TheInfoList

In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, the kinetic energy of an object is the
energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ...

that it possesses due to its
motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position (mathematics), position over time. Motion is mathematically described in terms of Displacem ...
. It is defined as the
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking * Work (physics), the product of ...

needed to accelerate a body of a given mass from rest to its stated
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (p ...

. Having gained this energy during its
acceleration In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement ( ...

, 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 In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

with respect to
time Time is the continued of and that occurs in an apparently succession from the , through the , into the . It is a component quantity of various s used to events, to compare the duration of events or the intervals between them, and to of ...

. In classical mechanics, the kinetic energy of a non-rotating object of
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
''m'' traveling at a
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude (mathematics), magnitude of the rate of change of its Position (vector), position with time or the magnitude of the change of its posit ...

''v'' is $\fracmv^2$. In
relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanics, quantum mechanical description of a system of particles, or of a fluid, in cases where ...
, this is a good approximation only when ''v'' is much less than the
speed of light The speed of light in vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure m ...
. The standard unit of kinetic energy is the
joule The joule ( ; symbol: J) is a derived unit of energy In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform W ...

, while the
English unit English units are the units of measurement A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of ...
of kinetic energy is the
foot-pound The foot-pound force (symbol: ft⋅lbf, ft⋅lbf, or ft⋅lb ) is a unit of work or energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...
.

# History and etymology

The adjective ''kinetic'' has its roots in the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
word κίνησις ''kinesis'', meaning "motion". The dichotomy between kinetic energy and
potential energy In physics, potential energy is the energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter ...

can be traced back to
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

's concepts of
actuality and potentiality In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, l ...
. The principle in classical mechanics that ''E'' ∝ ''mv''2 was first developed by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
and
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss Swiss may refer to: * the adjectival form of Switzerland ,german: Schweizer(in),french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Svizra , government ...

, who described kinetic energy as the ''living force'', ''
vis viva ''Vis viva'' (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
''.
Willem 's Gravesande Willem Jacob 's Gravesande (26 September 1688 – 28 February 1742) was a Dutch mathematician and natural philosopher, chiefly remembered for developing experimental demonstrations of the laws of classical mechanics. As professor of mathematic ...
of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay,
Willem 's Gravesande Willem Jacob 's Gravesande (26 September 1688 – 28 February 1742) was a Dutch mathematician and natural philosopher, chiefly remembered for developing experimental demonstrations of the laws of classical mechanics. As professor of mathematic ...
determined that their penetration depth was proportional to the square of their impact speed.
Émilie du Châtelet Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (; 17 December 1706 – 10 September 1749) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officiall ...
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 Coriolis Gaspard-Gustave de Coriolis (; 21 May 1792 – 19 September 1843) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...
, 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. Rankine, who had introduced the term "potential energy" in 1853, and the phrase "actual energy" to complement it, later cites William Thomson and
Peter TaitPeter Tait may refer to: * Peter Tait (physicist) (1831–1901), Scottish mathematical physicist * Peter Tait (footballer) (1936–1990), English professional footballer * Peter Tait (mayor) (1915–1996), New Zealand politician * Peter Tait (radio ...
as substituting the word "kinetic" for "actual".

# Overview

Energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

occurs in many forms, including
chemical energy Chemical energy is the energy of chemical substance A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects th ...
,
thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concepts, such as the internal energy of a system; heat or sensible heat, which are defined as types of energy transfer (as is ...
,
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

,
gravitational energy Gravitational energy or gravitational potential energy is the potential energy 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 othe ...
,
electric energy Electrical energy is energy derived as a result of movement of electrically charged particles. When used loosely, ''electrical energy'' refers to energy that has been converted ''from'' electric potential energy. This energy is supplied by the comb ...
,
elastic energy Elastic energy is the mechanical potential energy 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. Common types of p ...
,
nuclear energy Nuclear energy may refer to: *Nuclear power, the use of sustained nuclear fission or nuclear fusion to generate heat and electricity *Nuclear binding energy, the energy required to split a nucleus of an atom *Nuclear potential energy, the potential ...
, and rest energy. These can be categorized in two main classes:
potential energy In physics, potential energy is the energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter ...

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. 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 Cycling, also called bicycling or biking, is the use of bicycle A bicycle, also called a bike or cycle, is a human-powered transport, human-powered or motorized bicycle, motor-powered, bicycle pedal, pedal-driven, single-track vehic ...

uses chemical energy provided by food to accelerate a
bicycle A bicycle, also called a bike or cycle, is a human-powered transport, human-powered or motorized bicycle, motor-powered, bicycle pedal, pedal-driven, single-track vehicle, having two bicycle wheel, wheels attached to a bicycle frame, frame, ...

to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome
air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding f ...
and
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

. 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 A dynamo is an that creates using a . Dynamos were the first electrical generators capable of delivering power for industry, and the foundation upon which many other later devices were based, including the , the , and the . Today, the simple ...
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 In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

. 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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. Thus, the kinetic energy of an object is not invariant.
Spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite alt=, A full-size model of the Earth observation satellite ERS 2 ">ERS_2.html" ;"title="Earth observation satellite ERS 2">Earth obse ...

