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Momentum

2007 Schools Wikipedia Selection. Related subjects: General Physics

   In classical mechanics, momentum ( pl. momenta; SI unit kg m/s) is the
   product of the mass and velocity of an object. For more accurate
   measures of momentum, see the section "modern definitions of momentum"
   on this page.

   In general, the momentum of an object can be conceptually thought of as
   how difficult it is to stop the object, as determined by multiplying
   two factors: its inertia (the resistance of an object to being
   accelerated) and its velocity. As such, it is a natural consequence of
   Newton's first and second laws of motion. Having a lower speed or
   having less mass (how we measure inertia) results in having less
   momentum.

   Momentum is a conserved quantity, meaning that the total momentum of
   any closed system (one not affected by external forces, and whose
   internal forces are not dissipative in nature) cannot be changed.

   The concept of momentum in classical mechanics was originated by a
   number of great thinkers and experimentalists. René Descartes referred
   to mass times velocity as the fundamental force of motion. Galileo in
   his Two New Sciences used the term "impeto" (Italian), while Newton's
   Laws of Motion uses motus (Latin), which has been interpreted by
   subsequent scholars to mean momentum.

Momentum in Newtonian mechanics

   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.

   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 symbol for momentum is usually denoted by
   a small bold p (bold because it is a vector); so this can be written:

          \mathbf{p}= m \mathbf{v}

   where:

          p is the momentum
          m is the mass
          v the velocity

   (using bold text for vectors).

   The origin of the use of p for momentum is unclear. It has been
   suggested that, since m had already been used for "mass", the p may be
   derived from the Latin petere ("to go") or from "progress" (a term used
   by Leibniz).

   The velocity of an object at a particular instant is given by its speed
   and the direction of its motion at that instant. Because momentum
   depends on and includes the physical quantity of velocity, it too has a
   magnitude and a direction and is a vector quantity. For example the
   momentum of a 5-kg bowling ball would have to be described by the
   statement that it was moving westward at 2 m/s. It is insufficient to
   say that the ball has 10 kg m/s of momentum because momentum is not
   fully described unless its direction is given.

Momentum for a system

Relating to mass and velocity

   The momentum of a system of objects is the vector sum of the momenta of
   all the individual objects in the system.

          \mathbf{p}= m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + m_3
          \mathbf{v}_3 + ... + m_n \mathbf{v}_n

   where

          \mathbf{p} is the momentum
          m[i] is the mass of object i
          \mathbf{v}_i the velocity of object i
          n\ is the number of objects in the system

Relating to force

   Force is equal to the rate of change of momentum:

          \mathbf{F} = {\mathrm{d}\mathbf{p} \over \mathrm{d}t} .

   In the case of constant mass, and velocities much less than the speed
   of light, this definition results in the equation \mathbf{F} =
   m\mathbf{a} , commonly known as Newton's second law.

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

          \mathbf{F} = {\mathrm{d}\mathbf{p} \over \mathrm{d}t}=\
          m\mathbf{a}= 0

Conservation of momentum

   The principle of conservation of momentum 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 centre of mass of any system of objects will always continue with
   the same velocity unless acted on by a force outside the system.

   Conservation of momentum is a consequence of the homogeneity (shift
   symmetry) of space.

   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 momentum is a vector quantity it has direction. Thus, when a gun
   is fired, although overall movement has increased compared to before
   the shot was fired, the momentum of the bullet in one direction is
   equal in magnitude, but opposite in sign, to the momentum of the gun in
   the other direction. These then sum to zero which is equal to the zero
   momentum that was present before either the gun or the bullet was
   moving.

Conservation of momentum and collisions

   Momentum has the special property that, in a closed system, it is
   always conserved, even in collisions. 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 unknown velocities
   following a collision.

