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Mōmentum was not merely the motion, which was mōtus, but was the power

residing in a moving object, captured by today's mathematical definitions. A mōtus,

"movement", was a stage in any sort of change,[1] while velocitas, "swiftness",

captured only speed. The concept of momentum in classical mechanics was

originated by a number of great thinkers and experimentalists. The first of these

was Byzantine philosopher John Philoponus, in his commentary to Aristotle´sPhysics. As regards the natural motion of bodies falling through a medium,

Aristotle's verdict that the speed is proportional to the weight of the moving bodies

and indirectly proportional to the density of the medium is disproved by Philoponus

through appeal to the same kind of experiment that Galileo was to carry out

centuries later.[2] This idea was refined by the European philosophers Peter Olivi  and

 Jean Buridan. Buridan referred to impetus being proportional to the weight times

the speed.[3][4] Moreover, Buridan´s theory was different to 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.[5] 

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.[6] Galileo, later, in

his Two New Sciences, used the Italian word "impeto."

 The question has been much debated as to what Isaac Newton contributed to the

concept. The answer is apparently nothing, except to state more fully and with

better mathematics what was already known. Yet for scientists, this was the death

knell for Aristotelian physics and supported other progressive scientific theories

(i.e., Kepler's laws of planetary motion). Conceptually, the first and second of 

Newton's Laws of Motion had already been stated by 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".[7] 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[8] defines quantitas motus, "quantity of motion", as "arising from thevelocity and quantity of matter conjointly", which identifies it as momentum.[9] 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.[10] 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

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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.[11] 

Some languages, such as French still lack a single term for momentum, and use a

phrase such as the literal translation of "quantity of motion".

[edit] Linear momentum of a particle

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.

If an object is moving in any reference frame, then it has momentum in that frame.

It is important to note that momentum is frame dependent. That is, the same object

may have a certain momentum in one frame of reference, but a different amount in

another frame. For example, a moving object has momentum in a reference frame

fixed to a spot on the ground, while at the same time having 0 momentum in areference frame attached to the object's center of mass.

 The amount of momentum that an object has depends on two physical quantities:

the mass and the velocity of the moving object in the frame of reference. In physics,

the usual symbol for momentum is a bold p (bold because it is a vector); so this can

be written

where p is the momentum, m is the mass and v is the velocity.

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. To thedummy pilot in the cockpit it has a velocity and momentum of zero.

According to Newton's second law, the rate of change of the momentum of a

particle is proportional to the resultant force acting on the particle and is in the

direction of that force. The derivation of force from momentum is given below,

however because mass is constant the second term of the derivative is 0 so it is

ignored.

(if mass is constant)

or just simply

where F is understood to be the resultant.

Example: a model airplane of 1 kg accelerates from rest to a velocity of 1 m/s due

north in 1 s. The thrust required to produce this acceleration is 1 newton. The

change in momentum is 1 kg m/s. To the dummy pilot in the cockpit there is no

change of momentum. Its pressing backward in the seat is a reaction to the

unbalanced thrust, shortly to be balanced by the drag.

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Impulse and Momentum

Momentum

•

 The momentum of a body is equal to its mass multiplied by its velocity.

Momentum is measured in N s. Note that momentum is a vector quantity, in other words thedirection is important.

Impulse

The impulse of a force (also measured in N s) is equal to the change in momentum of a bodywhich a force causes. This is also equal to the magnitude of the force multiplied by the length of 

time the force is applied.

•

Impulse = change in momentum = force × time

Conservation of Momentum

When there is a collision between two objects, Newton's Third Law states that the force on one

of the bodies is equal and opposite to the force on the other body.

Therefore, if no other forces act on the bodies (in the direction of collision), then the total

momentum of the two bodies will be unchanged. Hence the total momentum before collision in a particular direction = total momentum after in a particular direction.

This can be used to solve problems involving colliding spheres (see also: restitution).

Example

We have the following scenario (a ball of mass 3kg is moving to the right with velocity 3m/s and

a ball of mass 1kg is moving to the left with velocity 2m/s):

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Suppose we are told that after the collision, the ball of mass 1kg moves away with velocity 2m/s,

then we can use the principle of conservation of momentum to determine the velocity of the

other ball after the collision.

