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| | Social Media as promoting method entails more than only traditional marketing. It is about using social media to widen and extend the news about your website. People always love the concept of being a part of an social network. Each of us is networking socially for different reasons. The following are some guidelines that will help maximize your social networking efforts that will create an effective online strategy for your small business.<br><br> |
| {{About|momentum in physics}}
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| {{pp-move-indef}}{{Infobox physical quantity
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| |image=[[File:Billard.JPG|thumb]]
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| |caption=In a game of [[Pool (cue sports)|pool]], momentum is [[Conservation of linear momentum|conserved]]; that is, if one ball stops dead after the collision, the other ball will continue away with all the momentum. If the moving ball continues or is deflected then both balls will carry a portion of the momentum from the collision,
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| |unit=kg m/s or N s
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| |symbols=''p'', '''p'''}}
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| {{Classical mechanics|cTopic=Fundamental concepts}}
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| In [[classical mechanics]], '''linear momentum''' or '''translational momentum''' ([[plural|pl.]] momenta; [[SI]] unit [[kilogram|kg]] [[meters per second|m/s]], or equivalently, [[newton (unit)|N]] [[second|s]]) is the product of the [[mass]] and [[velocity]] of an object. For example, a heavy truck moving fast has a large momentum—it takes a large and prolonged force to get the truck up to this speed, and it takes a large and prolonged force to bring it to a stop afterwards. If the truck were lighter, or moving more slowly, then it would have less momentum.
| | It's certainly a different way of performing things and very user centered- with each interaction should keep in mind- what's in it for my prospects?<br><br>On my website I optimize for the keywords, ensuring everything is within the right section. I link back to my website with each and every my Social Media presentations as well.<br><br><br><br>So, as it is important not to under-estimate Google, it crucial not to be able to scared of it either. An enormous it may be, but in essence an amiable one. Going on a white hat approach to SEO and hand available with Google will merely help these types of provide good search results, but your ranking results will reap the benefits. This is important to any industry sector, and not just those working towards travel marketing solutions online.<br><br>To begin, how can a new artist like myself sell records without record tirechains? Sadly over the last 10 years, records stores have become obsolete. In fact, even Wal-Mart and Target just offer a minimal selection of physical compact discs. The answer turns to self promotion through the help of a radio promoter. By this i mean that the artist will actually have to leave out and knock on doors of radio to have their music recognized. It's called a Radio Tour. The air promoter calls stations and the artist will literally pay a visit to as many stations while can, convince them to play their music so that they will build their listener fan base and drive that group of followers to online music stores such as iTunes, CD Baby, . . .. Without the help of radio, there is little change hope a new artist to gain exposure and sell their items.<br><br>Commercial speech works both verbally and orally. So, yes, it can certainly be effective via written correspondence. But, because people do business with people they know, like and trust and also the trust factor increases just because they get realize you, ultimately you in order to engage in a live conversation to speed up the pathway.<br><br>Chatting in Twitter is termed a "tweeting". Faster you have an open tweet, you use "@(the person's twitter account name) (message)". By using that, you send out a message to that individual. Direct messages (DMs) are tweets submitted in private - you associated with that by having "D (person's twitter account name) (message)" - but take note, you can simply send direct messages once the other person is following your organization.<br><br>Enter their posts to social content. This way you does to strengthen the romance. They'll be pleased find out that honesty spread their blog posts social media.<br><br>Should you loved this information and you would like to receive much more information about [http://www.mavsocial.com/ Social Media Campaign Planning Software] i implore you to visit our own page. |
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| Like velocity, linear momentum is a [[Euclidean vector|vector]] quantity, possessing a direction as well as a magnitude:
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| :<math>\mathbf{p} = m \mathbf{v}.</math>
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| Linear momentum is also a ''conserved'' quantity, meaning that if a [[closed system]] is not affected by external forces, its total linear momentum cannot change. In classical mechanics, [[#Conservation|conservation of linear momentum]] is implied by [[Newton's laws]]; but it also holds in [[special relativity]] (with a modified formula) and, with appropriate definitions, a (generalized) linear momentum [[conservation law]] holds in [[electrodynamics]], [[quantum mechanics]], [[quantum field theory]], and [[general relativity]].
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| {{TOC limit|3}}
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| ==Newtonian mechanics==
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| Momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities. Because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see [[Momentum#Multiple dimensions|multiple dimensions]]).
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| ===Single particle===
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| The momentum of a particle is traditionally represented by the letter {{math|''p''}}. It is the product of two quantities, the [[mass]] (represented by the letter {{math|''m''}}) and [[velocity]] ({{math|''v''}}):<ref name=FeynmanCh9>{{harvnb|Feynman Vol. 1|loc=Chapter 9}}</ref>
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| :<math>p = m v. </math>
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| The units of momentum are the product of the units of mass and velocity. In [[SI units]], if the mass is in kilograms and the velocity in meters per second, then the momentum is in kilograms meters/second (kg m/s). Being a vector, momentum has magnitude and direction. For example, a model airplane of 1 kg, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground.
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| ===Many particles===
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| The momentum of a system of particles is the sum of their momenta. If two particles have masses {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}, and velocities {{math|''v''<sub>1</sub>}} and {{math|''v''<sub>2</sub>}}, the total momentum is
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| :<math> \begin{align} p &= p_1 + p_2 \\
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| &= m_1 v_1 + m_2 v_2\,. \end{align} </math>
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| The momenta of more than two particles can be added in the same way.
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| A system of particles has a [[center of mass]], a point determined by the weighted sum of their positions:
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| :<math> r_\text{cm} = \frac{m_1 r_1 + m_2 r_2 + \cdots}{m_1 + m_2 + \cdots}.</math>
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| If all the particles are moving, the center of mass will generally be moving as well. If the center of mass is moving at velocity {{math|''v''<sub>cm</sub>}}, the momentum is:
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| :<math>p= mv_\text{cm}.</math>
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| This is known as [[Euler's laws of motion|Euler's first law]].<ref name="BookRags">{{cite web
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| |url=http://www.bookrags.com/research/eulers-laws-of-motion-wom/
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| |title=Euler's Laws of Motion
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| |accessdate=2009-03-30}}</ref><ref name="McGillKing">{{cite book
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| |title=Engineering Mechanics, An Introduction to Dynamics
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| |edition=3rd
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| |last=McGill and King
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| |publisher=PWS Publishing Company
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| |year=1995
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| |isbn=0-534-93399-8}}</ref>
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| ===Relation to force===
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| If a force {{math|''F''}} is applied to a particle for a time interval {{math|Δ''t''}}, the momentum of the particle changes by an amount
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| :<math>\Delta p = F \Delta t\,.</math>
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| In differential form, this gives [[Newton's second law]]: the rate of change of the momentum of a particle is equal to the force {{math|''F''}} acting on it:<ref name=FeynmanCh9/>
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| :<math>F = \frac{dp }{d t}. </math>
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| If the force depends on time, the change in momentum (or [[impulse (physics)|impulse]]) between times {{math|''t''<sub>1</sub>}} and {{math|''t''<sub>2</sub>}} is
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| :<math> \Delta p = \int_{t_1}^{t_2} F(t)\, dt\,.</math>
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| The second law only applies to a particle that does not exchange matter with its surroundings,<ref name="plastino">{{cite journal|last=Plastino|first=Angel R. |coauthors=Muzzio, Juan C.|year=1992|title=On the use and abuse of Newton's second law for variable mass problems|journal=Celestial Mechanics and Dynamical Astronomy|publisher=Kluwer Academic Publishers|location=Netherlands|volume= 53|issue= 3|pages=227–232|issn=0923-2958|bibcode=1992CeMDA..53..227P|doi=10.1007/BF00052611}} "We may conclude emphasizing that Newton's second law is valid for constant mass only. When the mass varies due to accretion or ablation, [an alternate equation explicitly accounting for the changing mass] should be used."</ref> and so it is equivalent to write
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| :<math>F = m\frac{dv}{d t} = m a,</math>
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| so the force is equal to mass times [[acceleration]].<ref name=FeynmanCh9/>
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| ''Example'': a model airplane of 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The thrust required to produce this acceleration is 3 [[newton (unit)|newton]]. The change in momentum is 6 kg m/s. The rate of change of momentum is 3 (kg m/s)/s = 3 N.
