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Momentum

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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 isn't asymptotically Minkowski, momentum isn't defined at all.

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

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.

The SI unit of momentum is newton-seconds, which can alternatively be expressed with the units kg.m/s.

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

$I=\int F\,dt$

using the definition of force yields:

$I=\int\frac{dp}{dt}\,dt$

$I=\int dp$

I = Δp

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:

E = γmc²

p = γmv

and

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

Setting velocity to zero, one derives the result that objects have a rest mass which is related by the experession E=mc^2

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

$p\cdot p-E^2$

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.

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.

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