# Kinetic energy

The kinetic energy (from the Greek kinesis = movement) or kinetic energy or rarely kinetic energy is the energy that an object due to its motion contains. It corresponds to the work that has to be expended to move the object from rest to momentary movement. It depends on the mass and the speed of the moving body.

Often or is used as a symbol for the kinetic energy . The SI - unit of kinetic energy is the joule . ${\ displaystyle T}$${\ displaystyle E _ {\ mathrm {kin}}}$

The concept of kinetic energy as a quantity that is retained in the event of elastic collisions and many other mechanical processes was introduced as vis viva ( "living force" ) by Gottfried Wilhelm Leibniz , who argued with the followers of René Descartes about the correct conservation quantity saw in mechanics (1686). However, this size was larger by a factor of 2 than the kinetic energy that is valid today. The factor 12 in the formula for the kinetic energy can be found as early as 1726 by Daniel Bernoulli . The actual energy concept only emerged in the 19th century, especially in the applied mathematics school in France and with the advent of thermodynamics . In the mechanics of the 18th century, the main subject of investigation was celestial mechanics , it did not yet play a major role.

## Kinetic energy in classical mechanics

### Mass point

In classical mechanics , the kinetic energy of a mass point depends on its mass and its speed : ${\ displaystyle E}$${\ displaystyle m}$${\ displaystyle v}$

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} mv ^ {2}}$

Driving a car, for example, the mass at a rate of , it has therefore a kinetic energy of (the Joule , is the SI unit of energy). ${\ displaystyle m = 1000 \, \ mathrm {kg}}$${\ displaystyle v = 100 \, \ mathrm {km} / \ mathrm {h}}$${\ displaystyle E = {\ frac {1} {2}} \ cdot 1000 \, \ mathrm {kg} \ cdot \ left (100 \, {\ frac {\ mathrm {km}} {\ mathrm {h}} } \ right) ^ {2} \ approx {\ frac {1} {2}} \ cdot 1000 \, \ mathrm {kg} \ cdot \ left (27 {,} 78 \, {\ frac {\ mathrm {m }} {\ mathrm {s}}} \ right) ^ {2} = 385 \, 800 \, \ mathrm {J}}$${\ displaystyle \ mathrm {J}}$

If you describe the state of motion of the body not by its speed , but by its impulse , as u. a. is common in Hamiltonian mechanics , the following applies to the kinetic energy (because of ): ${\ displaystyle v}$ ${\ displaystyle p}$${\ displaystyle p = mv}$

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {p ^ {2}} {2m}}}$

#### Simple derivation

If a body of mass from rest to the speed accelerates, we must ensure the acceleration work inflict. With constant force: ${\ displaystyle m}$${\ displaystyle v}$ ${\ displaystyle W}$

${\ displaystyle W = Fs}$

The force imparted to the body a smooth acceleration , according to the fundamental equation of mechanics is . After a while the speed is up and the way has been covered. Inserted everything above, results in the acceleration work ${\ displaystyle a}$${\ displaystyle F = ma}$${\ displaystyle t}$${\ displaystyle v = at}$${\ displaystyle s = {\ tfrac {1} {2}} at ^ {2}}$

${\ displaystyle W = ma \ cdot {\ frac {1} {2}} at ^ {2} = {\ frac {1} {2}} mv ^ {2}}$.

