Conservation of linear momentum:

A Newton's cradle demonstrates conservation of momentum.The law of conservation of linear momentum is a fundamental law of nature, and it 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 center of mass of any system of objects will always continue with the same velocity unless acted on by a force from outside the system.

Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). So, momentum conservation can be philosophically stated as "nothing depends on location per se".

In analytical mechanics the conservation of momentum is a consequence of translational invariance of Lagrangian in the absence of external forces. It can be proven that the total momentum is a constant of motion by making an infinitesmial translation of Lagrangian and then equating it with non translated Lagrangian. This is a special case of Noether's theorem .

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 position in space is a vector quantity, momentum (being the canonical conjugate of position) is a vector quantity as well - it has direction. Thus, when a gun is fired, the final total momentum of the system (the gun and the bullet) is the vector sum of the momenta of these two objects. Assuming that the gun and bullet were at rest prior to firing (meaning the initial momentum of the system was zero), the final total momentum must also equal 0.
In an isolated system with only two objects, the change in momentum of one object must be equal and opposite to the change in momentum of the other object.

Mathematically,
Momentum has the special property that, in a closed system, it is always conserved, even in collisions and separations caused by explosive forces. 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 an unknown velocity following a collision or a separation if all the other masses and velocities are known.

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:
where:
u signifies vector velocity before the collision. v signifies vector velocity after the collision. Usually, we only know the velocities either 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 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:
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 obeys

Special case: m1>>m2Now consider the case when the mass of one body, say m1, is far greater than that of the other, m2 (m1>>m2). In that case, both m1+m2 and m1-m2 are approximately equal to m1.
Using these approximations, the above formula reduces to

Its physical interpretation is that in the case of a collision between two bodies, one of which is much more massive than the other, the lighter body ends up moving in the opposite direction with twice the original speed of the more massive body.
Special case: m1=m2Another special case is when the collision is between two bodies of equal mass.

Say body m1 moving at velocity v1 strikes body m2. Putting this case in the equation derived above we will see that after the collision, the body that was moving (m1) will start moving with velocity v2 and the mass m2 will start moving with velocity v1. So there will be an exchange of velocities.

Now suppose one of the masses, say m2, was at rest. In that case after the collision the moving body, m1, will come to rest and the body that was at rest, m2, will start moving with the velocity that m1 had before the collision. That is that all of these observations are for an elastic collision.
This phenomenon is demonstrated by Newton's cradle, one of the best known examples of conservation of momentum, a real life example of this special case.

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.