Momentum    [!copyright and waiver^SectionEdit.png!]

Introduction to Linear Momentum

Although Newton is famous for the law F-ma, he actually stated his Second Law in terms of momentum and its change due to impressed forces or impulses (the time integral of the force).   He defined Momentum as
The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. - DEFINITiON II, Principia (Motte and Cajori).  

Momentum

Mass times velocity, or, equivalently, a quantity whose time rate of change is equal to the net force applied to a system.

Motivation for Concept

Forces are actions which cause a change in the velocity of an object, but a given application of force will have very different results when applied to objects of very different mass. Consider the force imparted by a baseball player swinging a bat. When delivered to a baseball, the change in velocity is dramatic. A 95 mph fasball might be completely reversed and exit the bat moving 110 mph in the other direction. When delivered to a car, however, the change in velocity is miniscule. A car moving 95 mph will not be slowed noticeably by the action of a bat. Thus, although the change in velocity of a system is proportional to the force applied, it is not equal to the force applied. To define a quantity whose rate of change is equal to the force applied, we must include both the mass and velocity of the system subject to the force.

Mathematical Definition

Momentum of a Point Particle

The momentum (p) of a point particle with mass m and velocity v is defined as:

Momentum of a System

For a system composed of N objects which are approximated as point particles with their position specified by the objects' centers of mass, the system momentum is defined as the vector sum of the momentum of the constituents:

This definition is completely equivalent to

where M^sys is the total mass of the system and v^CM is the velocity of the system's center of mass.

Momentum and Newton's Laws

Momentum and Newton's Second Law

One way of stating Newton's Second Law is that the rate of change of a system's momentum is equal to the vector sum of the forces applied to the object:

Momentum and Newton's Third Law

By Newton's 3rd Law, internal forces cancel from the vector sum above, leaving only the contribution of external forces:

Momentum and Impulse

The integrated change in momentum can be found explicitly by using the net external impulse (J^ext):

Conservation of Momentum

Conditions for True Conservation

In the absence of a net external force, the momentum of a system is constant:

This equation is normally broken up to explicitly show the system constituents and the vector components:

Approximate Conservation in Collisions

Because the change in momentum is proportional to the impulse, which involves a time integral, for instantaneous events:

For approximately instantaneous events such as collisions, it is often reasonable to approximate the external impulse as zero by considering a system composed of all the objects involved in the collision. The key to the utility of this assumption is that often during collisions the change in momentum of any individual system constituent being analyzed is dominated by the internal collision forces (the external forces make a negligible contribution to that constituent's change in momentum during the collision).

Newton emphasizes the linearity of momentum with both mass and velocity.  He specifies our sum over different bodies by further stating that the motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the momentum is double; with twice the velocity also, it is quadruple.

Newton's Second Law is the fundamental law of change for momentum,
Frac

=\Sigma_j\vec

Where the sum goes over all the forces acting on any body in the system.

The mathematical relationship between force and momentum, or, for systems with constant mass, the relationship between force and acceleration.

Motivation for Concept

When you push, kick or use some other means to apply force to an object, its velocity will change. It is of value to be able to quantitatively define the strength of such a push, kick or other force by examining the effects of the force on the object which is the target. Such a quantitative understanding of force is the basis of the science of dynamics.

Statement of the Law

Newton's Statement

"A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed." (The Principia by I. Newton, translated by I.B. Cohen and A. Whitman.)

Modern Statement

The modern form of the Law which is perhaps most consistent with Newton's statement is the integral formulation:

It is more common to express the Law in a differential form:

and, since it is rare to consider a system with changing mass, this form is often reduced to:

Use of the Law in Problem Solving

Form of the Second Law for Multiple Impressed Forces

When more than one force is impressed, the change in momentum is proportional to the vector sum of the forces. Thus, Newton's 2nd Law is usually expanded to state:

"Writing" Newton's Second Law

When a physics teacher or student says they are "writing Newton's 2nd Law" for a system, the form used should be that of the previous subsection, but expressed as several equations separated by vector component:

Free Body Diagrams

Free body diagrams are pictorial guides to the specific form of Newton's 2nd Law for a given system. Drawing an accurate free body diagram shows visually what forces should be included in the statement of the law, and also gives information about the sign of the vector components of these forces.

Technically, the concept of linear momentum applies only to collections of point objects.  The momentum of a rigid body is simply the sum of the momentum of each of the atoms in the body, which turns out to be the body's mass times its center of mass velocity.  The momentum of a system composed of many rigid bodies and point particles is then the sum of their individual momenta, which again can be expressed as the total mass of this system times the velocity of the system's center of mass.  This is true, and a valuable concept, especially when the system is composed of disparate objects going in different directions - for example a system of two cars about to have a collision, or after they have just had a collision.  