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| ===Conservation===
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| [[Image:Newtons cradle animation book 2.gif|right|thumb|A [[Newton's cradle]] demonstrates conservation of momentum.]]
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| In a [[closed system]] (one that does not exchange any matter with the outside and is not acted on by outside forces) the total momentum is constant. This fact, known as the ''law of conservation of momentum'', is implied by [[Newton's laws of motion]].<ref name=FeynmanCh10>{{harvnb|Feynman Vol. 1|loc=Chapter 10}}</ref> Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that {{math|''F''<sub>1</sub> {{=}} ''dp''<sub>1</sub>/''dt''}} and {{math|''F''<sub>2</sub> {{=}} ''dp''<sub>2</sub>/''dt''}}. Therefore
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| :<math> \frac{d p_1}{d t} = - \frac{d p_2}{d t}, </math>
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| or
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| :<math> \frac{d}{d t} \left(p_1+ p_2\right)= 0. </math>
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| If the velocities of the particles are {{math|''u''<sub>1</sub>}} and {{math|''u''<sub>2</sub>}} before the interaction, and afterwards they are {{math|''v''<sub>1</sub>}} and {{math|''v''<sub>2</sub>}}, then
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| :<math>m_1 u_{1} + m_2 u_{2} = m_1 v_{1} + m_2 v_{2}.</math>
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| This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including [[collision]]s and separations caused by explosive forces.<ref name=FeynmanCh10/> It can also be generalized to situations where Newton's laws do not hold, for example in the [[theory of relativity]] and in [[Classical electromagnetism|electrodynamics]].<ref name=Goldstein54/>
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| ===Dependence on reference frame===
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| [[File:Relativity an apple in a lift.svg|thumb|Newton's apple in Einstein's elevator. In person A's frame of reference, the apple has non-zero velocity and momentum. In the elevator's and person B's frames of reference, it has zero velocity and momentum.]]
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| Momentum is a measurable quantity, and the measurement depends on the motion of the observer. For example, if an apple is sitting in a glass elevator that is descending, an outside observer looking into the elevator sees the apple moving, so to that observer the apple has a nonzero momentum. To someone inside the elevator, the apple does not move, so it has zero momentum. The two observers each have a [[frame of reference]] in which they observe motions, and if the elevator is descending steadily they will see behavior that is consistent with the same physical laws.
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| Suppose a particle has position {{math|''x''}} in a stationary frame of reference. From the point of view of another frame of reference moving at a uniform speed {{math|''u''}}, the position (represented by a primed coordinate) changes with time as
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| :<math> x' = x - ut\,.</math>
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| This is called a [[Galilean transformation]]. If the particle is moving at speed {{math|''dx/dt'' {{=}} ''v''}} in the first frame of reference, in the second it is moving at speed
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| :<math> v' = \frac{dx'}{dt} = v-u\,.</math>
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| Since {{math|''u''}} does not change, the accelerations are the same:
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| :<math> a' = \frac{dv'}{dt} = a\,.</math>
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| Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or [[Galilean invariance]].<ref>{{harvnb|Goldstein|1980|p=276}}</ref>
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| A change of reference frame can often simplify calculations of motion. For example, in a collision of two particles a reference frame can be chosen where one particle begins at rest. Another commonly used reference frame is the [[center of mass frame]], one that is moving with the center of mass. In this frame, the total momentum is zero.
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| ===Application to collisions===
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| By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision. Another property of the motion, [[kinetic energy]], must be known. This is not necessarily conserved. If it is conserved, the collision is called an ''[[elastic collision]]''; if not, it is an ''[[inelastic collision]]''.
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| ====Elastic collisions====
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| {{Main|Elastic collision}}
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| [[File:Elastischer stoß.gif|thumb|right|Elastic collision of equal masses]]
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| [[File:Elastischer stoß3.gif|thumb|right|Elastic collision of unequal masses]]
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| An elastic collision is one in which no kinetic energy is lost. Perfectly elastic "collisions" can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps them apart. A [[gravity assist|slingshot maneuver]] of a satellite around a planet can also be viewed as a perfectly elastic collision from a distance. A collision between two [[Pool billiards|pool]] balls is a good example of an ''almost'' totally elastic collision, due to their high [[stiffness|rigidity]]; but when bodies come in contact there is always some [[dissipation]].<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html |title=Elastic and inelastic collisions |work=Hyperphysics |author=Carl Nave |year=2010 |accessdate=2 August 2012}}</ref>
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| A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are {{math|''u''<sub>1</sub>}} and {{math|''u''<sub>2</sub>}} before the collision and {{math|''v''<sub>1</sub>}} and {{math|''v''<sub>2</sub>}} after, the equations expressing conservation of momentum and kinetic energy are:
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| :<math>\begin{align} m_1 u_1 + m_2 u_2 &= m_1 v_1 + m_2 v_2\\
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| \tfrac{1}{2} m_1 u_1^2 + \tfrac{1}{2} m_2 u_2^2 &= \tfrac{1}{2} m_1 v_1^2 + \tfrac{1}{2} m_2 v_2^2\,.\end{align}</math>
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| A change of reference frame can often simplify the analysis of a collision. For example, suppose there are two bodies of equal mass {{math|''m''}}, one stationary and one approaching the other at a speed {{math|''v''}} (as in the figure). The center of mass is moving at speed {{math|''v''/2}} and both bodies are moving towards it at speed {{math|''v''/2}}. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed {{math|''v''}}. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by<ref name=FeynmanCh10/>
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| :<math>\begin{align} v_1 &= u_2\\
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| v_2 &= u_1\,. \end{align}</math>
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|
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| In general, when the initial velocities are known, the final velocities are given by<ref>{{cite book|last=Serway|first=Raymond A.|coauthors=John W. Jewett, Jr|title=Principles of physics : a calculus-based text|year=2012|publisher=Brooks/Cole, Cengage Learning|location=Boston, MA|isbn=9781133104261|page=245|edition=5th}}</ref>
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| :<math> v_{1} = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) u_{1} + \left( \frac{2 m_2}{m_1 + m_2} \right) u_{2}\,</math>
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| :<math> v_{2} = \left( \frac{m_2 - m_1}{m_1 + m_2} \right) u_{2} + \left( \frac{2 m_1}{m_1 + m_2} \right) u_{1}\,.</math>
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| If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.