Since the kinetic energy has the value zero at rest, it reaches exactly this value after the acceleration process . Hence for a body of mass with velocity : ${\ displaystyle W}$${\ displaystyle m}$${\ displaystyle v}$

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} mv ^ {2}}$

#### Special coordinate systems

In special coordinate systems this expression has the form:

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} m \ left ({\ dot {x}} ^ {2} + {\ dot {y}} ^ {2} + {\ dot {z}} ^ {2} \ right)}$
• Plane polar coordinates ( ):${\ displaystyle r, \ varphi}$
${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} m \ left ({\ dot {r}} ^ {2} + r ^ {2} {\ dot {\ varphi} } ^ {2} \ right)}$
• Spherical coordinates ( ):${\ displaystyle r, \ varphi, \ vartheta}$
${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} m \ left (r ^ {2} \ left [{\ dot {\ vartheta}} ^ {2} + {\ dot {\ varphi}} ^ {2} \ sin ^ {2} \ vartheta \ right] + {\ dot {r}} ^ {2} \ right)}$
${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} m \ left ({\ dot {r}} ^ {2} + r ^ {2} {\ dot {\ varphi} } ^ {2} + {\ dot {z}} ^ {2} \ right)}$

The point above the coordinate means its change over time, the derivation with respect to time.

### Rigid bodies

The kinetic energy of a rigid body with the total mass and the speed of its center of gravity is the sum of the energy from the movement of the center of gravity ( translational energy ) and the rotational energy from the rotation around the center of gravity: ${\ displaystyle M}$${\ displaystyle v _ {\ mathrm {s}}}$

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} M {v _ {\ mathrm {s}}} ^ {2} + {\ frac {1} {2}} J_ { \ mathrm {s}} \ omega ^ {2}}$

Here is the body's moment of inertia with respect to its center of gravity and the angular speed of rotation. ${\ displaystyle J _ {\ mathrm {s}}}$${\ displaystyle \ omega}$

With the inertia tensor this is generally written as: ${\ displaystyle I}$

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} M {v _ {\ mathrm {s}}} ^ {2} + {\ frac {1} {2}} {\ boldsymbol {\ omega}} ^ {T} I {\ boldsymbol {\ omega}}}$

### Hydrodynamics

In hydrodynamics , the kinetic energy density is often given instead of the kinetic energy . This is usually expressed by a small or : ${\ displaystyle e}$${\ displaystyle \ epsilon}$

${\ displaystyle e _ {\ mathrm {kin}} = {\ frac {1} {2}} \ rho v ^ {2}}$

Here is the density . ${\ displaystyle \ rho}$

## Kinetic energy in relativistic mechanics

Relativistic and classical kinetic energy in comparison

In relativistic physics , the above-mentioned dependence of the kinetic energy on the speed only applies approximately to speeds that are significantly lower than the speed of light . From the approach that the kinetic energy is the difference between total energy and rest energy , it follows: ${\ displaystyle E _ {\ mathrm {kin}}}$

${\ displaystyle E _ {\ mathrm {kin}} = \ gamma mc ^ {2} -mc ^ {2} = \ left (\ gamma -1 \ right) mc ^ {2}}$

Here is the speed of light, the mass and the Lorentz factor${\ displaystyle c}$${\ displaystyle m}$${\ displaystyle \ gamma}$

${\ displaystyle \ gamma = {\ frac {1} {\ sqrt {1- (v / c) ^ {2}}}}.}$

The Taylor expansion according to gives ${\ displaystyle v / c}$

${\ displaystyle E _ {\ mathrm {kin}} = {\ frac {1} {2}} mv ^ {2} + {\ frac {3} {8}} {\ frac {mv ^ {4}} {c ^ {2}}} + \ cdots}$,

thus for the Newtonian kinetic energy again. ${\ displaystyle v \ ll c}$

Since the energy would have to grow beyond all limits if the speed goes against the speed of light, it is not possible to accelerate a mass-laden body to the speed of light. ${\ displaystyle \ lim _ {v \ to c} E _ {\ mathrm {kin}} = \ infty,}$

The diagram on the right shows the relativistic and Newtonian kinetic energy as a function of speed (measured in multiples of the speed of light) for a body with the mass of . ${\ displaystyle m = 1 \, \ mathrm {kg}}$

Since the speed of a moving body depends on the reference system, this also applies to its kinetic energy. This is true in Newtonian and in relativistic physics.

