Dictionary Definition
momentum
Noun
1 an impelling force or strength; "the car's
momentum carried it off the road" [syn: impulse]
2 the product of a body's mass and its velocity;
"the momentum of the particles was deduced from meteoritic
velocities" [also: momenta (pl)]
User Contributed Dictionary
English
Pronunciation
 (US) /ˌmoʊˈmɛntəm/
Noun
 (of a body in motion) the product of its mass and velocity.
 The impetus, either
of a body in motion, or of an idea or course of events.
 1843, Nathaniel Hawthorne, "The Old Apple Dealer", in Mosses
from an Old Manse
 The travellers swarm forth from the cars. All are full of the momentum which they have caught from their mode of conveyance.
 1882, Thomas Hardy, Two on a Tower
 Their intention to become husband and wife, at first halting and timorous, had accumulated momentum with the lapse of hours, till it now bore down every obstacle in its course.
 1843, Nathaniel Hawthorne, "The Old Apple Dealer", in Mosses
from an Old Manse
Translations
 Chinese: 動量 (1), 衝力 (2)
 Czech: hybnost
 Dutch: impuls
 Finnish: liikemäärä, vauhti, voima, liikevoima
 German: impuls (1,2), moment (2)
 Italian: quantità di moto, impulso, momento
 Persian: اندازه حرکت
 Portuguese: ritmo, impulso
 Swedish: rörelsemängd
Extensive Definition
In classical
mechanics, momentum (pl. momenta; SI unit kg·m/s,
or, equivalently, N·s) is the product of the mass and velocity of an object (p=mv).
For more accurate measures of momentum, see the section
"modern definitions of momentum" on this page. It is sometimes
referred to as linear momentum to distinguish it from the related
subject of angular
momentum. Linear momentum is a vector quantity, since it has a
direction as well as a magnitude. Angular momentum is a pseudovector quantity
because it gains an additional sign flip under an improper
rotation. The total momentum of any group of objects remains
the same unless outside forces act on the objects.
Momentum is a conserved
quantity, meaning that the total momentum of any closed
system (one not affected by external forces) cannot
change.
History of the concept
The word for the general concept of mōmentum was used in the Roman Republic primarily to mean "a movement, motion (as an indwelling force ...)." A fish was able to change velocity (velocitas) through the mōmentum of its tail. The word is formed by an accretion of suffices on the stem of Latin movēre, "to move." A movimen is the result of the movēre just as fragmen is the result of frangere, "to break." Extension by to obtains mōvimentum and fragmentum, the former contracting to mōmentum.The 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
speed. The Romans, due to
limitations inherent in the Roman
numeral system, were unable to go further with the
perception.
The concept of momentum in classical mechanics
was originated by a number of great thinkers and experimentalists.
The first of these was Ibn Sina
(Avicenna) circa 1000, who referred to impetus
as proportional to weight
times velocity.
René
Descartes later referred to mass times velocity as the
fundamental force of motion. Galileo in his
Two New
Sciences used the Italian
word "impeto."
The question has been much debated as to what Sir
Isaac
Newton's contribution to the concept was. Apparently nothing,
except to state more fully and with better mathematics what was
already known. The first and second of Newton's
Laws of Motion had already been stated by John Wallis
in his 1670 work, Mechanica slive 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 "Mathematical Principles of Natural
History" when it first came out in 1686 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.
Linear momentum of a particle
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 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 small
bold p (bold because it is a vector);
so this can be written:
 \mathbf= m \mathbf
 \ \mathbf is the momentum
 \ m is the mass
 \ \mathbf the velocity
Example: a model airplane of 1 kg travelling 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. In the case of constant mass, and
velocities much less than the speed of light, this definition
results in the equation
 \ \sum = = \mathbf+ m=0+ m = m\mathbf
or just simply
 \mathbf= m \mathbf
Example: a model airplane of 1 kg accelerates
from rest to a velocity of 1 m/s due north in 1 sec. The thrust
required to produce this acceleration is 1 newton. The change in momentum is
1 kgm/sec. 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.
Linear momentum of a system of particles
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. \mathbf= \sum_^n m_i \mathbf_i = m_1 \mathbf_1 + m_2 \mathbf_2 + m_3 \mathbf_3 + \cdots + m_n \mathbf_n
 \mathbf is the momentum of the particle system
 \ m_i is the mass of object i
 \mathbf_i the vector velocity of object i
 \ 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 respect to that frame is simply:
 \mathbf= M\mathbf_\text
 M=\sum_^n m_i.
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 of the center of mass \mathbf_
 \ \sum = = M \frac=M\mathbf_
This is commonly known as Newton's
second law.