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| ====Inelastic collisions====
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| {{Main|Inelastic collision}}
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| [[Image:Inelastischer stoß.gif|thumb|right|a perfectly inelastic collision between equal masses]]
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| In an inelastic collision, some of the [[kinetic energy]] of the colliding bodies is converted into other forms of energy such as [[heat]] or [[sound]]. Examples include [[traffic collisions]],<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1 |title=Forces in car crashes |work=Hyperphysics |author=Carl Nave |year=2010 |accessdate=2 August 2012}}</ref> in which the effect of lost kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the [[Franck–Hertz experiment]]);<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html |title=The Franck-Hertz Experiment |work=Hyperphysics |author=Carl Nave |year=2010 |accessdate=2 August 2012}}</ref> and [[particle accelerator]]s in which the kinetic energy is converted into mass in the form of new particles.
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| In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. If one body is motionless to begin with, the equation for conservation of momentum is
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| :<math>m_1 u_1 = \left( m_1 + m_2 \right) v\,,</math>
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| so
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| :<math> v = \frac{m_1}{m_1+m_2} u_1\,.</math>
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| In a frame of reference moving at the speed {{math|''v'')}}, the objects are brought to rest by the collision and 100% of the kinetic energy is converted.
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| One measure of the inelasticity of the collision is the [[coefficient of restitution]] {{math|''C''<sub>R</sub>}}, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to ball sports, this can be easily measured using the following formula:<ref>{{cite book|last=McGinnis|first=Peter M.|title=Biomechanics of sport and exercise|year=2005|publisher=Human Kinetics|location=Champaign, IL [u.a.]|isbn=9780736051019|page=85|edition=2nd |url=http://books.google.com/books?id=PrOKEcZXJ58C&pg=PA85&lpg=PA85&dq=coefficient+of+restitution+bounciness Biomechanics of sport and exercise}}</ref>
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| :<math>C_\text{R} = \sqrt{\frac{\text{bounce height}}{\text{drop height}}}\,.</math>
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| The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an [[explosion]] is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. [[Rocket]]s also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.<ref>{{Citation | last = Sutton | first = George | title = Rocket Propulsion Elements |edition=7th |url=http://books.google.com/?id=LQbDOxg3XZcC&printsec=frontcover | publisher = John Wiley & Sons | location = Chichester | year = 2001 | isbn = 978-0-471-32642-7 |chapter=1}}</ref>
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| ===Multiple dimensions===
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| [[Image:Elastischer stoß 2D.gif|thumb|right|Two-dimensional elastic collision. There is no motion perpendicular to the image, so only two components are needed to represent the velocities and momenta. The two blue vectors represent velocities after the collision and add vectorially to get the initial (red) velocity.]]
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| Real motion has both direction and magnitude and must be represented by a [[Vector (geometry)|vector]]. In a coordinate system with {{math|''x, y, z''}} axes, velocity has components {{math|''v''<sub>x</sub>}} in the {{math|''x''}} direction, {{math|''v''<sub>y</sub>}} in the {{math|''y''}} direction, {{math|''v''<sub>z</sub>}} in the {{math|''z''}} direction. The vector is represented by a boldface symbol:<ref name=FeynmanCh11>{{harvnb|Feynman Vol. 1|loc=Chapter 11}}</ref>
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| :<math>\mathbf{v} = \left(v_x,v_y,v_z \right). </math>
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| Similarly, the momentum is a vector quantity and is represented by a boldface symbol:
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| :<math>\mathbf{p} = \left(p_x,p_y,p_z \right). </math>
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| The equations in the previous sections work in vector form if the scalars {{math|''p''}} and {{math|''v''}} are replaced by vectors {{math|'''p'''}} and {{math|'''v'''}}. Each vector equation represents three scalar equations. For example,
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| :<math>\mathbf{p}= m \mathbf{v}</math>
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| represents three equations:<ref name=FeynmanCh11/>
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| :<math>\begin{align} p_x &= m v_x\\ p_y &= m v_y \\ p_z &= m v_z. \end{align} </math>
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| The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the [[Magnitude (mathematics)#Euclidean vectors|magnitude of the vector]], for example,
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| :<math> v^2 = v_x^2+v_y^2+v_z^2\,.</math>
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| Each vector equation represents three scalar equations. Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.<ref name=FeynmanCh11/>
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| A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision (as in the figure).<ref>{{harvnb|Rindler|1986|pp=26–27}}</ref>
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| === Objects of variable mass ===
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| The concept of momentum plays a fundamental role in explaining the behavior of [[variable-mass system|variable-mass objects]] such as a [[rocket]] ejecting fuel or a [[star]] [[accretion (astrophysics)|accreting]] gas. In analyzing such an object, one treats the object's mass as a function that varies with time: {{math|''m''(''t'')}}. The momentum of the object at time {{math|''t''}} is therefore {{math|''p''(''t'') {{=}} ''m''(''t'')''v''(''t'')}}. One might then try to invoke Newton's second law of motion by saying that the external force {{math|''F''}} on the object is related to its momentum {{math|''p''(''t'')}} by {{math|''F'' {{=}} ''dp''/''dt''}}, but this is incorrect, as is the related expression found by applying the product rule to {{math|''d''(''mv'')/''dt''}}:<ref name="kleppner135">{{cite book|last1=Kleppner|last2=Kolenkow|title=An Introduction to Mechanics|page=135–39}}</ref>
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| :<math> F = m(t) \frac{dv}{dt} + v(t) \frac{dm}{dt}. \qquad \mathrm{(wrong)}</math>
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| This equation does not correctly describe the motion of variable-mass objects. The correct equation is
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| :<math> F = m(t) \frac{dv}{dt} - u \frac{dm}{dt},</math>
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| where {{math|''u''}} is the velocity of the ejected/accreted mass ''as seen in the object's rest frame''.<ref name="kleppner135" /> This is distinct from {{math|''v''}}, which is the velocity of the object itself as seen in an inertial frame.
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| This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass. When considered together, the object and the mass constitute a closed system in which total momentum is conserved.
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| ==Generalized coordinates==
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| {{See also|Analytical mechanics}}
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| Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by ''constraints''. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints can be incorporated by changing the normal [[Cartesian coordinates]] to a set of ''[[generalized coordinates]]'' that may be fewer in number.<ref>{{harvnb|Goldstein|1980|pp=11–13}}</ref> Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a ''generalized momentum'', also known as the ''canonical'' or ''conjugate momentum'', that extends the concepts of both linear momentum and [[angular momentum]]. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as ''mechanical'', ''kinetic'' or ''kinematic momentum''.<ref name=Goldstein54>{{harvnb|Goldstein|1980|pp=54–56}}</ref><ref>{{harvnb|Jackson|1975|p=574}}</ref><ref name=FeynmanQM>{{harvnb|Feynman Vol. 3|loc=Chapter 21-3}}</ref> The two main methods are described below.
