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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the kinetic energy of an object is the energy that it possesses due to its motion. 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 ** Working animal, an animal t ...
needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
, 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 In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
to a state of
rest Rest or REST may refer to: Relief from activity * Sleep ** Bed rest * Kneeling * Lying (position) * Sitting * Squatting position Structural support * Structural support ** Rest (cue sports) ** Armrest ** Headrest ** Footrest Arts and enter ...
. Formally, a kinetic energy is any term in a system's
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
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 (input value). Derivatives are a fundamental tool of calculus. ...
with respect to
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, the kinetic energy of a non-rotating object of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
''m'' traveling at a
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
''v'' is \fracmv^2. 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 English units are the units of measurement used in England up to 1826 (when they were replaced by Imperial units), which evolved as a combination of the Anglo-Saxon and Roman systems of units. Various standards have applied to English units at d ...
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 the engineering and gravitational systems in United States customary and imperial units of measure. It is the energy transferred upon applying a fo ...
.


History and etymology

The adjective ''kinetic'' has its roots in the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
word κίνησις ''kinesis'', meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
's concepts of
actuality and potentiality In philosophy, potentiality and actuality are a pair of closely connected principles which Aristotle used to analyze motion, causality, ethics, and physiology in his ''Physics'', ''Metaphysics'', ''Nicomachean Ethics'', and ''De Anima''. The c ...
. The principle in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
that ''E'' ∝ ''mv''2 was first developed by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
and
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...
, who described kinetic energy as the ''living force'', ''
vis viva ''Vis viva'' (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energy in an early formulation of the principle of conservation of energy. Overview Proposed by Gottfried L ...
''.
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 and the first experimental m ...
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 and the first experimental m ...
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. 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, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference, l ...
, 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 Tait Peter 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 occurs in many forms, including chemical energy,
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
, electromagnetic radiation,
gravitational energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
,
electric energy Electrical energy is energy related to forces on electrically charged particles and the movement of electrically charged particles (often electrons in wires, but not always). This energy is supplied by the combination of electric current and electr ...
, 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. 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, when on a two-wheeled bicycle, called bicycling or biking, is the use of cycles for transport, recreation, exercise or sport. People engaged in cycling are referred to as "cyclists", "bicyclists", or "bikers". Apart from two ...
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 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 flu ...
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 "Dynamo Electric Machine" (end view, partly section, ) A dynamo is an electrical generator that creates direct current using a commutator. Dynamos were the first electrical generators capable of delivering power for industry, and the foundati ...
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, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
. 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 A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, p ...
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 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 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 are a wide variety of games of skill played with a cue, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as . There are three major subdivisions ...
, 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 collision In physics, an elastic collision is an encounter ( collision) between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into ...
s, 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).
Flywheel A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, as ...
s 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
effects are significant, and a quantum mechanical model must be employed.


Newtonian kinetic energy


Kinetic energy of rigid bodies

In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, 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, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
depends on the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
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 Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
. The kinetic energy is equal to 1/2 the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
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 (magnitude of the velocity) of the body. In SI units, mass is measured in kilograms, speed in
metres per second The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in a ...
, 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 :E_\text = \frac \cdot 80 \,\text \cdot \left(18 \,\text\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 ** Working animal, an animal t ...
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 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 along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
, of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
with constant
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
m, whose center of mass 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 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, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver ...
. 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.


Derivation


=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 basic 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: # A body remains at rest, or in moti ...
:F = m a with ''m'' the mass and ''a'' the
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
of the object and :s = \frac the distance traveled 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 (m \mathbf)\,, where we have assumed the relationship p = ''m'' v and the validity of
Newton's Second Law Newton's laws of motion are three basic 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: # A body remains at rest, or in moti ...
. (However, also see the special relativistic derivation below.) 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. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
we see that: :d(\mathbf \cdot \mathbf) = (d \mathbf) \cdot \mathbf + \mathbf \cdot (d \mathbf) = 2(\mathbf \cdot d\mathbf). Therefore, (assuming constant mass so that ''dm'' = 0), we have, :\mathbf \cdot d (m \mathbf) = \frac d (\mathbf \cdot \mathbf) = \frac d v^2 = d \left(\frac\right). Since this is a
total differential In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is the ...
(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 (m \mathbf) = \int_0^t d \left(\frac\right) = \frac. This equation states that the kinetic energy (''E''k) is equal to the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
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).