Initial momentum = 3.3 - 2.1 = 7 [the minus sign is important: it is there because the velocity of 

the 1kg body is in the opposite direction to the velocity of the 3kg body].

Final momentum = 2 - 3x

Hence 7 = 2 - 3x (since momentum is conserved)

x = -5/3

Which means that my arrow was pointing in the wrong direction (because of the minus sign),

hence the velocity of the 3kg body after the collision is 5/3 ms-1 to the right.

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History of the concept

Mōmentum was not merely the motion, which was mōtus, but was the power residing in amoving object, captured by today's mathematical definitions. A mōtus, "movement", was a stage

in any sort of change,[1] while velocitas, "swiftness", captured only speed. The concept of 

momentum in classical mechanics was originated by a number of great thinkers and

experimentalists. The first of these was Byzantine philosopher John Philoponus, in his

commentary to Aristotle´s Physics. As regards the natural motion of bodies falling through amedium, Aristotle's verdict that the speed is proportional to the weight of the moving bodies and

indirectly proportional to the density of the medium is disproved by Philoponus through appealto the same kind of experiment that Galileo was to carry out centuries later.[2] This idea was

refined by the European philosophers Peter Olivi and Jean Buridan. Buridan referred to impetus 

 being proportional to the weight times the speed.[3][4] Moreover, Buridan´s theory was different tohis 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.[5]

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 asa 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 isconserved. 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.[6] Galileo, later, in

his Two New Sciences, used the Italian word "impeto."

The question has been much debated as to what Isaac Newton contributed to the concept. Theanswer is apparently nothing, except to state more fully and with better mathematics what was

already known. Yet for scientists, this was the death knell for Aristotelian physics and supported

other progressive scientific theories (i.e., Kepler's laws of planetary motion). Conceptually, the

first and second of  Newton's Laws of Motion had already been stated by John Wallis in his 1670work, 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".[7] Wallis uses momentum and vis for force. Newton's Philosophiæ Naturalis PrincipiaMathematica, when it was first published in 1687, showed a similar casting around for words to

use for the mathematical momentum. His Definition II[8] defines quantitas motus, "quantity of 

motion", as "arising from the velocity and quantity of matter conjointly", which identifies it asmomentum.[9] Thus when in Law II he refers to mutatio motus, "change of motion", being

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 proportional to the force impressed, he is generally taken to mean momentum and not motion.[10]

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'sMiscellanea 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 .[11]

Some languages, such as French still lack a single term for momentum, and use a phrase such asthe literal translation of "quantity of motion".

[edit] Linear momentum of a particle

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.

If an object is moving in any reference frame, then it has momentum in that frame. It is importantto note that momentum is frame dependent. That is, the same object may have a certain

momentum in one frame of reference, but a different amount in another frame. For example, a

moving object has momentum in a reference frame fixed to a spot on the ground, while at the

same time having 0 momentum in a reference frame attached to the object's center of mass.

The amount of momentum that an object has depends on two physical quantities: the mass and

the velocity of the moving object in the frame of reference. In physics, the usual symbol for 

momentum is a bold p (bold because it is a vector ); so this can be written

where p is the momentum, m is the mass and v is the velocity.

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. To the dummy pilot in the cockpitit has a velocity and momentum of zero.

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According to Newton's second law, the rate of change of the momentum of a particle is

 proportional to the resultant force acting on the particle and is in the direction of that force. The

derivation of force from momentum is given below, however because mass is constant thesecond term of the derivative is 0 so it is ignored.

(if mass is constant)

or just simply

where F is understood to be the resultant.

Example: a model airplane of 1 kg accelerates from rest to a velocity of 1 m/s due north in 1 s.The thrust required to produce this acceleration is 1 newton. The change in momentum is 1 kg

m/s. To the dummy pilot in the cockpit there is no change of momentum. Its pressing backward

in the seat is a reaction to the unbalanced thrust, shortly to be balanced by the drag.

[edit] Linear momentum of a system of particles

[edit] Relating to mass and velocity

The linear momentum of a system of particles is the vector sum of the momenta of all the

individual objects in the system:

where P is the total momentum of the particle system, mi and vi are the mass and the velocity

vector of the i-th object, and n is the number of objects in the system.