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| ===Lagrangian mechanics===
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| In [[Lagrangian mechanics]], a Lagrangian is defined as the difference between the [[kinetic energy]] {{math|''T''}} and the [[potential energy]] {{math|''V''}}:
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| :<math> \mathcal{L} = T-V\,.</math>
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| If the generalized coordinates are represented as a vector {{math|'''q''' {{=}} (''q''<sub>1</sub>, ''q''<sub>2</sub>, ... , ''q''<sub>''N''</sub>) }} and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or [[Euler–Lagrange equation]]s) are a set of {{math|''N''}} equations:<ref>{{harvnb|Goldstein|1980|pp=20–21}}</ref>
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| :<math> \frac{d}{d t}\left(\frac{\partial \mathcal{L} }{\partial\dot{q}_j}\right) - \frac{\partial \mathcal{L}}{\partial q_j} = 0\,.</math>
| |
| If a coordinate {{math|''q''<sub>''i''</sub>}} is not a Cartesian coordinate, the associated generalized momentum component {{math|''p''<sub>''i''</sub>}} does not necessarily have the dimensions of linear momentum. Even if {{math|''q''<sub>i</sub>}} is a Cartesian coordinate, {{math|''p''<sub>''i''</sub>}} will not be the same as the mechanical momentum if the potential depends on velocity.<ref name=Goldstein54/> Some sources represent the kinematic momentum by the symbol {{math|'''Π'''}}.<ref name=Lerner>{{cite book|editor-last=Lerner|editor-first=Rita G.|title=Encyclopedia of physics|year=2005|publisher=Wiley-VCH-Verl.|location=Weinheim|isbn=978-3527405541|edition=3rd |editor2-last=Trigg |editor2-first=George L.}}</ref>
| |
| | |
| In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as
| |
| :<math> p_j = \frac{\partial \mathcal{L} }{\partial \dot{q}_j}\,.</math>
| |
| Each component {{math|''p''<sub>''j''</sub>}} is said to be the ''conjugate momentum'' for the coordinate {{math|''q''<sub>''j''</sub>}}.
| |
| | |
| Now if a given coordinate {{math|''q''<sub>''i''</sub>}} does not appear in the Lagrangian (although its time derivative might appear), then
| |
| :<math> p_j = \text{constant}\,.</math>
| |
| This is the generalization of the conservation of momentum.<ref name=Goldstein54/>
| |
| | |
| Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism.
| |
| | |
| ===Hamiltonian mechanics===
| |
| In [[Hamiltonian mechanics]], the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as
| |
| :<math> \mathcal{H}\left(\mathbf{q},\mathbf{p},t\right) = \mathbf{p}\cdot\dot{\mathbf{q}} - \mathcal{L}\left(\mathbf{q},\dot{\mathbf{q}},t\right)\,,</math>
| |
| where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are<ref>{{harvnb|Goldstein|1980|pp=341–342}}</ref>
| |
| :<math> \begin{align}
| |
| \dot{q}_i &= \frac{\partial\mathcal{H}}{\partial p_i}\\
| |
| -\dot{p}_i &= \frac{\partial\mathcal{H}}{\partial q_i}\\
| |
| -\frac{\partial \mathcal{L}}{\partial t} &= \frac{d \mathcal{H}}{d t}\,.
| |
| \end{align}</math>
| |
| As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.<ref>{{harvnb|Goldstein|1980|p=348}}</ref>
| |
| | |
| ===Symmetry and conservation===
| |
| Conservation of momentum is a mathematical consequence of the [[Homogeneity (physics)|homogeneity]] (shift [[symmetry]]) of space (position in space is the [[canonical conjugate]] quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of [[Noether's theorem]].<ref>{{cite book|last1=Hand|first1=Louis N. |last2=Finch |first2=Janet D. |title=Analytical mechanics|year=1998|publisher=Cambridge University Press|location=Cambridge, England|isbn=9780521575720|edition=7th print |pages=Chapter 4 |nopp=true}}</ref>
| |
| | |
| ==Relativistic mechanics==
| |
| {{See also|Mass in special relativity|Tests of relativistic energy and momentum}}
| |
| | |
| ===Lorentz invariance===
| |
| Newtonian physics assumes that [[absolute time and space]] exist outside of any observer; this gives rise to the [[Galilean invariance]] described earlier. It also results in a prediction that the [[speed of light]] can vary from one reference frame to another. This is contrary to observation. In the [[special theory of relativity]], Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light {{math|''c''}} is invariant. As a result, position and time in two reference frames are related by the [[Lorentz transformation]] instead of the [[Galilean transformation]].<ref name=RindlerCh2>{{harvnb|Rindler|1986|loc=Chapter 2}}</ref>
| |
| | |
| Consider, for example, a reference frame moving relative to another at velocity {{math|''v''}} in the {{math|''x''}} direction. The Galilean transformation gives the coordinates of the moving frame as
| |
| :<math>\begin{align}
| |
| t' &= t \\
| |
| x' &= x - v t
| |
| \end{align}</math>
| |
| while the Lorentz transformation gives<ref name=FeynmanCh15>{{harvnb|Feynman Vol. 1|loc=Chapter 15-2}}</ref>
| |
| :<math>\begin{align}
| |
| t' &= \gamma \left( t - \frac{v x}{c^2} \right) \\
| |
| x' &= \gamma \left( x - v t \right)\,,
| |
| \end{align}</math>
| |
| where {{math|''γ''}} is the [[Lorentz factor]]:
| |
| :<math>\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}\,.</math>
| |
| | |
| Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the ''inertial mass'' {{math|''m''}} of an object a function of velocity:
| |
| :<math>m = \gamma m_0\,;</math>
| |
| {{math|''m''<sub>0</sub>}} is the object's [[invariant mass]].<ref name=Rindler>{{harvnb|Rindler|1986|pp=77–81}}</ref>
| |
| | |
| The modified momentum,
| |
| :<math> \mathbf{p} = \gamma m_0 \mathbf{v}\,,</math>
| |
| obeys Newton's second law:
| |
| :<math> \mathbf{F} = \frac{d \mathbf{p}}{dt}\,.</math>
| |
| | |
| Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, {{math|''γm''<sub>0</sub>'''v'''}} is approximately equal to {{math|''m''<sub>0</sub>'''v'''}}, the Newtonian expression for momentum.
