Momentum

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 the tendency for an object to continue to move in its direction of travel. As such, it is a natural consequence of Newton's first law.

Momentum is a conserved quantity, meaning that the total momentum of any closed system cannot be changed.

Momentum in classical 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 p (bolded because it is a vector), so this can be written:

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

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

The velocity of an object is given by its speed and its direction. Because momentum depends on 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.

Conservation of momentum

As far as we know, momentum is a conserved quantity. Conservation of momentum states that the total amount of momentum of all the things in the universe will never change. 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 outside the system.

Conservation of momentum is a consequence of the homogeneity 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 momentum before the collision must equal the sum of the momentum after the collision:

m_1 \mathbf v_{1,i} + m_2 \mathbf v_{2,i} = m_1 \mathbf v_{1,f} + m_2 \mathbf v_{2,f} \,

where the subscript i signifies initial, before the collision, and f signifies final, 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
Inelastic collisions don't conserve kinetic energy

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 $M_1 U_1 + M_2 U_2 = M_1 V_1 + M_2 V_2$

=Multi dimensional collisions

= In any other case of objects colliding in more than one dimension the velocity is split into its axes with one axis going through the point of collision and the other axis/axes tangent to the point of collision. The velocity components that are tangent to the point of intersection remain the same, while the velocity through the point of collision is calculated the same as the one dimensional case.

Inelastic collisions

A common example of a perfectly inelastic collision is when two objects 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 \,

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 rest mass of the thing moving,

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

v is the relative velocity between an object and an observer, and

c is the speed of light.

As you may see relativistic momentum becomes Newtonian momentum: $m\mathbf{v}$ at low speed limit (v/c -> 0).

Relativistic four-momentum as proposed by Albert Einstein arises from the invariance of four-vectors under Lorentzian translation. These four-vectors appear spontaneously in the Green's function from quantum field theory. 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, and E is the total energy of the system:

E = \gamma mc^2 \;

Setting velocity to zero, one derives that the rest mass and the energy of an object are related by E=mc².

The "length" of the vector that remains constant is defined thus:

\mathbf{p} \cdot \mathbf{p} - E^2/c^2

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. 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 accomplised 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 of the momentum of a particle has to be changed if it interacts with the electromagnetic field. In this case, using the principle of least coupling, the momentum of the particle should be $\vec p = m\vec v + q\vec A$ , which replaces the customary $\vec p = m\vec v$ .

Here $\vec A$ is the electromagnetic potential, $m$ the charged particle's mass, $\vec v$ its velocity and $q$ 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.

References

Halliday, David, and Resnick, Robert (1970). Fundamentals of Physics,2nd ed. John Wiley & Sons.
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

External links

conservation of momentum - a chapter from an online textbook

Physical quantity | Mechanics | Introductory physics | Fundamental physics concepts | Conservation laws

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