In
classical mechanicsIn physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
,
linear momentum or
translational momentum (
pl.In linguistics, plurality or [a] plural is a concept of quantity representing a value of morethanone. Typically applied to nouns, a plural word or marker is used to distinguish a value other than the default quantity of a noun, which is typically one...
momenta;
SISi, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit
kgThe kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...
·m/s, or, equivalently, N·
sThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
) is the product of the
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and
velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of an object (
). Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude.
Linear momentum is a
conservedIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
quantity, meaning that if a
closed systemIn physics:In thermodynamics, a closed system can exchange energy , but not matter, with its surroundings.In contrast, an isolated system cannot exchange any of heat, work, or matter with the surroundings, while an open system can exchange all of heat, work and matter.For a simple system, with...
is not affected by external forces, its total linear momentum cannot change. Although originally expressed in Newton's second law, the conservation of linear momentum also holds in
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and, with appropriate definitions, a (generalized) linear momentum conservation law holds in electrodynamics,
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
,
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
, and
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. In
relativistic mechanicsSpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, nonrelativistic linear momentum is further multiplied by the
Lorentz factorThe Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
.
History of the concept
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, while
velocitas, "swiftness", captured only
speedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
.
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 PhiloponusJohn Philoponus , also known as John the Grammarian or John of Alexandria, was a Christian and Aristotelian commentator and the author of a considerable number of philosophical treatises and theological works...
, in his commentary to
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
´s
Physics. 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. This idea was refined by the European philosophers
Peter OliviPeter John Olivi, in his native French Pierre Jean Olivi and also Pierre Déjean, was a Franciscan theologian who, although he died professing the faith of the Roman Catholic Church, became a controversial figure in the arguments surrounding poverty at the beginning of the fourteenth century...
and
Jean BuridanJean Buridan was a French priest who sowed the seeds of the Copernican revolution in Europe. Although he was one of the most famous and influential philosophers of the late Middle Ages, he is today among the least well known...
. Buridan referred to
impetusThe theory of impetus was an auxiliary or secondary theory of Aristotelian dynamics, put forth initially to explain projectile motion against gravity...
being proportional to the weight times the speed. 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.
René DescartesRené Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
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. Galileo, later, in his
Two New SciencesThe Discourses and Mathematical Demonstrations Relating to Two New Sciences was Galileo's final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years.After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned...
, used the
ItalianItalian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...
word "impeto".
The extent to which
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
contributed to the concept has been much debated. 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 MotionNewton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
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". 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. 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. 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.
The
Oxford English Dictionary dates the first print use of the word to 1699: John Keill,
Reflections upon the Theory of the earth [written by himself] occasion'd by a late Examination of it, a letter (London).
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". In Bulgarian and in Dutch the linear momentum is typically referred to as impulse, while the angular momentum is called momentum and the impulse has no distinct name.
Linear momentum of a particle
If an object is moving in any
reference frameA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, 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 a reference frame attached to the object's
center of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
.
The amount of momentum that an object has depends on two physical quantities: the
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and the
velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of the moving object in the
frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
. 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 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.
Given that mass is constant, the second term of the derivative is zero (
). We can therefore write the following:
or just simply
where
F is understood to be the net force (or 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.
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,
m_{i} and
v_{i} are the respective mass and velocity of the
ith 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
v_{cm} with respect to that frame is simply:
where:
This is known as Euler's first law.
Relating to force – General equations of motion
The linear momentum of a system of particles can also be defined as the product of the total mass,
m, of the system times the velocity,
v_{cm}, 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
continuumContinuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
, occupying a volume
V, at a time
t, having a surface area
S, 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
F_{i} per unit of volume on every point within the volume
V, and a velocity field
v_{i}, prescribed throughout the body. Following the previous equation, the linear momentum of the system is:
By definition the stress vector is defined as
, then
Using the
Gauss's divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the 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 the second part we obtain:
Now the second term inside the integral is:
Plugging this 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 derivativeThe material derivative is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and classical mechanics...