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 * ''r'' is the distance of any mass ''dm'' from that line * I is the body's moment of inertia, equal to \int_Q dm. (In this equation the moment of
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
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 System Capitalization 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 "Solar ...
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.


Fluid dynamics

In fluid dynamics, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the
dynamic pressure In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5 :q = \frac\rho\, u^2 where (in SI units): * is the dynamic pressure in pascals ( ...
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 (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
. 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, 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 (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
: 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 \textstyle\mathbf be the relative velocity of the center of mass frame ''i'' in the frame ''k''. Since : v^2 = \left(v_i + V\right)^2 = \left(\mathbf_i + \mathbf\right) \cdot \left(\mathbf_i + \mathbf\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 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).


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 energy): :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

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 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 or system of objects that is independent of the overall motion of the system. More precisely, i ...
, 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.
Integrating by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
yields :E_\text = \int \mathbf \cdot d \mathbf = \int \mathbf \cdot d (m \gamma \mathbf) = m \gamma \mathbf \cdot \mathbf - \int m \gamma \mathbf \cdot d \mathbf = m \gamma v^2 - \frac \int \gamma d \left(v^2\right) Since \gamma = \left(1 - v^2/c^2\right)^, :\begin E_\text &= m \gamma v^2 - \frac \int \gamma d \left(1 - \frac\right) \\ &= m \gamma v^2 + m c^2 \left(1 - \frac\right)^\frac - E_0 \end E_0 is a constant of integration for the
indefinite integral In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
. Simplifying the expression we obtain :\begin E_\text &= m \gamma \left(v^2 + c^2 \left(1 - \frac\right)\right) - E_0 \\ &= m \gamma \left(v^2 + c^2 - v^2\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 = (\gamma - 1) 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 of sums of 1 and a small number ''x''. It states that : (1 + x)^\alpha \approx 1 + \alpha x. It is valid when , x, -1 and \alpha \geq 1. Derivations Using linear ...
or by taking the first two terms of the
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
for the reciprocal square root: :E_\text \approx m c^2 \left(1 + \frac \frac\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(1 + \frac \frac + \frac \frac\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, 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-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
of a particle is :u^ \, = \, \frac and \tau is the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
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 and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. 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(u^\right)^2 + g_ v^2 \left(u^\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( \sqrt - 1 \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( c^2 + 2\Phi \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 (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
. 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 in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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, :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 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 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(\mathbf), 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 rho= \frac \int \frac d^3r where T rho/math> is known as the von Weizsäcker kinetic energy functional.


See also

*
Escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
*
Foot-pound The foot-pound force (symbol: ft⋅lbf, ft⋅lbf, or ft⋅lb ) is a unit of work or energy in the engineering and gravitational systems in United States customary and imperial units of measure. It is the energy transferred upon applying a fo ...
* Joule * Kinetic energy penetrator * Kinetic energy per unit mass of projectiles *
Kinetic projectile A kinetic energy weapon (also known as kinetic weapon, kinetic energy warhead, kinetic warhead, kinetic projectile, kinetic kill vehicle) is a weapon based solely on a projectile's kinetic energy instead of an explosive or any other kind of payl ...
* Parallel axis theorem * Potential energy *
Recoil Recoil (often called knockback, kickback or simply kick) is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, as according to Newton's third law the force r ...


Notes


References

* * * * *


External links

* {{Authority control Dynamics (mechanics) Forms of energy