Momentum

In Newtonian mechanics, momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and is its velocity, then the object's momentum is:
In the International System of Units, the unit of measurement of momentum is the kilogram metre per second, which is dimensionally equivalent to the newton-second.
Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame of reference, it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total momentum does not change. Momentum is also conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. It is an expression of one of the fundamental symmetries of space and time: translational symmetry.
Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is generalized momentum, and in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function. The momentum and position operators are related by the Heisenberg uncertainty principle.
In continuous systems such as electromagnetic fields, fluid dynamics and deformable bodies, a momentum density can be defined as momentum per volume and is said to satisfy a conservation law. A continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.

Classical

Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations.

Single particle

The momentum of a particle is conventionally represented by the letter. It is the product of two quantities, the particle's mass and its velocity :
The unit of momentum is the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second.
Being a vector, momentum has magnitude and direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground.

Many particles

The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses and, and velocities and, the total momentum is
The momenta of more than two particles can be added more generally with the following:
A system of particles has a center of mass, a point determined by the weighted sum of their positions:
If one or more of the particles is moving, the center of mass of the system will generally be moving as well. If the total mass of the particles is, and the center of mass is moving at velocity, the momentum of the system is:
This is known as Euler's first law.

Relation to force

If the net force applied to a particle is constant, and is applied for a time interval, the momentum of the particle changes by an amount
In differential form, this is Newton's second law; the rate of change of the momentum of a particle is equal to the instantaneous force acting on it,
If the net force experienced by a particle changes as a function of time,, the change in momentum between times and is
Impulse is measured in the derived units of the newton second or dyne second
Under the assumption of constant mass, it is equivalent to write
hence the net force is equal to the mass of the particle times its acceleration.
Example: A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3 newtons due north. The change in momentum is 6 kg⋅m/s due north. The rate of change of momentum is 3 /s due north which is numerically equivalent to 3 newtons.

Conservation

In a closed system the total momentum remains constant. This fact, known as the law of conservation of momentum, is implied by Newton's laws of motion. Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If the particles are numbered 1 and 2, the second law states that and. Therefore,
with the negative sign indicating that the forces oppose. Equivalently,
If the velocities of the particles are and before the interaction, and afterwards they are and, then
This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds to zero, so the total change in momentum is zero. The conservation of the total momentum of a number of interacting particles can be expressed as
This conservation law applies to all interactions, including collisions and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, such as in relativistic situations and those involving electrodynamics.

Dependence on reference frame

Momentum is a measurable quantity, and the measurement depends on the frame of reference. For example, if an aircraft of mass 1,000kg flies through the air at a speed of 50m/s, its momentum is, but if the aircraft flies into a headwind of 5m/s, its speed relative to the Earth's surface is only 45m/s and its momentum drops to. Both calculations are equally correct. In both frames of reference, any change in momentum will be found to be consistent with the relevant laws of physics.
Suppose is a position in an inertial frame of reference. From the point of view of another frame of reference, moving at a constant speed relative to the other, the position changes with time as
This is called a Galilean transformation.
If a particle is moving at speed in the first frame of reference, in the second, it is moving at speed
Since does not change, the second reference frame is also an inertial frame and the accelerations are the same:
Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or Galilean invariance.
A change of reference frame can often simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen where one particle begins at rest. Another commonly used reference frame is the center of mass frame – one that is moving with the center of mass. In this frame, the total momentum is zero.

Application to collisions

If two particles, each of known momentum, collide and coalesce, the law of conservation of momentum can be used to determine the momentum of the coalesced body. If the outcome of the collision is that the two particles separate, the law is not sufficient to determine the momentum of each particle. If the momentum of one particle after the collision is known, the law can be used to determine the momentum of the other particle. Alternatively if the combined kinetic energy after the collision is known, the law can be used to determine the momentum of each particle after the collision. Kinetic energy is usually not conserved. If it is conserved, the collision is called an elastic collision; if not, it is an inelastic collision.

Elastic collisions

An elastic collision is one in which no kinetic energy is transformed into heat or some other form of energy. Perfectly elastic collisions can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps the objects apart. A slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls is a good example of an almost totally elastic collision, due to their high rigidity, but when bodies come in contact there is always some dissipation.
A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are and before the collision and and after, the equations expressing conservation of momentum and kinetic energy are:
A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass, one stationary and one approaching the other at a speed . The center of mass is moving at speed and both bodies are moving towards it at speed. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by
In general, when the initial velocities are known, the final velocities are given by
If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.