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 is mass is conserved, so this term is zero. Hence we obtain:
putting this back into the original equation:
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 mass sources 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 continua only then the second term of the convective derivative operator can be neglected, and we are left with the usual 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 mechanicsSolid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the EulerBernoulli beam equation...
for solving problems of
linear elasticityLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
.
In engineering notation, the equilibrium equations are expressed in Cartesian coordinates as



Conservation of linear momentum
The law of
conservation of linear momentum is a fundamental law of nature, and it states that if no external force acts on a
closed systemIn physics:In thermodynamics, a closed system can exchange energy , but not matter, with its surroundings.In contrast, an isolated system cannot exchange any of heat, work, or matter with the surroundings, while an open system can exchange all of heat, work and matter.For a simple system, with...
of objects, the momentum of the closed system remains constant. One of the consequences of this is that the
center of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
of any
systemIn physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
of
objectsIn physics, a physical body or physical object is a collection of masses, taken to be one...
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
homogeneityIn general, homogeneity is defined as the quality or state of being homogeneous . For instance, a uniform electric field would be compatible with homogeneity...
(shift
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is equivalent to the fact that the physical laws do not depend on position.
In
analytical mechanicsAnalytical mechanics is a term used for a refined, mathematical form of classical mechanics, constructed from the 18th century onwards as a formulation of the subject as founded by Isaac Newton. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with...
the conservation of momentum is a consequence of translational invariance of
LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
in the absence of external forces. It can be proven that the total momentum is a
constant of motionIn mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint...
by making an infinitesimal translation of Lagrangian and then equating it with non translated Lagrangian. This is a special case of
Noether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
.
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 motionInertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
. Newton's third law of motion, the law of 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 total momentum 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 systemIn the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy  mass stays constant...
with only two objects, the change in momentum of one object must be equal 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 in
collisionA collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...
s and separations caused by explosive forces.
Kinetic energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
, 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
u_{1} and
u_{2} are the velocities before collision, and
v_{1} and
v_{2} 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 collisionAn elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...
s, which also conserve kinetic energy, and
inelastic collisionAn inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...
s, which do not.
Elastic collisions
A collision between two pool balls is a good example of an
almost totally elastic collision, due to their high
rigidityStiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.Calculations:...
; a totally elastic collision exists only in theory, occurring between bodies with mathematically 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:
In one dimension
When the initial velocities are known, the final velocities for a headon collision are given by
When the first body is much more massive than the other (that is), the final velocities are approximately given by
Thus the more massive body does not change its velocity, and the less massive body travels at twice the velocity of the more massive body less its own original velocity. Assuming both masses 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.
In a headon collision between two bodies of equal mass (that is), the final velocities are given by
Thus the bodies simply exchange velocities. If the first body has nonzero initial velocity
u_{1} 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
u_{1}. This phenomenon is demonstrated by
Newton's cradleNewton's cradle, named after Sir Isaac Newton, is a device that demonstrates conservation of momentum and energy via a series of swinging spheres. When one on the end is lifted and released, the resulting force travels through the line and pushes the last one upward...
.
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 onedimensional case.
For example, in a twodimensional collision, the momenta can be resolved into
x and
y components. 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.
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 energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
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 is converted. This is true even in the relativistic case and utilized in particle accelerators to efficiently convert kinetic energy into new forms of massenergy (i.e. to create massive particles).
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:
for two colliding objects, where
 v_{1} is the scalar final velocity of the first object after impact,
 v_{2} is the scalar final velocity of the second object after impact,
 u_{1} is the scalar initial velocity of the first object before impact,
 u_{2} is the scalar initial velocity of the second object before impact.
A perfectly elastic collision implies that
C_{R} is 1. So the relative velocity of approach is same as the relative velocity of separation of the colliding bodies.
Inelastic collisions have (
C_{R} < 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.
Explosions
An explosion occurs as a result of a chain reaction that transforms potential energy into kinetic energy displacing the surrounding material. Explosions do not conserve potential energy. Instead, potential energy stored in chemical, mechanical, or nuclear form, is transformed into kinetic energy, acoustic energy, and electromagnetic radiation.
See the
inelastic collisionAn inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...
page for more details.
