Kinetic energy | |
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Common symbols | KE, E_{k}, K or T |

SI unit | joule (J) |

Derivations from other quantities | E_{k} = 1/2 m v ^{2}E_{k} = E_{t} + E_{r} |

Part of a series on |

Classical mechanics |
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In physics, the **kinetic energy** of an object is the energy that it possesses due to its motion.^{ [1] } 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. Formally, a kinetic energy is any term in a system's Lagrangian which includes a derivative with respect to time. ^{ [2] }^{ [3] }

- History and etymology
- Overview
- Newtonian kinetic energy
- Kinetic energy of rigid bodies
- Rotating bodies
- Kinetic energy of systems
- Fluid dynamics
- Frame of reference
- Rotation in systems
- Relativistic kinetic energy
- General relativity
- Kinetic energy in quantum mechanics
- See also
- Notes
- References
- External links

In classical mechanics, the kinetic energy of a non-rotating object of mass *m* traveling at a speed *v* is . 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 English unit of kinetic energy is the foot-pound.

The adjective *kinetic* has its roots in the Greek word κίνησις *kinesis*, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality.^{ [4] }

The principle in classical mechanics that *E* ∝ *mv*^{2} was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the *living force*, * vis viva *. Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation.^{ [5] }

The terms *kinetic energy* and *work* in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled *Du Calcul de l'Effet des Machines* outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–1851.^{ [6] }^{ [7] } Rankine, who had introduced the term "potential energy" in 1853, and the phrase "actual energy" to complement it,^{ [8] } later cites William Thomson and Peter Tait as substituting the word "kinetic" for "actual".^{ [9] }

Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy.^{ [10] }

Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist.

The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat.

Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant.

Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an entirely circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However, it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is elliptical or hyperbolic, then throughout the orbit kinetic and potential energy are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Disregarding loss or gain however, the sum of the kinetic and potential energy remains constant.

Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically, and the ball it hit accelerates its speed as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, sound and binding energy (breaking bound structures).

Flywheels have been developed as a method of energy storage. This illustrates that kinetic energy is also stored in rotational motion.

Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian (classical) mechanics is suitable. However, if the speed of the object is comparable to the speed of light, relativistic effects become significant and the relativistic formula is used. If the object is on the atomic or sub-atomic scale, quantum mechanical effects are significant, and a quantum mechanical model must be employed.

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:

where is the mass and is the speed (magnitude of 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

When a person throws a ball, the person does work on it to give it speed as it leaves the hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: **net force × displacement = kinetic energy**, i.e.,

Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed.

The kinetic energy of an object is related to its momentum by the equation:

where:

- is momentum
- is mass of the body

For the *translational kinetic energy,* that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass , whose center of mass is moving in a straight line with speed , as seen above is equal to

where:

- is the mass of the body
- is the speed of the center of mass of the body.

The kinetic energy of any entity depends on the reference frame in which it is measured. However, the total energy of an isolated system, i.e. one in which energy can neither enter nor leave, does not change over time in the reference frame in which it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames would however disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.

The work W done by a force *F* on an object over a distance *s* parallel to *F* equals

- .

Using Newton's Second Law

with *m* the mass and *a* the acceleration of the object and

the distance traveled by the accelerated object in time *t*, we find with for the velocity *v* of the object

The work done in accelerating a particle with mass *m* during the infinitesimal time interval *dt* is given by the dot product of *force***F** and the infinitesimal *displacement**d***x**

where we have assumed the relationship **p** = *m* **v** and the validity of Newton's Second Law. (However, also see the special relativistic derivation below.)

Applying the product rule we see that:

Therefore, (assuming constant mass so that *dm* = 0), we have,

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by the force to bring the object from rest to velocity *v* is equal to the work necessary to do the reverse:

This equation states that the kinetic energy (*E*_{k}) is equal to the integral of the dot product of the velocity (**v**) of a body and the infinitesimal change of the body's momentum (**p**). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

If a rigid body Q is rotating about any line through the center of mass then it has *rotational kinetic energy* () which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

where:

- ω is the body's angular velocity
*r*is the distance of any mass*dm*from that line- is the body's moment of inertia, equal to .

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).

A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the Solar System the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains.

A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's center of momentum) may have various kinds of internal energy at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However, all internal energies of all types contribute to a body's mass, inertia, and total energy.

In fluid dynamics, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the dynamic pressure at that point.^{ [11] }

Dividing by V, the unit of volume:

where is the dynamic pressure, and ρ is the density of the incompressible fluid.

The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy.^{ [12] } By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case, the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame.

The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.

This may be simply shown: let be the relative velocity of the center of mass frame *i* in the frame *k*. Since

Then,

However, let the kinetic energy in the center of mass frame, would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: . Substituting, we get:^{ [13] }

Thus the kinetic energy of a system is lowest to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the center of mass frame or any other center of momentum frame). In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the center of momentum frame is a quantity that is invariant (all observers see it to be the same).