| |
| | |
| ===Four-vector formulation===
| |
| {{Main|Four-momentum}}
| |
| In the theory of relativity, physical quantities are expressed in terms of [[four-vector]]s that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example {{math|'''R'''}} for position. The expression for the ''four-momentum'' depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of [[proper time]], {{math|''τ''}}, defined by<ref>{{harvnb|Rindler|1986|p=66}}</ref>
| |
| :<math>c^2d\tau^2 = c^2dt^2-dx^2-dy^2-dz^2\,,</math>
| |
| is [[Invariant (physics)|invariant]] under Lorentz transformations (in this expression and in what follows the {{nowrap|(+ − − −)}} [[metric signature]] has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as [[Euclidean vector]]s and multiplying time by the [[Imaginary unit|square root of {{math|-1}}]]; or by keeping time a real quantity and embedding the vectors in a [[Minkowski space]].<ref>{{cite book|last=Misner|first=Charles W.|coauthors=Kip S. Thorne, John Archibald Wheeler |title=Gravitation|year=1973|publisher=W. H. Freeman|location=New York|isbn=9780716703440|page=51|others=24th printing.}}</ref> In a Minkowski space, the [[scalar product]] of two four-vectors {{math|'''U''' {{=}} (''U''<sub>0</sub>,''U''<sub>1</sub>,''U''<sub>2</sub>,''U''<sub>3</sub>)}} and {{math|'''V''' {{=}} (''V''<sub>0</sub>,''V''<sub>1</sub>,''V''<sub>2</sub>,''V''<sub>3</sub>)}} is defined as
| |
| :<math> \mathbf{U} \cdot \mathbf{V} = U_0 V_0 - U_1 V_1 - U_2 V_2 - U_3 V_3\,. </math>
| |
| | |
| In all the coordinate systems, the ([[Covariance and contravariance of vectors|contravariant]]) relativistic four-velocity is defined by
| |
| :<math> \mathbf{U} \equiv \frac{d \mathbf{R}}{d \tau} = \gamma \frac{d \mathbf{R}}{dt}\,,</math>
| |
| and the (contravariant) [[four-momentum]] is
| |
| :<math>\mathbf{P} = m_0\mathbf{U}\,,</math>
| |
| where {{math|m<sub>0</sub>}} is the invariant mass. If {{math|'''R''' {{=}} (''ct,x,y,z'')}} (in Minkowski space), then<ref group="note">Here the time coordinate comes first. Several sources put the time coordinate at the end of the vector.</ref>
| |
| :<math>\mathbf{P} = \gamma m_0 \left(c,\mathbf{v}\right) = (m c,\mathbf{p})\,.</math>
| |
| Using Einstein's [[mass-energy equivalence]], {{math|''E'' {{=}} ''mc''<sup>2</sup>}}, this can be rewritten as
| |
| :<math>\mathbf{P} = \left(\frac{E}{c}, \mathbf{p}\right)\,.</math>
| |
| Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.
| |
| | |
| The magnitude of the momentum four-vector is equal to {{math|''m''<sub>0</sub>''c''}}:
| |
| :<math>||\mathbf{P}||^2 = \mathbf{P}\cdot\mathbf{P} = \gamma^2m_0^2(c^2-v^2) = (m_0c)^2\,,</math>
| |
| and is invariant across all reference frames.
| |
| | |
| The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting {{math|''m''<sub>0</sub> {{=}} 0}} it follows that
| |
| :<math>E = pc\,.</math>
| |
| | |
| In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.<ref>{{harvnb|Rindler|1986|pp=86–87}}</ref>
| |
| | |
| ==Classical electromagnetism==
| |
| In Newtonian mechanics, the law of conservation of momentum can be derived from the [[law of action and reaction]], which states that the forces between two particles are equal and opposite. Electromagnetic forces violate this law. Under some circumstances one moving charged particle can exert a force on another without any return force.<ref>{{harvnb|Goldstein|1980|pp=7–8}}</ref> Moreover, [[Maxwell's equations]], the foundation of classical electrodynamics, are Lorentz-invariant. However, momentum is still conserved.
| |
| | |
| ===Vacuum===
| |
| In Maxwell's equations, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (''[[Lorentz force]]'') on a particle with charge {{math|''q''}} due to a combination of [[electric field]] {{math|'''E'''}} and [[magnetic field]] (as given by the "B-field" {{math|'''B'''}}) is
| |
| :<math>\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}).</math>
| |
| This force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.<ref name="Jackson238" />
| |
| | |
| In a vacuum, the momentum per unit volume is
| |
| :<math> \mathbf{g} = \frac{1}{\mu_0 c^2}\mathbf{E}\times\mathbf{B}\,,</math>
| |
| where {{math|μ<sub>0</sub>}} is the [[vacuum permeability]] and {{math|''c''}} is the [[speed of light]]. The momentum density is proportional to the [[Poynting vector]] {{math|'''S'''}} which gives the directional rate of energy transfer per unit area:<ref name="Jackson238" /><ref name=FeynmanCh27>{{harvnb|Feynman Vol. 1|loc=Chapter 27-6}}</ref>
| |
| :<math> \mathbf{g} = \frac{\mathbf{S}}{c^2}\,.</math>
| |
| | |
| If momentum is to be conserved in a volume {{math|''V''}}, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If {{math|'''P'''<sub>mech</sub>}} is the momentum of all the particles in a volume {{math|''V''}}, and the particles are treated as a continuum, then Newton's second law gives
| |
| :<math> \frac{d \mathbf{P}_\text{mech}}{d t} = \int_V \left(\rho\mathbf{E} + \mathbf{J}\times\mathbf{B}\right) dV\,.</math>
| |
| The electromagnetic momentum is
| |
| :<math> \mathbf{P}_\text{field} = \frac{1}{\mu_0c^2} \int_V \mathbf{E}\times\mathbf{B} dV\,,</math>
| |
| and the equation for conservation of each component {{math|''i''}} of the momentum is
| |
| :<math> \frac{d}{d t}\left(\mathbf{P}_\text{mech}+ \mathbf{P}_\text{field} \right)_i = \oint_S \sum_j T_{ij} n_j dS\,.</math>
| |
| The term on the right is an integral over the surface {{math|''S''}} representing momentum flow into and out of the volume, and {{math|''n''<sub>j</sub>}} is a component of the surface normal of {{math|''S''}}. The quantity {{math|''T''<sub>''i j''</sub>}} is called the [[Maxwell stress tensor]], defined as
| |
| :<math>T_{i j} \equiv \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0} \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)\,.</math><ref name=Jackson238>{{harvnb|Jackson|1975|pp=238–241}} Expressions, given in [[Gaussian units]] in the text, were converted to SI units using Table 3 in the Appendix.</ref>
| |
| | |
| ===Media===
| |
| The above results are for the ''microscopic'' Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density is modified to
| |
| :<math> \mathbf{g} = \frac{1}{c^2}\mathbf{E}\times\mathbf{H} = \frac{\mathbf{S}}{c^2}\,,</math>
| |
| where the H-field {{math|'''H'''}} is related to the B-field and the [[magnetization]] {{math|'''M'''}} by
| |
| :<math> \mathbf{B} = \mu_0 \left(\mathbf{H} + \mathbf{M}\right)\,.</math>
| |
| The electromagnetic stress tensor depends on the properties of the media.<ref name=Jackson238/>
| |
| | |
| ===Particle in field===
| |
| | |
| If a charged particle {{math|''q''}} moves in an electromagnetic field, its kinematic momentum {{math|''m'' '''v'''}} is not conserved. However, it has a canonical momentum that is conserved.
| |
| | |
| ====Lagrangian and Hamiltonian formulation====<!--this section links to [[Lorentz force#Lorentz force and analytical mechanics]]-->
| |
| | |
| The ''kinetic momentum'' '''p''' is different from the ''[[canonical momentum]]'' '''P''' (synonymous with the [[generalized momentum]]) conjugate to the ordinary position coordinates '''r''', because '''P''' includes a contribution from the [[electric potential]] φ('''r''', ''t'') and [[vector potential]] '''A'''('''r''', ''t''):<ref name=Lerner>{{cite book|editor-last=Lerner|editor-first=Rita G.|title=Encyclopedia of physics|year=2005|publisher=Wiley-VCH-Verl.|location=Weinheim|isbn=978-3527405541|edition=3rd, completely revised. and enlarged |coauthors=Trigg, George L.}}</ref>
| |
| | |
| :{| class="wikitable"
| |
| |-
| |
| !
