The

**kinetic energy** of an object is the

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

which it possesses due to its

motionIn physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

.

It is defined as the work needed to accelerate a body of a given mass from rest to its stated

velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. Having gained this energy during its

accelerationIn physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest.

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 referenceIn physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary from the point of view of an observer moving with the same velocity as the bullet, and so has zero kinetic energy. 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 massThe invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...

, which is independent of the reference frame.

In

classical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, the kinetic energy of a non-rotating object of

massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

*m* traveling at a

speedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

*v* is

*½ mv²*. In

relativistic mechanicsSpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, this is only a good approximation when

*v* is much less than the

speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

.

## History and etymology

The adjective

*kinetic* has its roots in the

GreekAncient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

word

*κίνησις* (

kinesisKinesis, like a taxis, is a movement or activity of a cell or an organism in response to a stimulus. However, unlike taxis, the response to the stimulus provided is non-directional....

) meaning

*motion*, which is the same root as in the word

cinemaA film, also called a movie or motion picture, is a series of still or moving images. It is produced by recording photographic images with cameras, or by creating images using animation techniques or visual effects...

, referring to motion pictures.

The principle in

classical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

that

*E ∝ mv²* was first developed by

Gottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

and

Johann BernoulliJohann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...

, who described kinetic energy as the

*living force*,

*vis viva*In the history of science, vis viva is an obsolete scientific theory that served as an elementary and limited early formulation of the principle of conservation of energy...

.

Willem 's GravesandeWillem Jacob 's Gravesande was a Dutch philosopher and mathematician.-Life:Born in 's-Hertogenbosch, he studied law in Leiden and wrote a thesis on suicide. He was praised by John Bernoulli when he published his book Essai de perspective. In 1715, he visited London and King George I. He became a...

of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay,

's GravesandeWillem Jacob 's Gravesande was a Dutch philosopher and mathematician.-Life:Born in 's-Hertogenbosch, he studied law in Leiden and wrote a thesis on suicide. He was praised by John Bernoulli when he published his book Essai de perspective. In 1715, he visited London and King George I. He became a...

determined that their penetration depth was proportional to the square of their impact speed.

Émilie du Châtelet-Early life:Du Châtelet was born on 17 December 1706 in Paris, the only daughter of six children. Three brothers lived to adulthood: René-Alexandre , Charles-Auguste , and Elisabeth-Théodore . Her eldest brother, René-Alexandre, died in 1720, and the next brother, Charles-Auguste, died in 1731...

recognized the implications of the experiment and published an explanation.

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 CoriolisGaspard-Gustave de Coriolis or Gustave Coriolis was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference. See the Coriolis Effect...

, who in 1829 published the paper titled

*Du Calcul de l'Effet des Machines* outlining the mathematics of kinetic energy.

William ThomsonWilliam Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...

, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51.

## Introduction

EnergyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

occurs in many forms, including

chemical energyChemical energy is the potential of a chemical substance to undergo a transformation through a chemical reaction or, to transform other chemical substances...

,

thermal energyThermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

,

electromagnetic radiationElectromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

,

gravitational energyGravitational energy is the energy associated with the gravitational field. This phrase is found frequently in scientific writings about quasars and other active galaxies.-Newtonian mechanics:...

, electric energy,

elastic energyElastic energy is the potential mechanical energy stored in the configuration of a material or physical system as work is performed to distort its volume or shape....

,

nuclear energyNuclear binding energy is the energy required to split a nucleus of an atom into its component parts. The component parts are neutrons and protons, which are collectively called nucleons...

, rest energy. These can be categorized in two main classes:

potential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

and kinetic energy.

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 foodFood energy is the amount of energy obtained from food that is available through cellular respiration.Food energy is expressed in food calories or kilojoules...

to accelerate a

bicycleA bicycle, also known as a bike, pushbike or cycle, is a human-powered, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A person who rides a bicycle is called a cyclist, or bicyclist....

to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome

air resistanceIn fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

and

frictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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

dynamoIn electricity generation, an electric generator is a device that converts mechanical energy to electrical energy. A generator forces electric charge to flow through an external electrical circuit. It is analogous to a water pump, which causes water to flow...

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

heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

.

Like any physical quantity which is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's

frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...

. Thus, the kinetic energy of an object is not

invariantGalilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames...

.

SpacecraftA spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....

use chemical energy to launch and gain considerable kinetic energy to reach

orbital velocityOrbital velocity can refer to the following:* The orbital speed of a body in a gravitational field.* The velocity of particles due to wave motion, in particular in wind waves....

. This kinetic energy remains constant while in orbit 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.

Kinetic energy can be passed from one object to another. In the game of

billiardsCue sports , also known as billiard sports, are a wide variety of games of skill generally played with a cue stick which is used to strike billiard balls, moving them around a cloth-covered billiards table bounded by rubber .Historically, the umbrella term was 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 collided with accelerates to a speed as the kinetic energy is passed on to it. Collisions in billiards are effectively

elastic collisionAn elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...

s, in which kinetic energy is preserved. In

inelastic collisionAn inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...

s, kinetic energy is dissipated in various forms of energy, such as heat, sound, binding energy (breaking bound structures).

FlywheelA flywheel is a rotating mechanical device that is used to store rotational energy. Flywheels have a significant moment of inertia, and thus resist changes in rotational speed. The amount of energy stored in a flywheel is proportional to the square of its rotational speed...

s have been developed as a method of

energy storageFlywheel energy storage works by accelerating a rotor to a very high speed and maintaining the energy in the system as rotational energy...

. This illustrates that kinetic energy is also stored in rotational motion.

Several mathematical description 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 effectsSpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

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.