   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_{1} + m_2 \mathbf u_{2} = m_1 \mathbf v_{1} + m_2
          \mathbf v_{2} \,

   where:

          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 or snooker 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{matrix}\frac{1}{2}\end{matrix} m_1 v_{1,i}^2 +
                \begin{matrix}\frac{1}{2}\end{matrix} m_2 v_{2,i}^2 =
                \begin{matrix}\frac{1}{2}\end{matrix} m_1 v_{1,f}^2 +
                \begin{matrix}\frac{1}{2}\end{matrix} m_2 v_{2,f}^2 \,

   Since the 1/2 factor is common to all the terms, it can be taken out
   right away.

Head-on collision (1 dimensional)

   In the case of two objects colliding head on we find that the final
   velocity

                v_{1,f} = \left( \frac{m_1 - m_2}{m_1 + m_2} \right)
                v_{1,i} + \left( \frac{2 m_2}{m_1 + m_2} \right) v_{2,i}
                \,

                v_{2,f} = \left( \frac{2 m_1}{m_1 + m_2} \right) v_{1,i} +
                \left( \frac{m_2 - m_1}{m_1 + m_2} \right) v_{2,i} \,

   which can then easily be rearranged to

                m_{1,f} \cdot v_{1,f} + m_{2,f} \cdot v_{2,f} = m_{1,i}
                \cdot v_{1,i} + m_{2,i} \cdot v_{2,i}\,

   Now consider if mass of one body say m1 is far more than m2 (m1>>m2).
   In that case m1+m2 is approximately equal to m1. And m1-m2 is
   approximately equal to m1.

   Put these values in the above equation to calculate the value of v2
   after collsion. The expression changes to v2 final is 2*v1-v2. Its
   physical interpretation is in case of collison between two body one of
   which is very heavy, the lighter body moves with twice the velocity of
   the heavier body less its actual velocity but in opposite direction.

Multi-dimensional 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 one-dimensional case.

   For example, in a two-dimensional 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.

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_{1,i} + m_2 \mathbf v_{2,i} = \left( m_1 +
                m_2 \right) \mathbf v_f \,

   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 is converted. 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).

   See the inelastic collision page for more details.

Modern definitions of momentum

Momentum in relativistic mechanics

   In relativistic mechanics, momentum is defined as:

          \mathbf{p} = \gamma m\mathbf{v}

   where

          m is the invariant mass of the object moving,
          \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} is the Lorentz factor
          v is the relative velocity between an object and an observer
          c is the speed of light.

   Relativistic momentum becomes Newtonian momentum: m\mathbf{v} at low
   speed (v/c -> 0).

   Relativistic four-momentum as proposed by Albert Einstein arises from
   the invariance of four-vectors under Lorentzian translation. The
   four-momentum is defined as:

          \left( {E \over c} , 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 mc^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 = (mc)^2

   The diagram given alongside can serve as a useful mnemonic for
   remembering the above relations involving relativistic energy(E),
   invariant mass (m[0]), and relativistic momentum(p).

   (Please note that in the notation used by the diagram's creator, the
   invariant mass m is subscripted with a zero, m[0].)

   Momentum of massless objects

   Massless objects such as photons also carry momentum. The formula is:

          p = \frac{h}{\lambda} = \frac{E}{c}

   where

          h is Planck's constant,
          λ 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 space-time 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{p}={\hbar\over i}\nabla=-i\hbar\nabla

   where:

          \nabla is the gradient operator
          \hbar is the reduced Planck constant.

   This is a commonly encountered form of the momentum operator, though
   not the most general one.

Momentum in electromagnetism

   When electric and/or magnetic fields move, they carry momenta. 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 so-called energy-momentum 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 ,

   instead of the customary

          \mathbf p = m\mathbf v ,

   where:

          \mathbf A is the electromagnetic vector potential
          m the charged particle's invariant mass
          \mathbf v its velocity
          q its charge.

Figurative use

   A process may be said to gain momentum. The terminology implies that it
   requires effort to start such a process, but that it is relatively easy
   to keep it going. Alternatively, the expression can be seen to reflect
   that the process is adding adherents, or general acceptance, and thus
   has more mass at the same velocity; hence, it gained momentum.

   Retrieved from " http://en.wikipedia.org/wiki/Momentum"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
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