It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass (rotational energy):

where:

*E*_{k}is the total kinetic energy*E*_{t}is the translational kinetic energy*E*_{r}is the*rotational energy*or*angular kinetic energy*in the rest frame

Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.

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 , where .

Integrating by parts yields

Since ,

is a constant of integration for the indefinite integral.

Simplifying the expression we obtain

is found by observing that when and , giving

resulting in the formula

This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary.

The mathematical by-product of this calculation is the mass-energy equivalence formula—the body at rest must have energy content

At a low speed (*v* ≪ *c*), the relativistic kinetic energy is approximated well by the classical kinetic energy. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root:

So, the total energy can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds.

When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the Taylor series approximation

is small for low speeds. For example, for a speed of 10 km/s (22,000 mph) the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg).

The relativistic relation between kinetic energy and momentum is given by

This can also be expanded as a Taylor series, the first term of which is the simple expression from Newtonian mechanics:^{ [14] }

This suggests that the formulae for energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity.

Using the convention that

where the four-velocity of a particle is

and is the proper time of the particle, there is also an expression for the kinetic energy of the particle in general relativity.

If the particle has momentum

as it passes by an observer with four-velocity *u*_{obs}, then the expression for total energy of the particle as observed (measured in a local inertial frame) is

and the kinetic energy can be expressed as the total energy minus the rest energy:

Consider the case of a metric that is diagonal and spatially isotropic (*g*_{tt}, *g*_{ss}, *g*_{ss}, *g*_{ss}). Since

where *v*^{α} is the ordinary velocity measured w.r.t. the coordinate system, we get

Solving for *u*^{t} gives

Thus for a stationary observer (*v* = 0)

and thus the kinetic energy takes the form

Factoring out the rest energy gives:

This expression reduces to the special relativistic case for the flat-space metric where

In the Newtonian approximation to general relativity

where Φ is the Newtonian gravitational potential. This means clocks run slower and measuring rods are shorter near massive bodies.

In quantum mechanics, observables like kinetic energy are represented as operators. For one particle of mass *m*, the kinetic energy operator appears as a term in the Hamiltonian and is defined in terms of the more fundamental momentum operator . The kinetic energy operator in the non-relativistic case can be written as

Notice that this can be obtained by replacing by in the classical expression for kinetic energy in terms of momentum,

In the Schrödinger picture, takes the form where the derivative is taken with respect to position coordinates and hence

The expectation value of the electron kinetic energy, , for a system of *N* electrons described by the wavefunction is a sum of 1-electron operator expectation values:

where is the mass of the electron and is the Laplacian operator acting upon the coordinates of the *i*^{th} electron and the summation runs over all electrons.

The density functional formalism of quantum mechanics requires knowledge of the electron density *only*, i.e., it formally does not require knowledge of the wavefunction. Given an electron density , the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as

where is known as the von Weizsäcker kinetic energy functional.

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*Mechanics*(Third ed.). p. 15. ISBN 0-7506-2896-0. - ↑ Goldstein, Herbert.
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*Energy and Empire: A Biographical Study of Lord Kelvin*. Cambridge University Press. p. 866. ISBN 0-521-26173-2. - ↑ John Theodore Merz (1912).
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In physics, the **Maxwell–Boltzmann distribution**, or **Maxwell(ian) distribution**, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.

In Newtonian mechanics, **linear momentum**, **translational momentum**, or simply **momentum** is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If *m* is an object's mass and **v** is its velocity, then the object's momentum **p** is :

In physics, **potential energy** is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.

In mechanics, the **virial theorem** provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. Mathematically, the theorem states

In particle physics, the **Dirac equation** is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

**Compton scattering**, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a stationary charged particle, usually an electron. If it results in a decrease in energy of the photon, it is called the **Compton effect**. Part of the energy of the photon is transferred to the recoiling electron. **Inverse Compton scattering** occurs when a charged particle transfers part of its energy to a photon.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.

In special relativity, **four-momentum** is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum **p** = = *γm***v**, where **v** is the particle's three-velocity and γ the Lorentz factor, is

In fluid dynamics, the **Euler equations** are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.

The **Lorentz factor** or **Lorentz term** is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz.

The **rigid rotor** is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the *linear rotor* requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

In differential geometry, the **four-gradient** is the four-vector analogue of the gradient from vector calculus.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

In physics, **relativistic mechanics** refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light *c*. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at *any* speed, including faster than light. The foundations of relativistic mechanics are the postulates of special relativity and general relativity. The unification of SR with quantum mechanics is relativistic quantum mechanics, while attempts for that of GR is quantum gravity, an unsolved problem in physics.

In physics, the **energy–momentum relation**, or **relativistic dispersion relation**, is the relativistic equation relating total energy to invariant mass and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum. It can be written as the following equation:

In relativity, **proper velocity****w** of an object relative to an observer is the ratio between observer-measured displacement vector and proper time τ elapsed on the clocks of the traveling object:

In fluid dynamics, **Airy wave theory** gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

**Velocity** is the directional speed of a object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

In physics, **relativistic angular momentum** refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.

In mathematical physics, the **Gordon decomposition** of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

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