| |
| ! [[Classical mechanics]]
| |
| ! [[Relativistic mechanics]]
| |
| |-
| |
| ! [[Lagrangian]]
| |
| | <math>\mathcal{L}=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+e\mathbf{A}\cdot\mathbf{\dot{r}}-e\phi\,\!</math>
| |
| | <math>\mathcal{L} = -mc^2\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2} + e \mathbf{A}\cdot\dot{\mathbf{r}} - e \phi \,\!</math>
| |
| |-
| |
| ! ''Canonical'' momentum
| |
| | |
| <math>\bold{P} = \frac{\partial L}{\partial \dot{\mathbf{r}} }</math>
| |
| | <math>\mathbf{P} = m\mathbf{v} + e\mathbf{A} </math>
| |
| | <math> \mathbf{P} = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} + e\mathbf{A} \,\!</math>
| |
| |-
| |
| ! ''Kinetic'' momentum
| |
| | |
| <math>\bold{p} = m\mathbf{\dot{r}}</math>
| |
| | <math>m\mathbf{v} = \mathbf{P} - e\mathbf{A} </math>
| |
| | <math> \mathbf{P} - e\mathbf{A} = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} \,\!</math>
| |
| |-
| |
| ! [[Hamiltonian mechanics|Hamiltonian]]
| |
| |<math>\begin{align}
| |
| \mathcal{H} & = T + V \\
| |
| & = \frac{\mathbf{p}^2}{2m} + V \\
| |
| & = \frac{(\mathbf{P}-e\mathbf{A})^2}{2m} + e\phi
| |
| \end{align}</math>
| |
| |<math>\begin{align}
| |
| \mathcal{H} & = \mathbf{P}\cdot\dot{\mathbf{r}} - L \\
| |
| & = {mc^2\over \sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} + e \phi \\
| |
| & = \sqrt{c^2(\mathbf{P} -e\mathbf{A})^2 + (mc^2)^2} + e \phi
| |
| \end{align}</math>
| |
| |}
| |
| | |
| where '''ṙ''' = '''v''' is the velocity (see [[time derivative]]) and ''e'' is the [[electric charge]] of the particle. See also [[momentum#Electromagnetism|Electromagnetism (momentum)]]. If neither φ nor '''A''' depends on position, '''P''' is conserved.<ref name=Goldstein54/>
| |
| | |
| The classical [[Hamiltonian mechanics|Hamiltonian]] <math>\mathcal{H}</math> for a particle in any field equals the total energy of the system - the [[kinetic energy]] ''T'' = '''p'''<sup>2</sup>/2''m'' (where '''p'''<sup>2</sup> = '''p·p''', see [[dot product]]) plus the [[potential energy]] ''V''. For a particle in an [[electromagnetic field]], the potential energy is ''V'' = ''eφ'', and since the kinetic energy ''T'' always corresponds to the kinetic momentum '''p''', replacing the kinetic momentum by the above equation ('''p''' = '''P''' − ''e'''''A''') leads to the Hamiltonian in the table.
| |
| | |
| These Lagrangian and Hamiltonian expressons can derive the [[Lorentz force#Lorentz force and analytical mechanics|Lorentz force]].
| |
| | |
| ====Canonical commutation relations====
| |
| | |
| The kinetic momentum ('''p''' above) satisfies the [[Commutator|commutation relation]]:<ref name=Lerner/>
| |
| | |
| :<math>\left [ p_j , p_k \right ] = \frac{i\hbar e}{c} \epsilon_{jk\ell } B_\ell</math>
| |
| | |
| where: ''j, k, {{ell}}'' are indices labelling vector components, ''B<sub>{{ell}}</sub>'' is a component of the [[magnetic field]], and ''ε<sub>kj{{ell}}</sub>'' is the [[Levi-Civita symbol]], here in 3-dimensions.
| |
| | |
| ==Quantum mechanics==
| |
| {{further2|[[Momentum operator]]}}
| |
| In [[quantum mechanics]], momentum is defined as an [[operator (physics)|operator]] on the [[wave function]]. The [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are [[conjugate variables]].
| |
| | |
| For a single particle described in the position basis the momentum operator can be written as
| |
| :<math>\mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla\,,</math>
| |
| | |
| where ∇ is the [[gradient]] operator, ''ħ'' is the [[reduced Planck constant]], and ''i'' is the [[imaginary unit]]. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in [[momentum space]] the momentum operator is represented as
| |
| :<math>\mathbf{p}\psi(p) = p\psi(p)\,,</math>
| |
| | |
| where the operator '''p''' acting on a wave function ''ψ''(''p'') yields that wave function multiplied by the value ''p'', in an analogous fashion to the way that the position operator acting on a wave function ''ψ''(''x'') yields that wave function multiplied by the value ''x''.
| |
| | |
| For both massive and massless objects, relativistic momentum is related to the [[Matter wave|de Broglie wavelength]] {{math|''λ''}} by
| |
| :<math> p = h/\lambda.\,</math>
| |
| [[Electromagnetic radiation]] (including [[light|visible light]], [[ultraviolet]] light, and [[radio waves]]) is carried by [[photons]]. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the [[solar sail]]. The calculation of the momentum of light within [[dielectric]] media is somewhat controversial (see [[Abraham–Minkowski controversy]]).<ref>{{cite journal|last=Barnett|first=Stephen M.|title=Resolution of the Abraham-Minkowski Dilemma|journal=Physical Review Letters|year=2010|volume=104|issue=7|doi=10.1103/PhysRevLett.104.070401|bibcode = 2010PhRvL.104g0401B|pmid=20366861}}</ref>
| |
| | |
| ==Deformable bodies and fluids==
| |
| | |
| ===Conservation in a continuum===
| |
| {{Main|Cauchy momentum equation}}
| |
| [[File:Equation motion body.svg|right|thumb|Motion of a material body]]
| |
| In fields such as [[fluid dynamics]] and [[solid mechanics]], it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a [[Continuum mechanics|continuum]] in which there is a particle or [[fluid parcel]] at each point that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density {{math|''ρ''}} and velocity {{math|'''v'''}} that depend on time {{math|''t''}} and position {{math|'''r'''}}. The momentum per unit volume is {{math|''ρ'''''v'''}}.<ref>{{harvnb|Tritton|2006|pages=48–51}}</ref>
| |
| | |
| Consider a column of water in [[hydrostatic equilibrium]]. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is {{math|''ρ'''''g'''}}, where {{math|'''g'''}} is the [[gravitational acceleration]]. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the [[pressure]] {{math|''p''}}. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is<ref name=FeynmanCh40>{{harvnb|Feynman Vol. 2|loc=Chapter 40}}</ref>
| |
| :<math>-\nabla p +\rho \mathbf{g} = 0\,.</math>
| |
| | |
| If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative {{math|''∂'''''v'''/''∂t''}} because the fluid in a given volume changes with time. Instead, the [[material derivative]] is needed:<ref>{{harvnb|Tritton|2006|pages=54}}</ref>
| |
| :<math>\frac{D}{Dt} \equiv \frac{\partial}{\partial t} + \mathbf{v}\cdot\boldsymbol{\nabla}\,.</math>
| |
| Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes dues to [[advection]] as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to {{math|''ρD'''''v'''/''Dt''}}. This is equal to the net force on the droplet.
