In classical mechanics, momentum (pl. momenta; SI unit kg·m/s, or, equivalently, N·s) is the product of the mass and velocity of an object ( p = m v ). In relativistic mechanics, this quantity is multiplied by the Lorentz factor. Momentum is sometimes referred to as linear momentum to distinguish it from the related subject of angular momentum . Linear momentum is a vector quantity, since it has a direction as well as a magnitude. Angular momentum is a pseudovector quantity because it gains an additional sign flip under an improper rotation. The total momentum of any group of objects remains the same unless outside forces act on the objects (law of conservation of momentum).Momentum is a conserved quantity, meaning that the total momentum of any closed system (one not affected by external forces) cannot change. Although originally seen to be due to Newton's laws, this law is also true in special relativity, and with appropriate definitions a (generalized) momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory, and general relativity.
History of the concept
Mōmentum was not merely the motion, which was mōtus , but was the power residing in a moving object, captured by today's mathematical definitions. A mōtus , "movement", was a stage in any sort of change, while velocitas , "swiftness", captured only speed. The concept of momentum in classical mechanics was originated by a number of great thinkers and experimentalists. The first of these was Ibn Sina (Avicenna) circa 1000, during the Islamic Renaissance who referred to impetus as proportional to the mass times the velocity.
René Descartes believed that the total "quantity of motion" in the universe is conserved, where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more importantly he believed that it is speed rather than velocity that is conserved. So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion. Galileo, later, in his Two New Sciences, used the Italian word "impeto."
The question has been much debated as to what Isaac Newton contributed to the concept. The answer is apparently nothing, except to state more fully and with better mathematics what was already known. Yet for scientists, this was the death knell for Aristotelian physics and supported other progressive scientific theories (i.e., Kepler's laws of planetary motion). Conceptually, the first and second of Newton's Laws of Motion had already been stated by John Wallis in his 1670 work, Mechanica sive De Motu, Tractatus Geometricus : "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result". Wallis uses momentum and vis for force. Newton's Philosophiæ Naturalis Principia Mathematica , when it was first published in 1686, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines quantitas motus , "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum. Thus when in Law II he refers to mutatio motus , "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion. It remained only to assign a standard term to the quantity of motion. The first use of "momentum" in its proper mathematical sense is not clear but by the time of Jenning's Miscellanea in 1721, four years before the final edition of Newton's Principia Mathematica , momentum M or "quantity of motion" was being defined for students as "a rectangle", the product of Q and V where Q is "quantity of material" and V is "velocity", s/t.
Some languages, such as French and Italian, still lack a single term for momentum, and use a phrase such as the literal translation of "quantity of motion".
Linear momentum of a particle
If an object is moving in any reference frame, then it has momentum in that frame. It is important to note that momentum is frame dependent. That is, the same object may have a certain momentum in one frame of reference, but a different amount in another frame. For example, a moving object has momentum in a reference frame fixed to a spot on the ground, while at the same time having 0 momentum in a reference frame attached to the object's center of mass.
The amount of momentum that an object has depends on two physical quantities: the mass and the velocity of the moving object in the frame of reference. In physics, the usual symbol for momentum is a bold p (bold because it is a vector); so this can be written
where p is the momentum, m is the mass and v is the velocity.
Example: a model airplane of 1 kg traveling due north at 1 m/s in straight and level flight has a momentum of 1 kg m/s due north measured from the ground. To the dummy pilot in the cockpit it has a velocity and momentum of zero.
According to Newton's second law, the rate of change of the momentum of a particle is proportional to the resultant force acting on the particle and is in the direction of that force. The derivation of force from momentum is given below, however because mass is constant the second term of the derivative is 0 so it is ignored.
or just simply
where F is understood to be the resultant.
Example: a model airplane of 1 kg accelerates from rest to a velocity of 1 m/s due north in 1 s. The thrust required to produce this acceleration is 1 newton. The change in momentum is 1 kg m/s. To the dummy pilot in the cockpit there is no change of momentum. Its pressing backward in the seat is a reaction to the unbalanced thrust, shortly to be balanced by the drag.
Linear momentum of a system of particles
Relating to mass and velocity
The linear momentum of a system of particles is the vector sum of the momenta of all the individual objects in the system:
where P is the total momentum of the particle system, m i and v i are the mass and the velocity vector of the i -th object, and n is the number of objects in the system.
It can be shown that, in the center of mass frame the momentum of a system is zero. Additionally, the momentum in a frame of reference that is moving at a velocity v cm with respect to that frame is simply:
where:
This is known as Euler's first law.
Relating to force - General equations of motion
The linear momentum of a system of particles can also be defined as the product of the total mass
of the system times the velocity of the center of mass
This is commonly known as Newton's second law.
For a more general derivation using tensors, we consider a moving body (see Figure), assumed as a continuum, occupying a volume
at a time
, having a surface area
, with defined traction or surface forces
acting on every point of the body surface, body forces
per unit of volume on every point within the volume
prescribed throughout the body. Following the previous equation, The linear momentum of the system is:
By definition the stress vector is
, then
Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives (we denote
as the differential operator):
Now we only need to take care of the right side of the equation. We have to be careful, since we c
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