Elastic collisions:

A collision between two pool balls is a good example of an almost totally elastic collision; a totally elastic collision exists only in theory. 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:
In one dimensionWhen the initial velocities are known, the final velocities for a head-on collision are given by

Thus the more massive body does not change its velocity, and the less massive body travels in the opposite direction with twice the original speed of the more massive body.

A Newton's cradle demonstrates conservation of momentum.In a collision between two bodies of equal mass (that is, m1 = m2), the final velocities are given by

Thus the bodies simply exchange velocities. If the first body has nonzero initial velocity u1 and the second body is at rest, then after collision the first body will be at rest and the second body will travel with velocity u1. This phenomenon is demonstrated by Newton's cradle.

In multiple dimensions: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.

Perfectly 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:
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).

Coefficient of Restitution:

Coefficient of RestitutionThe coefficient of restitution 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. The coefficient of restitution is given by:

V1f is the scalar final velocity of the first object after impact V2f is the scalar final velocity of the second object after impact V1 is the scalar initial velocity of the first object before impact V2 is the scalar initial velocity of the second object before impact A perfectly elastic collision implies that CR is 1. So the relative velocity of approach is same as the relative velocity of separation of the colliding bodies.

Inelastic collisions have (CR < 1). 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.

For both massive and massless objects, relativistic momentum is related to the de Broglie wavelength ? by
where h is the Planck constant.
Four-vector formulation;Relativistic four-momentum as proposed by Albert Einstein arises from the invariance of four-vectors under Lorentzian translation. The four-momentum P is defined as:
where E = ?m0c2 is the total relativistic energy of the system, and px, py, and pz represent the x-, y-, and z-components of the relativistic momentum, respectively.
The magnitude P of the momentum four-vector is equal to m0c, since
which is invariant across all reference frames.
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:

Further information:

Momentum operatorIn 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
where ? is the gradient operator, h is the reduced Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, though not the most general one.

Momentum in electromagnetismElectric 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 that the pressure of an electrostatic (magnetostatic) field upon a metal sphere, cylindrical capacitor or ferromagnetic bar is:

where , , , are the electromagnetic energy density, electric field, and magnetic field respectively. The electromagnetic pressure may be sufficiently high to explode the capacitor. Thus electric and magnetic fields do carry momentum.

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:
instead of the customary
where:
is the electromagnetic vector potential the charged particle's invariant mass its velocity its charge.