For a more general derivation using tensors, we
consider a moving body (see Figure), assumed as a continuum,
occupying a volume \ V at a time \ t, having a surface area \ S,
with defined traction or surface forces \ T_i^ acting on every
point of the body surface, 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:
 \ \int_S T_i^dS + \int_V F_i dV = \int_V \rho \frac \, dV
 \ \int_S \sigma_n_j \, dS + \int_V F_i \, dV = \int_V \rho \frac \, dV
 \ \int_V \sigma_ \, dV + \int_V F_i \, dV = \int_V \rho \frac \, dV
 \ \int_V \sigma_ + F_i \, dV = \int_V \rho \frac \, dV
 \ \sigma_ + F_i = \rho \frac
If a system is in equilibrium, the change in
momentum with respect to time is equal to 0, as there is no
acceleration.
 \ \sum = =\ M\mathbf_= 0
 \ \sigma_ + F_i = 0
 \frac + \frac + \frac + F_x = 0
 \frac + \frac + \frac + F_y = 0
 \frac + \frac + \frac + F_z = 0
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 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 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
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 system 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,
\Delta \mathbf_1 = \Delta \mathbf_2
Momentum has the special property that, in a
closed
system, it is always conserved, even in collisions 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:
 m_1 \mathbf u_ + m_2 \mathbf u_ = m_1 \mathbf v_ + m_2 \mathbf v_ \,
 u signifies vector velocity before the collision
 v signifies vector velocity after the collision.
Usually, we either only know the velocities
before or after a collision and would like to also find out the
opposite. Correctly solving this problem means you have to know
what kind of collision took place. There are two basic kinds of
collisions, both of which conserve momentum:
 Elastic collisions conserve kinetic energy as well as total momentum before and after collision.
 Inelastic collisions don't conserve kinetic energy, but total momentum before and after collision is conserved.
Elastic collisions
A collision between two Pool balls is a good example of an almost totally elastic collision. 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:
 \begin\frac\end m_1 v_^2
Since the 1/2 factor is common to all the terms,
it can be taken out right away.
Headon collision (1 dimensional)
In the case of two objects colliding head on we find that the final velocity
 v_ = \left( \frac \right) v_ + \left( \frac \right) v_ \,

 v_ = \left( \frac \right) v_ + \left( \frac \right) v_ \,
which can then easily be rearranged to

 m_ \cdot v_ + m_ \cdot v_ = m_ \cdot v_ + m_ \cdot v_\,
Special Case: m1>>m2
Now consider the case when the mass of one body,
say m1, is far greater than that of the other, m2 (m1>>m2).
In that case m1+m2 is approximately equal to m1 and m1m2 is
approximately equal to m1.
Using these approximations, the above formula for
v_ reduces to v_=2v_v_. Its physical interpretation is that in the
case of a collision between two bodies, one of which is much more
massive than the other, the lighter body ends up moving in the
opposite direction with twice the original speed of the more
massive body.
Special Case: m1=m2
Another special case is when the collision is
between two bodies of equal mass.
Say body m1 moving at velocity v1 strikes body m2
that is at rest (v2). Putting this case in the equation derived
above we will see that after the collision, the body that was
moving (m1) will start moving with velocity v2 and the mass m2 will
start moving with velocity v1. So there will be an exchange of
velocities.
Now suppose one of the masses, say m2, was at
rest. In that case after the collision the moving body, m1, will
come to rest and the body that was at rest, m2, will start moving
with the velocity that m1 had before the collision.
Note that all of these observations are for an
elastic
collision.
This phenomenon is demonstrated by Newton's
cradle, one of the best known examples of conservation of
momentum, a real life example of this special case.
Multidimensional collisions
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.
See the elastic
collision page for more details. x=2a
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:
 m_1 \mathbf v_ + m_2 \mathbf v_ = \left( m_1 + m_2 \right) \mathbf v_f \,
In case of Inelastic collision, there is a
parameter attached called coefficient of restitution (denoted by
small 'e' or 'c' in many text books). It 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.
When we have an elastic collision the value of e
(= coefficient of restitution) is 1, i.e. the relative velocity of
approach is same as the relative velocity of separation of the
colliding bodies. In an elastic collision the Kinetic energy of the
system is conserved.
When a collision is not elastic (e<1) it is an
inelastic collision. In case of a perfectly inelastic collision the
relative velocity of separation of the centre of masses of the
colliding bodies is 0. Hence after collision the bodies stick
together after collision. In case of an inelastic collision the
loss of Kinetic energy is maximum as stated above.
In all types of collision if no external force is
acting on the system of colliding bodies, the momentum will always
be preserved.
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 involves energy transformation. (i.e. kinetic energy changes into heat and sound energy)In the exploding cannon demonstration, total
system momentum is conserved. The system consists of two objects 
a cannon and a tennis ball. Before the explosion, the total
momentum of the system is zero since the cannon and the tennis ball
located inside of it are both at rest. After the explosion, the
total momentum of the system must still be zero. If the ball
acquires 50 units of forward momentum, then the cannon acquires 50
units of backwards momentum. The vector sum of the individual
momenta of the two objects is 0. Total system momentum is
conserved.
See the inelastic
collision page for more details.