| |
| | |
| Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a [[shear stress]] {{math|''τ''}}, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or [[strain rate]]. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the {{math|''x''}} direction varies with {{math|''z''}}, the tangential force in direction {{math|''x''}} per unit area normal to the {{math|''z''}} direction is
| |
| :<math>\sigma_\text{zx} = -\mu\frac{\partial v_\text{x}}{\partial z}\,,</math>
| |
| where {{math|''μ''}} is the [[viscosity]]. This is also a [[flux]], or flow per unit area, of x-momentum through the surface.<ref>{{cite book|last=Bird|first=R. Byron|coauthors=Warren Stewart; Edwin N. Lightfoot |title=Transport phenomena|year=2007|publisher=Wiley|location=New York|isbn=9780470115398|page=13|edition=2nd revised}}</ref>
| |
| | |
| Including the effect of viscosity, the momentum balance equations for the [[incompressible flow]] of a [[Newtonian fluid]] are
| |
| :<math>\rho \frac{D \mathbf{v}}{D t} = -\boldsymbol{\nabla} p + \mu\nabla^2 \mathbf{v} + \rho\mathbf{g}.\,</math>
| |
| These are known as the [[Navier–Stokes equations]].<ref>{{harvnb|Tritton|2006|p=58}}</ref>
| |
| | |
| The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction {{math|''i''}} and force in direction {{math|''j''}}, there is a stress component {{math|''σ''<sub>''ij''</sub>}}. The nine components make up the [[Cauchy stress tensor]] {{math|'''σ'''}}, which includes both pressure and shear. The local conservation of momentum is expressed by the [[Cauchy momentum equation]]:
| |
| :<math>\rho \frac{D \mathbf{v}}{D t} = \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{f}\,,</math>
| |
| where {{math|'''f'''}} is the [[body force]].<ref>{{cite book
| |
| | last = Acheson
| |
| | first = D. J.
| |
| | title = Elementary Fluid Dynamics
| |
| | publisher = [[Oxford University Press]]
| |
| | year = 1990
| |
| |page = 205
| |
| | isbn = 0-19-859679-0}}</ref>
| |
| | |
| The Cauchy momentum equation is broadly applicable to [[Deformation (mechanics)|deformations]] of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see [[Viscosity#Types of viscosity|Types of viscosity]]).
| |
| | |
| ===Acoustic waves===
| |
| | |
| A disturbance in a medium gives rise to oscillations, or [[wave]]s, that propagate away from their source. In a fluid, small changes in pressure {{math|''p''}} can often be described by the [[acoustic wave equation]]:
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| :<math>\frac{\partial^2 p}{\partial t^2} = c^2 \nabla^2 p\,,</math>
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| where {{math|''c''}} is the [[speed of sound]]. In a solid, similar equations can be obtained for propagation of pressure ([[P-wave]]s) and shear ([[S-waves]]).<ref>{{cite book|last=Gubbins|first=David|title=Seismology and plate tectonics|year=1992|publisher=Cambridge University Press|location=Cambridge [England]|isbn=0521379954|page=59|edition=Repr. (with corr.)}}</ref>
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| The flux, or transport per unit area, of a momentum component {{math|''ρv<sub>j</sub>''}} by a velocity {{math|''v<sub>i</sub>''}} is equal to {{math|''ρ v<sub>j</sub>v<sub>j</sub>''}}. In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero. However, nonlinear effects can give rise to a nonzero average.<ref>{{cite book|last=LeBlond|first=Paul H.|title=Waves in the ocean|year=1980|publisher=Elsevier|location=Amsterdam [u.a.]|isbn=9780444419262|page=258|edition=2. impr.|coauthors=Mysak, Lawrence A.}}</ref> It is possible for momentum flux to occur even though the wave itself does not have a mean momentum.<ref>{{cite journal|last=McIntyre|first=M. E.|title=On the 'wave momentum' myth|journal=J. Fluid. Mech|year=1981|volume=106|pages=331–347}}</ref>
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| ==History of the concept==
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| {{see also|Theory of impetus}}
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| In about 530 A.D., working in Alexandria, Byzantine philosopher [[John Philoponus]] developed a concept of momentum in his commentary to [[Aristotle]]'s ''Physics''.
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| Aristotle claimed that everything that is moving must be kept moving by something. For example, a thrown ball must be kept moving by motions of the air. Most writers continued to accept Aristotle's theory until the time of Galileo, but a few were skeptical. Philoponus pointed out the absurdity in Aristotle's claim that motion of an object is promoted by the same air that is resisting its passage. He proposed instead that an impetus was imparted to the object in the act of throwing it.<ref>{{cite encyclopedia |url=http://plato.stanford.edu/entries/philoponus/#2.1 |encyclopedia=Standford Encyclopedia of Philosophy |title=John Philoponus |date=8 June 2007 |accessdate=26 July 2012}}</ref> Ibn Sīnā (also known by his Latinized name [[Avicenna]]) read Philoponus and published his own theory of motion in ''The Book of Healing'' in 1020. He agreed that an impetus is imparted to a projectile by the thrower; but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as a persistent, requiring external forces such as [[air resistance]] to dissipate it.<ref name=Espinoza>{{cite journal | last1 = Espinoza | first1 = Fernando | year = 2005 | title = An analysis of the historical development of ideas about motion and its implications for teaching | url = | journal = Physics Education | volume = 40 | issue = 2| page = 141 | doi=10.1088/0031-9120/40/2/002}}</ref><ref name=Nasr>{{Cite book |title=The Islamic intellectual tradition in Persia |author=[[Seyyed Hossein Nasr]] & Mehdi Amin Razavi |publisher=[[Routledge]] |year=1996 |isbn=0-7007-0314-4 |page=72}}</ref><ref name=Sayili>{{cite journal
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| |doi=10.1111/j.1749-6632.1987.tb37219.x
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| |author=[[Aydin Sayili]]
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| |year=1987
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| |title=Ibn Sīnā and Buridan on the Motion of the Projectile
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| |journal=Annals of the New York Academy of Sciences
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| |volume=500
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| |issue=1
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| |pages=477–482
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| |quote=
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| |bibcode=1987NYASA.500..477S}}</ref>
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| The work of Philoponus, and possibly that of Ibn Sīnā,<ref name=Sayili/> was read and refined by the European philosophers [[Peter Olivi]] and [[Jean Buridan]]. Buridan, who in about 1350 was made rector of the University of Paris, referred to [[Theory of impetus|impetus]] being proportional to the weight times the speed. Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.<ref>{{cite encyclopedia |encyclopedia=Medieval Science, Technology and Medicine:an Encyclopedia |page=107 |title=Buridian, John |author=T.F. Glick, S.J. Livesay, F. Wallis}}</ref><ref name=Park>{{cite book|last=Park|first=David|title=The how and the why : an essay on the origins and development of physical theory|year=1990|publisher=Princeton University Press|location=Princeton, N.J.|isbn=9780691025087|pages=139–141|edition=3rd print |others= With drawings by Robin Brickman}}</ref>
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| [[René Descartes]] believed that the total "quantity of motion" in the universe is conserved, where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more importantly he believed that it is speed rather than velocity that is conserved. So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion.<ref>{{Cite book
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| |author=Daniel Garber
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| |editor=John Cottingham
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| |chapter=Descartes' Physics
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| |pages=310–319
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| |title=The Cambridge Companion to Descartes
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| |year=1992
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| |place=Cambridge
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| |publisher=Cambridge University Press
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| |isbn=0-521-36696-8
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| |url=}}</ref><ref>{{cite book|last=Rothman|first=Milton A.|title=Discovering the natural laws : the experimental basis of physics|year=1989|publisher=Dover Publications|location=New York|isbn=9780486261782|pages=83–88|edition=2nd}}</ref> [[Galileo]], later, in his ''[[Two New Sciences]]'', used the [[Italian language|Italian]] word ''impeto''.