### Kinetic energy of rigid bodies

In

classical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, 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 bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

, is given by the equation

where

is the mass and

is the speed (or the velocity) of the body. In

SISi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units (used for most modern scientific work), mass is measured in

kilogramThe kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

s, speed in metres per

secondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

, and the resulting kinetic energy is in

jouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

s.

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

*E*_{k} = (1/2) · 80 · 18^{2} J = 12.96 kJ

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.

The kinetic energy of an object is related to its

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

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 bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

with constant

massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

, whose

center of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

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 massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

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 which energy can neither enter nor leave, does not change in whatever reference frame 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 effectIn astronautics, the Oberth effect is where the use of a rocket engine when travelling at high speed generates much more useful energy than one at low speed...

. 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 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 massThe invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...

of the system as a whole.

#### Derivation

The work done accelerating a particle during the infinitesimal time interval

*dt* is given by the dot product of

*force* and

*displacement*:

where we have assumed the relationship

**p** =

*m* **v**. (However, also see the special relativistic derivation below.)

Applying the

product ruleIn calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

we see that:

Therefore (assuming constant mass), the following can be seen:

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:

This equation states that the kinetic energy (

*E*_{k}) is equal to the

integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of the

dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

of the

velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

(

**v**) of a body and the

infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

change of the body's

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

(

**p**). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

### Rotating bodies

If a rigid body is rotating about any line through the center of mass then it has

*rotational kinetic energy*The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total 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
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

*r* is the distance of any mass *dm* from that line
- is the body's moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

, equal to .

(In this equation the moment of

inertiaInertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

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).

### Kinetic energy of systems

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 SystemThe Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

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 energyIn thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

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 body's mass, inertia, and total energy.

### Frame of reference

The total kinetic energy of a system depends on the

inertial frame of referenceIn physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

: it is the sum of the total kinetic energy in a

center of momentum frameA center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame...

and the kinetic energy the total mass would have if it were concentrated in the

center of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

.

This may be simply shown: let

*V* be the relative speed of the frame

*k* from the center of mass frame

*i* :

However, let

the kinetic energy in the center of mass frame,

would be simply the total momentum which is by definition zero in the center of mass frame, and let the total mass:

. Substituting, we get:

Thus the kinetic energy of a system is lowest with respect 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 frameA center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame...

). In any other 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 frameA center-of-momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame...

is a quantity which is both invariant (all observers see it to be the same) and is conserved (in an isolated system, it cannot change value, no matter what happens inside the system).

### Rotation in systems

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 energyThe rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic 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.

## Relativistic kinetic energy of rigid bodies

In

special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, we must change the expression for linear momentum.

Using

*m* for rest mass,

**v** and

*v* for the object's velocity and speed respectively, and

*c* for the speed of light in vacuum, we assume for linear momentum that

, where

.

Integrating by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

gives

Remembering that

, we get:

where

*E*_{0} serves as an integration constant.

Thus:

The constant of integration

*E*_{0} is found by observing that, when

and

, giving

and giving the usual formula:

If a body's speed is a significant fraction of the

speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, it is necessary to use relativistic mechanics (the theory of relativity as developed by

Albert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

) to calculate its kinetic energy.

For a relativistic object the momentum p is equal to:

.

Thus the work expended accelerating an object from rest to a relativistic speed is:

.

The equation shows that the energy of an object approaches infinity as the velocity

*v* approaches the speed of light

*c*, thus it is impossible to accelerate an object across this boundary.

The mathematical by-product of this calculation is the

mass-energy equivalenceIn physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant...

formula—the body at rest must have energy content equal to:

At a low speed (v<

binomial approximationThe binomial approximation is useful for approximately calculating powers of numbers close to 1. It states that if x is a real number close to 0 and \alpha is a real number, then...

. Indeed, taking Taylor expansion for the reciprocal square root and keeping first two terms we get:

,

So, the total energy E can be partitioned into the energy of the rest mass plus the traditional 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 approximation is small for low speeds, and can be found by extending the expansion into a Taylor series by one more term:

.

For example, for a speed of 10 km/s 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), etc.

For higher speeds, the formula for the relativistic kinetic energy is derived by simply subtracting the rest mass energy from the total energy:

.

The relation between kinetic energy and momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

is more complicated in this case, and is given by the equation:

.

This can also be expanded as a Taylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

, the first term of which is the simple expression from Newtonian mechanics.

What this suggests is that the formulas for energy and momentum are not special and axiomatic, but rather concepts which emerge from the equation of mass with energy and the principles of relativity.

### General relativity

Using the convention that

where the four-velocityIn physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classicalvelocity...

of a particle is

and is the proper timeIn relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

of the particle, there is also an expression for the kinetic energy of the particle in general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

.

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 which 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.

## Quantum mechanical kinetic energy of rigid bodies

In the realm of quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the expectation value of the electron kinetic energy, , for a system of 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. Notice that this is the quantized version of the non-relativistic expression for kinetic energy in terms of momentum:

The density functionalDensity functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...

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 WeizsäckerCarl Friedrich Freiherr von Weizsäcker was a German physicist and philosopher. He was the longest-living member of the research team which performed nuclear research in Germany during the Second World War, under Werner Heisenberg's leadership...

kinetic energy functional.

## See also

- Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

- Escape velocity
In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

- Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

- KE-Munitions
- Kinetic energy per unit mass of projectiles
- Kinetic projectile
- Parallel axis theorem
In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular...

- Recoil
Recoil is the backward momentum of a gun when it is discharged. In technical terms, the recoil caused by the gun exactly balances the forward momentum of the projectile and exhaust gasses, according to Newton's third law...