It can be shown that, in the center of mass frame the momentum of a system is zero.

Additionally, the momentum in a frame of reference that is moving at a velocity vcm with respectto that frame is simply:

where:

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.

This is known as Euler's first law.[12][13]

[edit] Relating to force - General equations of motion

Motion of a material body

The linear momentum of a system of particles can also be defined as the product of the total

mass of the system times the velocity of the center of mass

This is a special case of   Newton's second law. (If mass is constant)

For a more general derivation using tensors, we consider a moving body (see Figure), assumed

as a continuum, occupying a volume at a time , having a surface area , with defined traction

or surface forces per unit area represented by the stress vector  acting on every point of 

every body surface (external and internal), body forces  per unit of volume on every point

within the volume , and a velocity field prescribed throughout the body. Following the previous equation, the linear momentum of the system is:

By definition the stress vector is defined as , then

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Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives

(we denote as the differential operator):

 Now we only need to take care of the right side of the equation. We have to be careful, since we

cannot just take the differential operator under the integral. This is because while the motion of the continuum body is taking place (the body is not necessarily solid), the volume we are

integrating on can change with time too. So the above integral will be:

Performing the differentiation in the first part, and applying the divergence theorem on thesecond part we obtain:

  Now the second term inside the integral is: . Pluggingthis into the previous equation, and rearranging the terms, we get:

We can easily recognize the two integral terms in the above equation. The first integral contains

the Convective derivative of the velocity vector, and the second integral contains the change and

flow of mass in time. Now lets assume that there are no sinks and sources in the system, that ismass is conserved, so this term is zero. Hence we obtain:

 putting this back into the original equation:

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For an arbitrary volume the integrand itself must be zero, and we have the Cauchy's equation of 

motion

As we see the only extra assumption we made is that the system doesn't contain any masssources or sinks, which means that mass is conserved. So this equation is valid for the motion of 

any continuum, even for that of fluids. If we are examining elastic continuums only then the

second term of the convective derivative operator can be neglected, and we are left with theusual time derivative, of the velocity field.

If a system is in equilibrium, the change in momentum with respect to time is equal to 0, as there

is no acceleration.

or using tensors,

These are the equilibrium equations which are used in solid mechanics for solving problems of 

linear elasticity. In engineering notation, the equilibrium equations are expressed in Cartesiancoordinates as

[edit] Conservation of linear momentum

The law of conservation of linear momentum is a fundamental law of nature, and it states that

the total momentum of a closed system of objects (which has no interactions with external

agents) is constant. One of the consequences of this is that the center of mass of any system of 

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objects will always continue with the same velocity unless acted on by a force from outside the

system.

Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry)of space (position in space is the canonical conjugate quantity to momentum). So, momentum

conservation can be philosophically stated as "nothing depends on location per se".

In analytical mechanics the conservation of momentum is a consequence of  translational

invariance of Lagrangian in the absence of external forces. It can be proven that the totalmomentum is a constant of motion by making an infinitesimal translation of Lagrangian and then

equating it with non translated Lagrangian. This is a special case of  Noether's theorem [14].

In an isolated system (one where external forces are absent) the total momentum will be

constant: this is implied by Newton's first law of motion. Newton's third law of motion, the lawof reciprocal actions, which dictates that the forces acting between systems are equal in

magnitude, but opposite in sign, is due to the conservation of momentum.

Since position in space is a vector quantity, momentum (being the canonical conjugate of 

 position) is a vector quantity as well—it has direction. Thus, when a gun is fired, the final totalmomentum of the system (the gun and the bullet) is the vector sum of the momenta of these two

objects. Assuming that the gun and bullet were at rest prior to firing (meaning the initial

momentum of the system was zero), the final total momentum must also equal 0.

In an isolated system with only two objects, the change in momentum of one object must beequal and opposite to the change in momentum of the other object. Mathematically,

Momentum has the special property that, in a closed system, it is always conserved, even incollisions and separations caused by explosive forces. Kinetic energy, on the other hand, is not

conserved in collisions if they are inelastic. Since momentum is conserved it can be used to

calculate an unknown velocity following a collision or a separation if all the other masses and

velocities are known.

