate system of this expression has the form:

• Cartesian coordinates (x, y, z):

• Plane polar coordinates ():

• Cylindrical coordinates ():

• Spherical coordinates ():

This is the point on the coordinate their temporal change, the derivative with respect to time.

In the Hamiltonian mechanics of the motion state of a particle is expressed not by its speed, but by its momentum. It is thus:

Rigid body

The kinetic energy of a rigid body with the mass and the velocity of its center of gravity can be separated as the sum of its energy from the motion of its center of gravity ( translational energy ) and the rotational energy from the rotation around the center of gravity.

Here, the moment of inertia of the body with respect to its center of gravity and its angular velocity.

With the inertia tensor this is generally written as

Hydrodynamics

In hydrodynamics, the kinetic energy density is often specified place of the kinetic energy. This is usually expressed by a small or:

Herein, the density.

Kinetic energy in relativistic mechanics

In relativistic physics, the above mentioned dependence of the kinetic energy of the velocity is only approximately valid for speeds much less than the speed of light. The recognition that the kinetic energy T is the difference between total energy and rest energy, follows:

Where c is the speed of light, m is the rest mass and MREL the relativistic mass. With MREL = γ · m, the relationship:

γ is the Lorentz factor

From the Taylor expansion is obtained by

So again for the Newtonian kinetic energy.

As the energy grows beyond all limits, when the speed against the speed of light goes, it is not possible to accelerate a mass -prone body to the speed of light.

The chart on the right shows for a body with the mass of the relativistic and the Newtonian kinetic energy as a function of speed (measured in multiples of the speed of light).

As the speed of a moving body is dependent on the reference frame, the same applies to its kinetic energy. This is true in Newtonian and relativistic physics.

In the electric field, the kinetic energy of an electron of charge e and rest mass m linearly increases with the accelerating voltage U. The total energy E is:

Inducted into the relation above ( with E0 = mc ²) is obtained for the velocity v of an electron:

When accelerating voltages below 1 kV can be the speed of the classical approach for the kinetic energy estimate, at higher energies must be calculated relativistically. Already at a voltage of 10 kV, the electrons reach a speed of almost 20% the speed of light, at 1 MV 94%.

The Large Hadron Collider leads protons to an energy of 7 TeV. The proton ( rest energy 940 MeV) are accelerated to 0.999999991 times the speed of light.

Kinetic energy in quantum mechanics

In quantum mechanics, the expected value of the kinetic energy of a particle of the mass, which is described by the wave function is given by

Where the square of the momentum operator of the particle.

Formalism density functional theory is assumed that the electron density is well known, that is, the wave function not formally be known. With the electron density, the exact functional of the kinetic energy of electrons is unknown; If, however, in the case of a single electron is considered, then the kinetic energy as

Be written, with the Weizsäcker functional of the kinetic energy.