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 or
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
, then throughout the orbit kinetic and
potential energy In physics, potential energy is the energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter ...

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 Cue sports (sometimes written cuesports), also known as billiard sports, are a wide variety of generally played with a , which is used to strike s and thereby cause them to move around a -covered bounded by elastic bumpers known as . Histor ...
, 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 physics, a collision is any event in which two or more bodies exert Force, forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with gr ...
in billiards are effectively
elastic collision (not shown) doesn't escape a system, atoms in thermal agitation undergo essentially elastic collisions. On average, two atoms rebound from each other with the same kinetic energy as before a collision. Five atoms are colored red so their paths of ...

s, in which kinetic energy is preserved. In
inelastic collision captured with a stroboscopic flash at 25 images per second. Each impact of the ball is inelastic, meaning that energy dissipates at each bounce. Ignoring air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction ...
s, kinetic energy is dissipated in various forms of energy, such as heat, sound and binding energy (breaking bound structures).
Flywheel A flywheel is a mechanical device which uses the conservation of angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantit ...

s have been developed as a method of
energy storage Energy storage is the capture of energy In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform Work (thermody ...
. 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 Relativistic quantum chemistry combines relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanics, quantum mechanical desc ...
become significant and the relativistic formula is used. If the object is on the atomic or sub-atomic scale,
quantum mechanical Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
effects are significant, and a quantum mechanical model must be employed.

# Newtonian kinetic energy

## Kinetic energy of rigid bodies

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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
depends on the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
of the body as well as its
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude (mathematics), magnitude of the rate of change of its Position (vector), position with time or the magnitude of the change of its posit ...

. The kinetic energy is equal to 1/2 the of the mass and the square of the speed. In formula form: :$E_\text = \frac mv^2$ where $m$ is the mass and $v$ is the speed (or the velocity) of the body. In units, mass is measured in
kilogram The kilogram (also kilogramme) is the base unit of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
s, speed in
metres per second The metre ( Commonwealth spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure", and cognate with Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, no ...
, and the resulting kinetic energy is in
joule The joule ( ; symbol: J) is a derived unit of energy In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform W ...

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 :$E_\text = \frac \cdot 80 \,\text \cdot \left\left(18 \,\text\right\right)^2 = 12,960 \,\text = 12.96 \,\text$ When a person throws a ball, the person does
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking * Work (physics), the product of ...

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., :$Fs = \frac mv^2$ 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 In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

by the equation: :$E_\text = \frac$ where: *$p$ is momentum *$m$ is mass of the body For the ''translational kinetic energy,'' that is the kinetic energy associated with
rectilinear motion Linear motion, also called rectilinear motion, is one-dimensional motion (physics), motion along a line (mathematics), straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two ...
, of a
rigid body In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
with constant
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
$m$, whose
center of mass In physics, the center of mass of a distribution of mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...
is moving in a straight line with speed $v$, as seen above is equal to :$E_\text = \frac mv^2$ where: *$m$ is the mass of the body *$v$ is the speed of the
center of mass In physics, the center of mass of a distribution of mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...
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 In astronautics Astronautics (or cosmonautics) is the theory and practice of travel beyond Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie represents trac ...
. 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 Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, invariants used to prove properties of loops * A data type in method ove ...
of the system as a whole.

### =Without vectors and calculus

= The work W done by a force ''F'' on an object over a distance ''s'' parallel to ''F'' equals :$W = F \cdot s$. Using
Newton's Second Law Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
:$F = m a$ with ''m'' the mass and ''a'' the
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...
of the object and :$s = \frac$ the distance travelled by the accelerated object in time ''t'', we find with $v = a t$ for the velocity ''v'' of the object :$W = m a \frac = \frac = \frac.$