Introduction to the Model Description and Assumptions This model is generally applicable (assuming knowledge of the external forces and system constituents). The model is especially useful when describing the momentum of systems where external forces are absent (system momentum will be constant) or estimating the force in a process that occurs in a very short time interval such as collisions (impulse will be easier to determine than force). Learning Objectives Students will be assumed to understand this model who can: Define the momentum of a point particle. Define impulse in terms of momentum or in terms of force. Give an expression for the time-average force on a system in terms of its momentum. Calculate the net external force on a system containing several objects. Describe the conditions required for the momentum of a system to be conserved. Describe the system that must be considered and the assumptions that lead to approximate conservation of momentum in a collision. Relevant Definitions S.I.M. Structure of the Model Compatible Systems The system must be effectively composed of point particles, though rigid bodies may be treated as point particles with positions specified by the center of mass positions of the rigid body when this model is used. Relevant Interactions Only external forces need be considered, since internal forces do not change the system's momentum. Laws of Change Mathematical Representation Differential Form Integral Form Diagrammatic Representations initial-state final-state diagram free body diagram Relevant Examples Examples Involving Constant Momentum Page: Out of Bounds — A typical perfectly inelastic collision in 2-D. example_problem constant_momentum 2d_collision totally_inelastic Page: Let it Rain — Analyzing a continuous momentum flux (falling water). example_problem constant_momentum momentum_force Page: Cue the Right-Angle Bracket — Learn a valuable shortcut for dealing with a specific kind of elastic collision. example_problem constant_momentum 2d_collision elastic_collision elastic Page: Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision. example_problem constant_momentum 1d_collision impulse average_force totally_inelastic Page: A Walk on the Pond — How far will two children slide after a perfectly inelastic collision? example_problem 2d_collision totally_inelastic constant_momentum kinetic_friction non-conservative_work Page: Momentum Transport — Analyzing a continuous momentum flux (water from a hose). example_problem dynamics 1d_motion constant_momentum impulse 1d_collision momrntum_force momentum_force Examples Involving Impulse Page: Watch Your Head — Consider the impulse and average force delivered to the head of a player performing a "header" in soccer. example_problem impulse projectile_motion average_force Page: Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision. example_problem constant_momentum 1d_collision impulse average_force totally_inelastic Page: Momentum Transport — Analyzing a continuous momentum flux (water from a hose). example_problem dynamics 1d_motion constant_momentum impulse 1d_collision momrntum_force momentum_force Examples Involving 1-D Collisions Page: Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision. example_problem constant_momentum 1d_collision impulse average_force totally_inelastic Page: Momentum Transport — Analyzing a continuous momentum flux (water from a hose). example_problem dynamics 1d_motion constant_momentum impulse 1d_collision momrntum_force momentum_force Examples Involving 2-D Collisions Page: Out of Bounds — A typical perfectly inelastic collision in 2-D. example_problem constant_momentum 2d_collision totally_inelastic Page: Cue the Right-Angle Bracket — Learn a valuable shortcut for dealing with a specific kind of elastic collision. example_problem constant_momentum 2d_collision elastic_collision elastic Page: A Walk on the Pond — How far will two children slide after a perfectly inelastic collision? example_problem 2d_collision totally_inelastic constant_momentum kinetic_friction non-conservative_work Examples Involving Elastic Collisions Page: Cue the Right-Angle Bracket — Learn a valuable shortcut for dealing with a specific kind of elastic collision. example_problem constant_momentum 2d_collision elastic_collision elastic Examples Involving Totally Inelastic Collisions Page: Out of Bounds — A typical perfectly inelastic collision in 2-D. example_problem constant_momentum 2d_collision totally_inelastic Page: Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision. example_problem constant_momentum 1d_collision impulse average_force totally_inelastic Page: A Walk on the Pond — How far will two children slide after a perfectly inelastic collision? example_problem 2d_collision totally_inelastic constant_momentum kinetic_friction non-conservative_work Examples Involving Continuous Momentum Flux Page: Let it Rain — Analyzing a continuous momentum flux (falling water). example_problem constant_momentum momentum_force Page: Momentum Transport — Analyzing a continuous momentum flux (water from a hose). example_problem dynamics 1d_motion constant_momentum impulse 1d_collision momrntum_force momentum_force All Examples Using this Model Page: Momentum Transport — Analyzing a continuous momentum flux (water from a hose). example_problem dynamics 1d_motion constant_momentum impulse 1d_collision momrntum_force momentum_force Page: Let it Rain — Analyzing a continuous momentum flux (falling water). example_problem constant_momentum momentum_force Page: Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision. example_problem constant_momentum 1d_collision impulse average_force totally_inelastic Page: Watch Your Head — Consider the impulse and average force delivered to the head of a player performing a "header" in soccer. example_problem impulse projectile_motion average_force Page: Out of Bounds — A typical perfectly inelastic collision in 2-D. example_problem constant_momentum 2d_collision totally_inelastic Page: Cue the Right-Angle Bracket — Learn a valuable shortcut for dealing with a specific kind of elastic collision. example_problem constant_momentum 2d_collision elastic_collision elastic Page: A Walk on the Pond — How far will two children slide after a perfectly inelastic collision? example_problem 2d_collision totally_inelastic constant_momentum kinetic_friction non-conservative_work Search Search

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