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| [[Leibniz]], in his "[[Discourse on Metaphysics]]", gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances. He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.<ref>{{Cite book
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| |author=G. W. Leibniz
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| |editor=Roger Ariew and Daniel Garber
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| |chapter=Discourse on Metaphysics
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| |pages=49-51
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| |title=Philosophical Essays
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| |year=1989
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| |place=Indianapolis, IN
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| |publisher=Hackett Publishing Company, Inc.
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| |isbn=0-87220-062-0
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| |url=}}</ref>
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| The first correct statement of the law of conservation of momentum was by English mathematician [[John Wallis]] in his 1670 work, ''Mechanica sive De Motu, Tractatus Geometricus'': "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result".<ref>{{cite book |first=J.F. |last=Scott |title=The Mathematical Work of John Wallis, D.D., F.R.S. |publisher=Chelsea Publishing Company |year=1981 |isbn=0-8284-0314-7 |page=111}}</ref> Wallis uses ''momentum'' and ''vis'' for force. Newton's ''[[Philosophiæ Naturalis Principia Mathematica]]'', when it was first published in 1687, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines ''quantitas motus'', "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum.<ref>{{cite book |first=Ernst |last=Grimsehl |coauthor=Leonard Ary Woodward, Translator |title=A Textbook of Physics |publisher=Blackie & Son limited |year=1932 |location=London, Glasgow |page=78}}</ref> Thus when in Law II he refers to ''mutatio motus'', "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion.<ref>{{cite book |first=Aldo |last=Rescigno |title=Foundation of Pharmacokinetics |year=2003 |isbn=0306477041 |location=New York |publisher=Kluwer Academic/Plenum Publishers |page=19}}</ref> It remained only to assign a standard term to the quantity of motion. The first use of "momentum" in its proper mathematical sense is not clear but by the time of Jenning's ''Miscellanea'' in 1721, four years before the final edition of Newton's ''Principia Mathematica'', momentum M or "quantity of motion" was being defined for students as "a rectangle", the product of Q and V, where Q is "quantity of material" and V is "velocity", s/t.<ref>{{cite book |first=John |last=Jennings |authorlink=John Jennings (tutor) |title=Miscellanea in Usum Juventutis Academicae |publisher=R. Aikes & G. Dicey |year=1721 |location=Northampton |page=67}}</ref>
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| ==See also==
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| {{div col}}
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| * [[Crystal momentum]]
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| * [[Galilean cannon]]
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| * [[Momentum transfer]]
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| * [[Planck momentum]]
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| {{div col end}}
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| ==Notes==
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| {{Reflist|group=note}}
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| ==References==
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| {{Reflist|2}}
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| ==Further reading==
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| {{Refbegin}}
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| * {{cite book|last=[[David Halliday (physicist)|Halliday]]|first=David|coauthors=[[Robert Resnick]]|date=1960-2007|title=Fundamentals of Physics|publisher=John Wiley & Sons|pages=Chapter 9|nopp=true}}
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| *{{cite book|last=Dugas|first=René|title=A history of mechanics|year=1988|publisher=Dover Publications|location=New York|isbn=9780486656328|edition=Dover |others= Translated into English by J.R. Maddox |ref=harv}}
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| *{{cite book|last=Feynman|first=Richard P.|title=The Feynman lectures on physics, Volume 1: Mainly Mechanics, Radiation, and Heat |year=2005|first2= Robert B. |last2=Leighton |first3=Matthew |last3=Sands |publisher=Pearson Addison-Wesley|location=San Francisco, Calif. |isbn=978-0805390469|edition=Definitive |ref={{harvid|Feynman Vol. 1}} }}
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| * {{cite book|last=Feynman|first=Richard P.|first2= Robert B. |last2=Leighton |first3=Matthew |last3=Sands |title=The Feynman lectures on physics, Volume III: Quantum Mechanics |year=2005|publisher=BasicBooks|location=New York|isbn=978-0805390490 |edition=Definitive |ref={{harvid|Feynman Vol. 3}} }}
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| *{{cite book|last=Goldstein|first=Herbert|title=Classical mechanics|year=1980|publisher=Addison-Wesley Pub. Co.|location=Reading, Mass.|isbn=0201029189|edition=2d |ref=harv}}
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| * {{cite book|last1=Hand|first1=Louis N.|last2=Finch|first2=Janet D.|title=Analytical Mechanics|publisher=Cambridge University Press|pages=Chapter 4|nopp=true}}
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| *{{cite book|last=Jackson|first=John David|title=Classical electrodynamics|year=1975|publisher=Wiley|location=New York|pages= |isbn=047143132X|edition=2d|ref=harv}}
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| *{{cite book|last=Jammer|first=Max|title=Concepts of force : a study in the foundations of dynamics|year=1999|publisher=Dover Publications|location=Mineola, N.Y.|isbn=9780486406893|edition=Facsim.}}
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| *{{cite book|last=Landau|first=L.D.|title=The classical theory of fields|year=2000|publisher=Butterworth Heinemann|location=Oxford|isbn=9780750627689 |coauthors=E.M. Lifshitz |others=4th rev. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh|ref=harv}}
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| *{{cite book|last=Rindler|first=Wolfgang|title=Essential Relativity : Special, general and cosmological|year=1986|publisher=Springer|location=New York u.a.|isbn=0387100903|edition=Rev. 2. |ref=harv}}
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| * Serway, Raymond; Jewett, John (2003). ''Physics for Scientists and Engineers'' (6 ed.). Brooks Cole. ISBN 0-534-40842-7
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| * Stenger, Victor J. (2000). ''Timeless Reality: Symmetry, Simplicity, and Multiple Universes''. Prometheus Books. Chpt. 12 in particular.
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| * Tipler, Paul (1998). ''Physics for Scientists and Engineers: Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics'' (4th ed.). W. H. Freeman. ISBN 1-57259-492-6
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| *{{cite book|last=Tritton|first=D.J.|title=Physical fluid dynamics|year=2006|publisher=Claredon Press|location=Oxford|isbn=0198544936|page=58|edition=2nd. |ref=harv}}
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| {{Refend}}
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| ==External links==
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| {{Wiktionary|momentum}}
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| * [http://www.lightandmatter.com/html_books/lm/ch14/ch14.html Conservation of momentum] – A chapter from an online textbook
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| * [http://www.youtube.com/watch?v=4B_x1XjBhiI MiuntePhysics - Another Physics Misconception] - MinutePhysics explaining a common misconception of momentum
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| [[Category:Natural philosophy]]
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| [[Category:Physical quantities]]
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| [[Category:Mechanics]]
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| [[Category:Concepts in physics]]
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| [[Category:Conservation laws]]
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| [[Category:Continuum mechanics]]
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| [[Category:Mass]]
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| [[Category:Velocity]]
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