Application examples
Relativistic speed of an electron after passing through an electric field

In the electric field, the energy of an electron of charge and mass increases linearly with the acceleration voltage passed through . The kinetic energy is now the difference between the relativistic total energy and the rest energy 0 . So the kinetic energy is: ${\ displaystyle e}$${\ displaystyle m}$${\ displaystyle U}$${\ displaystyle E}$${\ displaystyle E}$${\ displaystyle eU}$

${\ displaystyle e \ cdot U = E-E_ {0}}$

Note that for the total energy

${\ displaystyle E ^ {2} = c ^ {2} p ^ {2} + E_ {0} ^ {2} \ quad (*)}$

applies ( : relativistic momentum) and the relationship between momentum and total energy ${\ displaystyle p}$

${\ displaystyle cp = E \ cdot {\ frac {v} {c}}}$

consists, it follows for the total energy from : ${\ displaystyle (*)}$

${\ displaystyle E (v) = {\ frac {E_ {0}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}}}}}}}$

If you now calculate the difference from and , set the expression equal and solve for , you finally get: ${\ displaystyle E (v)}$${\ displaystyle E_ {0}}$${\ displaystyle e \ cdot U}$${\ displaystyle v}$

${\ displaystyle v = c \ cdot {\ sqrt {1 - {\ left ({\ frac {1} {1 + {\ frac {eU} {E_ {0}}}}} \ right)} ^ {2} }}}$ with the rest energy of an electron ${\ displaystyle E_ {0} = 0 {,} 51 ​​\, \ mathrm {MeV}}$

With acceleration voltages below 1 kV, the speed can be estimated from the classical approach for the kinetic energy; with higher energies, relativistic calculations must be made. Even at a voltage of 10 kV, the electrons reach a speed of almost 20% of the speed of light, at 1 MV 94%.

The Large Hadron Collider supplies protons with a kinetic energy of 6.5 TeV. This energy is about 8 thousand times greater than the rest energy of a proton. A collision between oppositely accelerated protons can result in particles with a correspondingly high rest energy.

## Kinetic energy in quantum mechanics

In quantum mechanics , the expected value of the kinetic energy of a particle of mass , which is described by the wave function , is given by ${\ displaystyle \ langle {\ hat {E}} _ {\ mathrm {kin}} \ rangle}$${\ displaystyle m}$ ${\ displaystyle \ vert \ psi \ rangle}$

${\ displaystyle \ langle {\ hat {E}} _ {\ mathrm {kin}} \ rangle = {\ frac {1} {2m}} \ langle \ psi | {\ hat {P}} ^ {2} | \ psi \ rangle}$,

where is the square of the momentum operator of the particle. ${\ displaystyle {\ hat {P}} ^ {2}}$

In the formalism of density functional theory it is only assumed that the electron density is known, that is, that the wave function does not have to be known formally. With the electron density , the exact functional of the kinetic energy for electrons is unknown; if, however, a single electron is considered in the case , the kinetic energy can be as ${\ displaystyle \ rho (\ mathbf {r})}$${\ displaystyle N}$${\ displaystyle N = 1}$

${\ displaystyle E _ {\ mathrm {kin}} [\ rho] = \ int {\ frac {1} {8}} {\ frac {\ nabla \ rho (\ mathbf {r}) \ cdot \ nabla \ rho ( \ mathbf {r})} {\ rho (\ mathbf {r})}} \ mathrm {d} ^ {3} r}$

where the Weizsäcker functional is the kinetic energy. ${\ displaystyle E _ {\ mathrm {kin}} [\ rho]}$

## Individual evidence

1. ^ Szabo, History of Mechanical Principles, Birkhäuser, p. 71.
2. Max Jammer , Article Energy, in Donald Borchert (Ed.), Encyclopedia of Philosophy, Thomson Gale 2006.
3. AP French: The special theory of relativity - MIT introductory course in physics 1968, pp. 19-23.