View Source

Introduction to the Model

Description and Assumptions

System: Any. — Interactions: Any. — Note: Linear momentum evolves separately from angular momentum, so all system constituents are treated as point particles in this model.

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

\begin{large}\[ \vec{p} = m\vec{v}\]\end{large}

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

\begin{large}\[ \frac{d\vec{p}}{dt} = \sum \vec{F}^{\rm ext}\]\end{large}

Integral Form

\begin{large}\[ \vec{p}_{f} = \vec{p}_{i} + \sum \vec{J}^{\rm ext}_{fi} = \vec{p}_{i} + \int_{t_{i}}^{t_{f}} \sum \vec{F}^{\rm ext}\:dt  \]
\end{large}

Diagrammatic Representations

Relevant Examples

Examples Involving Constant Momentum

Examples Involving Impulse

Examples Involving 1-D Collisions

Examples Involving 2-D Collisions

Examples Involving Elastic Collisions

Examples Involving Totally Inelastic Collisions

Examples Involving Continuous Momentum Flux

All Examples Using this Model

Modeling Applied to Problem Solving > Momentum and External Force > skaters.jpg

Photos courtesy:

Wikimedia Commons by user No-w-ay in collaboration with H. Caps
Wikimedia Commons by user Uwe Langer
NASA Johnson Space Center (NASA-JSC)

{html}
<script type="text/javascript">
var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www.");
document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E"));
</script>
<script type="text/javascript">
try {
var pageTracker = _gat._getTracker("UA-11762009-2");
pageTracker._trackPageview();
} catch(err) {}</script>
{html}