Modern definitions of momentum
Momentum in relativistic mechanics
In relativistic mechanics, in order to be conserved, momentum must be defined as: \mathbf = \gamma m_0\mathbf
 m_0\, is the invariant mass of the object moving,
 \gamma = \frac is the Lorentz factor
 v\, is the relative velocity between an object and an observer
 c\, is the speed of light.
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).
Relativistic momentum becomes Newtonian momentum: m\mathbf at low
speed \big(\mathbf/c \rightarrow 0 \big).
Relativistic fourmomentum
as proposed by Albert
Einstein arises from the invariance of fourvectors
under Lorentzian translation. The fourmomentum is defined
as:
 \left( , p_x , p_y ,p_z \right)
where
 p_x\, is the x\, component of the relativistic momentum,
 E \, is the total energy of the system:
 E = \gamma m_0c^2 \,
The "length" of the vector is the mass times the
speed of light, which is invariant across all reference
frames:
 (E/c)^2  p^2 = (m_0c)^2\,
Momentum of massless objects
Objects without a rest mass, such as photons, also carry momentum. The
formula is:
 p = \frac = \frac
 h\, is Planck's constant,
 \lambda\, is the wavelength of the photon,
 E\, is the energy the photon carries and
 c\, is the speed of light.
Generalization of momentum
Momentum is the Noether
charge of translational invariance. As such, even fields as
well as other things can have momentum, not just particles.
However, in curved
spacetime which is not asymptotically Minkowski,
momentum isn't defined at all.
Momentum in quantum mechanics
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.For a single particle with no electric
charge and no spin, the
momentum operator can be written in the position basis as
 \mathbf=\nabla=i\hbar\nabla
where:
 \nabla is the gradient operator;
 \hbar is the reduced Planck constant;
 i = \sqrt is the imaginary unit.
This is a commonly encountered form of the
momentum operator, though not the most general one.
Momentum in electromagnetism
Electric and magnetic fields possess momentum regardless of whether they are static or they change in time. It is a great surprise for freshmen who are introduced to the well known fact of the pressure of an electrostatic (magnetostatic) field upon a metal sphere, cylindrical capacity or ferromagnetic bar: P_ = = \left[ + \right],
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.
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
vector integrated over some volume. This is a tensor field
which has components 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:
 \mathbf P = m\mathbf v + q\mathbf A,
 \mathbf p = m\mathbf v,
 \mathbf A is the electromagnetic vector potential
 m the charged particle's invariant mass
 \mathbf v its velocity
 q its charge.
See also
Notes
References
 Fundamentals of Physics
 Serway, Raymond; Jewett, John (2003). Physics for Scientists and Engineers (6 ed.). Brooks Cole. ISBN 0534408427
 Stenger, Victor J. (2000). Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Prometheus Books. Chpt. 12 in particular.
 Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics (4th ed.). W. H. Freeman. ISBN 1572594926
 'H C Verma' 'Concepts of Physics, Part 1' 'Bharti Bhawan'
 For numericals refer 'IE Irodov','Problems in General Physics'
External links
 Conservation of momentum  A chapter from an online textbook
momentum in Arabic: زخم الحركة
momentum in Belarusian: Імпульс
momentum in Bulgarian: Импулс (механика)
momentum in Bosnian: Količina kretanja
momentum in Catalan: Quantitat de moviment
momentum in Czech: Hybnost
momentum in Danish: Impuls (fysik)
momentum in German: Impuls
momentum in Modern Greek (1453): Ορμή
momentum in Esperanto: Movokvanto
momentum in Spanish: Cantidad de
movimiento
momentum in Estonian: Impulss
momentum in Basque: Momentu lineal
momentum in Persian: تکانه
momentum in Finnish: Liikemäärä
momentum in French: Quantité de mouvement
momentum in Galician: Cantidade de
movemento
momentum in Hebrew: תנע
momentum in Croatian: Količina gibanja
momentum in Hungarian: Impulzus
momentum in Indonesian: Momentum
momentum in Italian: Quantità di moto
momentum in Japanese: 運動量
momentum in Georgian: იმპულსი
momentum in Korean: 운동량
momentum in Lithuanian: Judesio kiekis
momentum in Malay (macrolanguage):
Momentum
momentum in Dutch: Impuls (natuurkunde)
momentum in Norwegian Nynorsk: Rørslemengd
momentum in Norwegian: Bevegelsesmengde
momentum in Polish: Pęd (fizyka)
momentum in Portuguese: Quantidade de movimento
linear
momentum in Russian: Импульс
momentum in Simple English: Momentum
momentum in Slovak: Hybnosť
momentum in Slovenian: Gibalna količina
momentum in Serbian: Импулс
momentum in Swedish: Rörelsemängd
momentum in Tamil: உந்தம்
momentum in Thai: โมเมนตัม
momentum in Turkish: Momentum
momentum in Ukrainian: Імпульс
momentum in Vietnamese: Động lượng
momentum in Chinese: 动量
momentum in Min Nan: Ūntōngliōng