A common problem in physics that requires the use of this fact is the collision of two particles.

Since momentum is always conserved, the sum of the momenta before the collision must equal

the sum of the momenta after the collision:

where u1 and u2 are the velocities before collision, and v1 and v2 are the velocities after collision.

Determining the final velocities from the initial velocities (and vice versa) depend on the type of 

collision. There are two types of collisions that conserve momentum: elastic collisions, whichalso conserve kinetic energy, and inelastic collisions, which do not.

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[edit] Elastic collisions

A collision between two pool  balls is a good example of an almost totally elastic collision, due to

their high rigidity; a totally elastic collision exists only in theory, occurring between bodies withmathematically infinite rigidity. In addition to momentum being conserved when the two balls

collide, the sum of kinetic energy before a collision must equal the sum of kinetic energy after:

[edit] In one dimension

When the initial velocities are known, the final velocities for a head-on collision are given by

When the first body is much more massive than the other (that is, m1 ≫ m2), the final velocities

are approximately given by

Thus the more massive body does not change its velocity, and the less massive body travels attwice the velocity of the more massive body less its own original velocity. Assuming bothmasses were heading towards each other on impact, the less massive body is now therefore

moving in the opposite direction at twice the speed of the more massive body plus its own

original speed.

A Newton's cradle demonstrates conservation of momentum.

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In a head-on collision between two bodies of equal mass (that is, m1 = m2), the final velocities are

given by

Thus the bodies simply exchange velocities. If the first body has nonzero initial velocity u1 and

the second body is at rest, then after collision the first body will be at rest and the second body

will travel with velocity u1. This phenomenon is demonstrated by Newton's cradle.

[edit] In multiple dimensions

In the case of objects colliding in more than one dimension, as in oblique collisions, the velocity

is resolved into orthogonal components with one component perpendicular to the plane of collision and the other component or components in the plane of collision. The velocity

components in the plane of collision remain unchanged, while the velocity perpendicular to the

 plane of collision is calculated in the same way as the one-dimensional case.

For example, in a two-dimensional collision, the momenta can be resolved into  x and ycomponents. We can then calculate each component separately, and combine them to produce a

vector result. The magnitude of this vector is the final momentum of the isolated system.

[edit] Perfectly inelastic collisions

A common example of a perfectly inelastic collision is when two snowballs collide and then stick together afterwards. This equation describes the conservation of momentum:

It can be shown that a perfectly inelastic collision is one in which the maximum amount of 

kinetic energy is converted into other forms. For instance, if both objects stick together after the

collision and move with a final common velocity, one can always find a reference frame in

which the objects are brought to rest by the collision and 100% of the kinetic energy isconverted. This is true even in the relativistic case and utilized in particle accelerators to

efficiently convert kinetic energy into new forms of mass-energy (i.e. to create massive

 particles).

[edit] Coefficient of Restitution

Main article: Coefficient of Restitution

The coefficient of restitution is defined as the ratio of relative velocity of separation to relative

velocity of approach. It is a ratio hence it is a dimensionless quantity. The coefficient of restitution is given by:

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for two colliding objects, where

V 1f  is the scalar final velocity of the first object after impact

V 2f  is the scalar final velocity of the second object after impact

V 1 is the scalar initial velocity of the first object before impact

V 2 is the scalar initial velocity of the second object before impact

A perfectly elastic collision implies that CR  is 1. So the relative velocity of approach is same as

the relative velocity of separation of the colliding bodies.

Inelastic collisions have (CR  < 1). In case of a perfectly inelastic collision the relative velocity of separation of the centre of masses of the colliding bodies is 0. Hence the bodies stick together 

after collision.

[edit] Explosions

An explosion occurs when an object is divided into two or more fragments due to a release of 

energy. Note that kinetic energy in a system of explosion is not conserved because it involvesenergy transformation (i.e. kinetic energy changes into heat and acoustic energy).

See the inelastic collision  page for more details.