### =With vectors and calculus

= 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 :$\mathbf \cdot d \mathbf = \mathbf \cdot \mathbf d t = \frac \cdot \mathbf d t = \mathbf \cdot d \mathbf = \mathbf \cdot d \left(m \mathbf\right)\,,$ where we have assumed the relationship p = ''m'' v and the validity of
Newton's Second Law Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
. (However, also see the special relativistic derivation
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
.) Applying the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more Functions (mathematics), functions. For two functions, it may be stated in Notation for differentiatio ...
we see that: :$d\left(\mathbf \cdot \mathbf\right) = \left(d \mathbf\right) \cdot \mathbf + \mathbf \cdot \left(d \mathbf\right) = 2\left(\mathbf \cdot d\mathbf\right).$ Therefore, (assuming constant mass so that ''dm'' = 0), we have, :$\mathbf \cdot d \left(m \mathbf\right) = \frac d \left(\mathbf \cdot \mathbf\right) = \frac d v^2 = d \left\left(\frac\right\right).$ Since this is a
total differential In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...
(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: :$E_\text = \int_0^t \mathbf \cdot d \mathbf = \int_0^t \mathbf \cdot d \left(m \mathbf\right) = \int_0^t d \left\left(\frac\right\right) = \frac.$ This equation states that the kinetic energy (''E''k) is equal to the
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of the
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (p ...

(v) of a body and the
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
change of the body's
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

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

## Rotating bodies

If a rigid body Q is rotating about any line through the center of mass then it has ''rotational kinetic energy'' ($E_\text\,$) which is simply the sum of the kinetic energies of its moving parts, and is thus given by: :$E_\text = \int_Q \frac = \int_Q \frac = \frac \int_Q dm = \frac I = \frac I \omega^2$ where: * ω is the body's
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

* ''r'' is the distance of any mass ''dm'' from that line * $I$ is the body's
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body In physics Physics is the natural science that studies matter, its ...

, equal to $\int_Q dm$. (In this equation the moment of
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

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 System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

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 The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
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.

## Fluid dynamics

In
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the
dynamic pressure In incompressible fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air a ...
at that point. :$E_\text = \frac mv^2$ Dividing by V, the unit of volume: :$\begin \frac &= \frac \fracv^2 \\ q &= \frac \rho v^2 \end$ where $q$ is the dynamic pressure, and ρ is the density of the incompressible fluid.

## Frame of reference

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 In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a const ...
. 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. 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 Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, invariants used to prove properties of loops * A data type in method ove ...
, which is independent of the reference frame. The total kinetic energy of a system depends on the
inertial frame of reference In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a const ...
: it is the sum of the total kinetic energy in a
center of momentum frame In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
and the kinetic energy the total mass would have if it were concentrated in the
center of mass In physics, the center of mass of a distribution of mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...
. This may be simply shown: let $\textstyle\mathbf$ be the relative velocity of the center of mass frame ''i'' in the frame ''k''. Since : $v^2 = \left\left(v_i + V\right\right)^2 = \left\left(\mathbf_i + \mathbf\right\right) \cdot \left\left(\mathbf_i + \mathbf\right\right) = \mathbf_i \cdot \mathbf_i + 2 \mathbf_i \cdot \mathbf + \mathbf \cdot \mathbf = v_i^2 + 2 \mathbf_i \cdot \mathbf + V^2,$ Then, :$E_\text = \int \frac dm = \int \frac dm + \mathbf \cdot \int \mathbf_i dm + \frac \int dm.$ However, let $\int \frac dm = E_i$ the kinetic energy in the center of mass frame, $\int \mathbf_i dm$ would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: $\int dm = M$. Substituting, we get: :$E_\text = E_i + \frac.$ 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 In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
or any other
center of momentum frame In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
). 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
is a quantity that is invariant (all observers see it to be the same).

## 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 energyRotational energy or angular kinetic energy is kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body o ...
): :$E_\text = E_\text + E_\text$ 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

If a body's speed is a significant fraction of the
speed of light The speed of light in vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure m ...
, it is necessary to use relativistic mechanics to calculate its kinetic energy. In
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
theory, the expression for linear momentum is modified. With ''m'' being an object's
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
, v and ''v'' its velocity and speed, and ''c'' the speed of light in vacuum, we use the expression for linear momentum $\mathbf = m\gamma \mathbf$, where $\gamma = 1/\sqrt$. yields :$E_\text = \int \mathbf \cdot d \mathbf = \int \mathbf \cdot d \left(m \gamma \mathbf\right) = m \gamma \mathbf \cdot \mathbf - \int m \gamma \mathbf \cdot d \mathbf = m \gamma v^2 - \frac \int \gamma d \left\left(v^2\right\right)$ Since $\gamma = \left\left(1 - v^2/c^2\right\right)^$, :$\begin E_\text &= m \gamma v^2 - \frac \int \gamma d \left\left(1 - \frac\right\right) \\ &= m \gamma v^2 + m c^2 \left\left(1 - \frac\right\right)^\frac - E_0 \end$ $E_0$ is a
constant of integration In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. Th ...

