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 weight times velocity. René Descartes
later referred to momentum as mass times velocity and as the fundamental force of motion.
This allowed Descartes to maintain that mass and velocity are fundamental and conserved,
everywhere and all the time.
• 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, 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.
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
• 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, mi and vi 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
vcm with respect to that frame is simply:
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. 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, and a velocity field prescribed
throughout the body. Following the
previous equation, The linear
momentum of the system is:
By definition the stress vector is
Motion of a material body