# Momentum

In physics, momentum is a physical quantity related to the velocity and mass of an object.

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.

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## Momentum in classical mechanics

In classical mechanics, momentum (traditionally written as p) is defined as the product of mass and velocity. It is thus a vector quantity.

[itex]\vec{p} = m \vec{v}[itex]

### Impulse

The change in momentum, called the impulse, is equal to force times the change in time.

[itex]\Delta \mathbf{p} = \mathbf{F} \cdot \Delta t[itex]
[itex] \mathbf{I} = \mathbf{F} \cdot t[itex]

The SI unit of momentum can be expressed as kg m/s, not to be confused with newton seconds as a newton is defined as mass by acceleration.

An impulse changes the momentum of an object. An impulse is calculated as the integral of force with respect to duration.

[itex] \mathbf{I} = \int \mathbf{F}\,dt [itex]

using the definition of force yields:

[itex] \mathbf{I} = \int\frac{d\mathbf{p}}{dt}\,dt [itex]
[itex] \mathbf{I} = \int d\mathbf{p} [itex]
[itex] \mathbf{I} = \Delta \mathbf{p} [itex]

## Conservation of Momentum & Collisions

Momentum has the special property that it is always conserved in collisions. Kinetic energy, on the other hand, is often not conserved in collisions.

A common problem in physics that requires the use of this fact is that of 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:

[itex]m_1 v_{1,i} + m_2 v_{2,i} = m_1 v_{1,f} + m_2 v_{2,f} \,[itex]
where i signifies the velocity initially (before) the collision, and f signifies the final (after) the collision.

Usually, we either only know the velocities before or after a collision and 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

A collision between two pool balls is a good example of an almost totally elastic collision. So in addition to momentum being conserved, when two pool balls collide, the sum of kinetic energy before a collision must equal the sum of kinetic energy after:

[itex]\frac{1}{2} m_1 v_{1,i}^2 + \frac{1}{2} m_2 v_{2,i}^2 = \frac{1}{2} m_1 v_{1,f}^2 + \frac{1}{2} m_2 v_{2,f}^2 \,[itex]

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

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

[itex]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} \,[itex]

[itex]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} \,[itex]

### Inelastic Collisions

A common example of an inelastic collision is when two objects collide and then stick together afterwards. So we end up with this equation describing the conservation of momentum:

[itex]m_1 v_{1,i} + m_2 v_{2,i} = \left( m_1 + m_2 \right) v_f \,[itex]

## Momentum in relativistic mechanics

It is commonly believed that the physical laws should be invariant under translations. Thus, the definition of momentum was changed when Einstein formulated Special relativity so that its magnitude would remain invariant under relativistic transformations. See physical conservation law. We now define a vector, called the 4-momentum thus:

[E/c p]

where E is the total energy of the system, and p is called the "relativistic momentum" defined thus:

[itex] E = \gamma mc^2 \;[itex]
[itex] \mathbf{p} = \gamma m\mathbf{v} [itex]

where

[itex] \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}[itex].

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:

[itex] \mathbf{p} \cdot \mathbf{p} - E^2 [itex]

Massless objects such as photons also carry momentum; the formula is p=E/c, where E is the energy the photon carries and c is the speed of light.

## 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 interchangeable.

For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as

[itex]\mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla[itex]

where Missing image
Del.gif
Image:del.gif

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

## Origin of momentum

Momentum arises from the condition that an experiment must give the same results regardless of the position or velocity of the observer. More formally it is the requirement of invariance under translation. Classical momentum is the result of the invariance of translation in three dimensions. Relativistic 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.

## 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.

## References

• Halliday, David; Resnick, Robert (1970). Fundamentals of Physics (2nd Ed). New York: John Wiley & Sons.

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