for the
indefinite integral In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This can ...
. Simplifying the expression we obtain :$\begin E_\text &= m \gamma \left\left(v^2 + c^2 \left\left(1 - \frac\right\right)\right\right) - E_0 \\ &= m \gamma \left\left(v^2 + c^2 - v^2\right\right) - E_0 \\ &= m \gamma c^2 - E_0 \end$ $E_0$ is found by observing that when $\mathbf = 0,\ \gamma = 1$ and $E_\text = 0$, giving :$E_0 = m c^2$ resulting in the formula :$E_\text = m \gamma c^2 - m c^2 = \frac - m c^2 = \left(\gamma - 1\right) m c^2$ 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 :$E_\text = E_0 = m c^2$ At a low speed (''v'' ≪ ''c''), the relativistic kinetic energy is approximated well by the classical kinetic energy. This is done by
binomial approximation The binomial approximation is useful for approximately calculating powers Powers (stylized as POWERS) is a musical duo composed of Mike Del Rio and Crista Ru. Their music has been described as alternative pop, electropop, and Progressive p ...
or by taking the first two terms of the
Taylor expansion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
for the reciprocal square root: :$E_\text \approx m c^2 \left\left(1 + \frac \frac\right\right) - m c^2 = \frac m v^2$ So, the total energy $E_k$ 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 :$E_\text \approx m c^2 \left\left(1 + \frac \frac + \frac \frac\right\right) - m c^2 = \frac m v^2 + \frac m \frac$ is small for low speeds. For example, for a speed of 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 :$E_\text = \sqrt - m c^2$ This can also be expanded as a
Taylor series In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
, the first term of which is the simple expression from Newtonian mechanics: :$E_\text \approx \frac - \frac.$ 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.

## General relativity

Using the convention that :$g_ \, u^ \, u^ \, = \, - c^2$ where the
four-velocityIn physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...

of a particle is :$u^ \, = \, \frac$ and $\tau$ is the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

of the particle, there is also an expression for the kinetic energy of the particle in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. If the particle has momentum :$p_ \, = \, m \, g_ \, u^$ 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 :$E \, = \, - \, p_ \, u_^$ and the kinetic energy can be expressed as the total energy minus the rest energy: :$E_ \, = \, - \, p_ \, u_^ \, - \, m \, c^2 \,.$ Consider the case of a metric that is diagonal and spatially isotropic (''g''''tt'', ''g''''ss'', ''g''''ss'', ''g''''ss''). Since :$u^ = \frac \frac = v^ u^$ where ''v''α is the ordinary velocity measured w.r.t. the coordinate system, we get :$-c^2 = g_ u^ u^ = g_ \left\left(u^\right\right)^2 + g_ v^2 \left\left(u^\right\right)^2 \,.$ Solving for ''u''t gives :$u^ = c \sqrt \,.$ Thus for a stationary observer (''v'' = 0) :$u_^ = c \sqrt$ and thus the kinetic energy takes the form :$E_\text = -m g_ u^t u_^t - m c^2 = m c^2 \sqrt - m c^2\,.$ Factoring out the rest energy gives: :$E_\text = m c^2 \left\left( \sqrt - 1 \right\right) \,.$ This expression reduces to the special relativistic case for the flat-space metric where :$\begin g_ &= -c^2 \\ g_ &= 1 \,. \end$ In the Newtonian approximation to general relativity :$\begin g_ &= -\left\left( c^2 + 2\Phi \right\right) \\ g_ &= 1 - \frac \end$ where Φ is the Newtonian
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical wo ...

. This means clocks run slower and measuring rods are shorter near massive bodies.

# Kinetic energy in quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, observables like kinetic energy are represented as
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
. 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 $\hat p$. The kinetic energy operator in the non-relativistic case can be written as :$\hat T = \frac.$ Notice that this can be obtained by replacing $p$ by $\hat p$ in the classical expression for kinetic energy in terms of
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

, :$E_\text = \frac.$ In the Schrödinger picture, $\hat p$ takes the form $-i\hbar\nabla$ where the derivative is taken with respect to position coordinates and hence :$\hat T = -\frac\nabla^2.$ The expectation value of the electron kinetic energy, $\left\langle\hat\right\rangle$, for a system of ''N'' electrons described by the Wave function, wavefunction $\vert\psi\rangle$ is a sum of 1-electron operator expectation values: :$\left\langle\hat\right\rangle = \left\langle \psi \left\vert \sum_^N \frac \nabla^2_i \right\vert \psi \right\rangle = -\frac \sum_^N \left\langle \psi \left\vert \nabla^2_i \right\vert \psi \right\rangle$ where $m_\text$ is the mass of the electron and $\nabla^2_i$ is the Laplacian operator acting upon the coordinates of the ''i''th electron and the summation runs over all electrons. The Density functional theory, 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 $\rho\left(\mathbf\right)$, 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 :$T\left[\rho\right] = \frac \int \frac d^3r$ where $T\left[\rho\right]$ is known as the Carl Friedrich von Weizsäcker, von Weizsäcker kinetic energy functional.