[edit] Modern definitions of momentum

[edit] Momentum in relativistic mechanics

In relativistic mechanics, in order to be conserved, the momentum of an object must be definedas

where m0 is the invariant mass of the object and γ is the Lorentz factor , given by

where v is the speed of the object and c is the speed of light.

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Relativistic momentum can also be written as invariant mass times the object's proper velocity,

defined as the rate of change of object position in the observer frame with respect to time elapsed

on object clocks (i.e. object proper time). Within the domain of classical mechanics, relativisticmomentum closely approximates Newtonian momentum: at low velocity, γm0v is approximately

equal to m0v, the Newtonian expression for momentum.

A graphical representation of the interrelation of relativistic energy E, invariant

mass m0, relativistic momentum p, and relativistic mass m = γm0.

The total energy E of a body is related to the relativistic momentum p by

where p denotes the magnitude of p. This relativistic energy-momentum relationship holds even

for massless particles such as photons; by setting m0 = 0 it follows that

For both massive and massless objects, relativistic momentum is related to the de Broglie

wavelength  λ by

where h is the Planck constant.

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[edit] Four-vector formulation

Relativistic four-momentum as proposed by Albert Einstein arises from the invariance of  four-

vectors under Lorentzian translation. The four-momentum P is defined as:

where E = γm0c2 is the total relativistic energy of the system, and p x, p y, and p z  represent the x-,

 y-, and z -components of the relativistic momentum, respectively.

The magnitude ||P|| of the momentum four-vector is equal to m0c, since

which is invariant across all reference frames. For a closed system, the total four-momentum is

conserved, which effectively combines the conservation of both momentum and energy into a

single equation. For example, in the radiationless collision of two particles with rest masses m1

and m2 with initial velocities and , the respective final velocities and may be found

from the conservation of four-momentum which states that:

where

For elastic collisions, the rest masses remain the same (m1 = m3 and m2 = m4), while for inelasticcollisions, the rest masses will increase after collision due to an increase in their heat energycontent. The conservation of four-momentum can be shown to be the result of the homogeneity

of spacetime.

[edit] Generalization of momentum

Momentum is the Noether charge of translational invariance. As such, even fields as well asother things can have momentum, not just particles. However, in curved space-time which is not

asymptotically Minkowski, momentum isn't defined at all.

[edit] Momentum in quantum mechanics

Further information: Momentum operator

In quantum mechanics, momentum is defined as an operator on the wave function. The

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.

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For a single particle described in the position basis the momentum operator can be written as

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 the momentum basis the momentumoperator is represented as

where the operator p acting on a wave function ψ(p) yields that wave function multiplied by thevalue 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.

[edit] Momentum in electromagnetism

Electric and magnetic fields possess momentum regardless of whether they are static or they

change in time. The pressure, P , of an electrostatic (magnetostatic) field upon a metal sphere,

cylindrical capacitor or ferromagnetic bar is:

where , , , are the electromagnetic energy density, electric field, and magnetic field

respectively. The electromagnetic pressure may be sufficiently high to explode thecapacitor. Thus electric and magnetic fields do carry momentum.

Light (visible, UV, radio) is an electromagnetic wave and also has momentum. 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 is controversial (see

Abraham–Minkowski controversy [2]).

Momentum is conserved in an electrodynamic system (it may change from momentum in thefields to mechanical momentum of moving parts). The treatment of the momentum of a field is

usually accomplished by considering the so-called energy-momentum tensor  and the change in

time of the Poynting vector integrated over some volume. This is a tensor field which hascomponents related to the energy density and the momentum density.

The definition canonical momentum corresponding to the momentum operator of quantum

mechanics when it interacts with the electromagnetic field is, using the principle of least

coupling:

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,

instead of the customary

,

where:

is the electromagnetic vector potential

the charged particle's invariant mass

its velocity

its charge.

[edit] Analogies between heat, mass, and momentum transferMain article: Transport phenomena

There are some notable similarities in equations for momentum, heat, and mass transfer [15]. The

molecular transfer equations of   Newton's law for fluid momentum, Fourier's law for heat, andFick's law for mass are very similar. A great deal of effort has been devoted to developing

analogies among these three transport processes so as to allow prediction of one from any of the

others.

[edit] See also