Momentum in relativistic mechanics
In relativistic mechanics, in order to be conserved, the momentum of an object must be defined as
where
m_{0} is the
invariant massThe invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
of the object and
γ is the
Lorentz factorThe Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
, given by
where
v is the
speedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
of the object and
c is the
speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. The inverse relation is given by:
where
is the magnitude of the momentum.
Relativistic momentum can also be written as invariant mass times the object's
proper velocityIn relativity, propervelocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observermeasured...
, 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 timeIn relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
). Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity,
γm_{0}v is approximately equal to
m_{0}v, the Newtonian expression for momentum.
The total energy
E of a body is related to the relativistic momentum
p by
where
p denotes the magnitude of
p. This relativistic energymomentum relationship holds even for massless particles such as photons; by setting 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 constantThe Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
.
Fourvector formulation
Relativistic
fourmomentumIn special relativity, fourmomentum is the generalization of the classical threedimensional momentum to fourdimensional spacetime. Momentum is a vector in three dimensions; similarly fourmomentum is a fourvector in spacetime...
as proposed by
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
arises from the invariance of
fourvectorIn the theory of relativity, a fourvector is a vector in a fourdimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the fourvector name tacitly assumes that its components refer to a standard basis...
s under Lorentzian translation. The fourmomentum
P is defined as:
where
E =
γm_{0}c^{2} is the total relativistic energy of the system, and
p_{x},
p_{y}, and
p_{z} represent the
x,
y, and
zcomponents of the relativistic momentum, respectively.
The magnitude 
P of the momentum fourvector is equal to
m_{0}c, since
which is invariant across all reference frames. For a closed system, the total fourmomentum 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
and
with initial velocities
and
, the respective final velocities
and
may be found from the conservation of fourmomentum which states that:
where
For elastic collisions, the rest masses remain the same (
and
), while for inelastic collisions, the rest masses will increase after collision due to an increase in their heat energy content. The conservation of fourmomentum can be shown to be the result of the homogeneity of space–time.
Generalization of momentum
Momentum is the Noether charge of translational invariance. As such, not just particles, but fields and other things can have momentum. However, where space–time is curved there is no Noether charge for translational invariance.
Momentum in quantum mechanics
In
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, momentum is defined as an
operatorIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
on the wave function. The
HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
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 variablesConjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals of oneanother, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty in physics called the Heisenberg uncertainty...
.
For a single particle described in the position basis the momentum operator can be written as
where ∇ is the
gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
operator,
ħ is the reduced Planck constant, and
i is the
imaginary unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
. 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 momentum operator is represented as
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.
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 the capacitor.
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 sailSolar sails are a form of spacecraft propulsion using the radiation pressure of light from a star or laser to push enormous ultrathin mirrors to high speeds....
. The calculation of the momentum of light within
dielectricA dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
media is somewhat controversial (see Abraham–Minkowski controversy
http://prl.aps.org/abstract/PRL/v104/i7/e070401).
Momentum is conserved in an electrodynamic system (it may change from momentum in the fields to mechanical momentum of moving parts). The treatment of the momentum of a field is usually accomplished by considering the socalled energymomentum tensor and the change in time of the
Poynting vectorIn physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently coinvented the Poynting vector...
integrated over some volume. This is a tensor field which has components related to the energy density and the momentum density.
The definition of canonical momentum corresponding to the momentum operator of quantum mechanics when it interacts with the electromagnetic field is, using the principle of least coupling:
,
instead of the customary
where:
is the electromagnetic vector potential
the charged particle's invariant mass
its velocity
its charge.
Analogies between heat, mass, and momentum transfer
There are some notable similarities in equations for momentum, heat, and mass transfer. The molecular transfer equations of
Newton's lawA Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.Definition:...
for fluid momentum,
Fourier's lawIn heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
for heat, and Fick'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.
See also
 Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
 Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
 Impulse
 Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
 Moment map
In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...
 Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
 Planck momentum
 Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
 Galilean cannon
A Galilean cannon is a device that demonstrates conservation of linear momentum. It comprises a stack of balls, starting with a large, heavy ball at the base of the stack and progresses up to a small, lightweight ball at the top...
External links