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Physics LibreTexts

7.3: Momentum Conservation Model Summary

One way of summarizing the main ideas in a model/approach is to list the \((1)\) constructs, i.e., the “things” or ideas that are “used” in the model, \((2)\) the relationships–in mathematical or sentence form–that connect the constructs in meaningful ways, and \((3)\) the ways of representing the relationships. During your study of the model/approach you should have developed a good understanding of the meaning of each of the constructs. Some of these constructs probably start out as nothing but memorized definitions, but eventually take on a deeper meaning. The relationships might also start out as nothing more meaningful than a simple equation relating some of the constructs, e.g., \(J = \sum F \Delta t = \Delta p\). By the time you finish this part of the course, however, you should understand this particular relationship, for example, as expressing one of the most fundamental, universal, and widely applicable principles in all of physics. Developing a deep and rich understanding of the relationships in a model/approach comes slowly. It is absolutely not something you can memorize. This understanding comes only with repeated mental effort over a period of time. A good test you can use to see if you are “getting it” is whether you can tell a full story about each of the relationships. It is the meaning behind the equations, behind the simple sentence relationships, that is important for you to acquire. With this kind of understanding, you can apply a model/approach to the analysis of phenomena you have not thought about before. You can reason with the model.

Listed here are the major, most important constructs, relationships, and representations of the momentum conservation model.

Constructs

Velocity, \(v\)

Momentum, \(p\)

Net Force, \(\sum F\)

Impulse, \(J\)

Newton’s 3rd law

Conservation of momentum

Relationships

The velocity is the time derivative of the displacement:

\(v =\frac{ dr}{ dt} ~or ~v_{average} = \frac{\Delta r}{ \Delta t} \tag{7.3.1}\)

The linear momentum of an object measured in some coordinate system is simply the product of the object’s mass and velocity:

\[p = mv \tag{7.3.2} \]

The linear momentum of a system of particles is the vector sum of the individual momenta:

\[p_{system} = \sum p_i \tag{7.3.3} \]

The net force acting on an object (physical system) is the vector sum of all forces acting on that object (physical system) due to the interactions with other objects (physical systems).

\[\sum F_A = F_{B~ on~ A} + F_{C~ on~ A} + F_{D ~on~ A} + … \tag{7.3.4} \]

The impulse of the total (or net) external force acting on a system equals the product of the average force and the time interval during which the force acted.

\[Net~ Impulse_{ext} = J = \sum F_{avg~ ext} \Delta t = \int \sum F_{ext}(t) dt \tag{7.3.5} \]

The force (impulse) exerted by object A on object B is equal and opposite to the force (impulse) exerted by object B on object A.

\[F_{A ~on~ B} = – F_{B~ on~ A}~~ and~~ J_{A ~on~ B} = – J_{B ~on ~A} \tag{7.3.6} \]

Conservation of Linear Momentum

If the net external impulse acting on a system is zero, then there is no change in the total linear momentum of that system; otherwise, the change in momentum is equal to the net external impulse.

\[Net~ Impulse_{ext} = J = \int \sum F_{ext}(t) dt = p_f - p_i = \Delta p_{system} \tag{7.3.7} \]

Collisions

The momentum of the system of objects (particles) remains constant if the external impulses are negligible. This is true whether the collision is elastic or inelastic

\[p_{tot_i} = p_{tot_f} \tag{7.3.8} \]

If a collision is elastic, then none of the mechanical energy is transferred to bond or thermal energies and both the total mechanical energy (all kinetic and elastic energies) and the momentum remain constant.

\[(mechanical ~energy)_i = (mechanical~ energy)_f~ and~ p_{tot_i} = p_{tot_f} \tag{7.3.9} \]

Representations

Graphical representation of all vector quantities and (vector relationships) as arrows whose length is proportional to the magnitude of the vector and whose direction is in the direction of the vector quantity.

Algebraic vector equations. Vectors denoted as bold symbols or with small arrows over the symbol.

Component algebraic equations, one equation for each of the three independent directions.

A useful way to organize and use the representations of the various quantities that occur in phenomena involving momentum, change in momentum, and impulse and forces is a momentum chart. The momentum chart, like an energy-system diagram, helps us keep track of what we know about the interaction, as well as helping us see what we don’t know.

The boxes are to be filled in with scaled arrows representing the various momenta and changes in momenta.

Closed System

Typically used for collisions/interactions involving two or more objects.

For total system: \(\Delta p = 0\)

For each object: \(p_i + \Delta p = p_f\)

(written as component equations, if useful)

Write expressions for each momentum vector, such as \(p = mv\)

Open System

Typically used when the phenomenon involves a net impulse acting on the system.

For total system: \(\Delta p = J \)

     \(p_i + \Delta p = p_f\)

(and for component equations, if useful)

Write expressions for each momentum vector, such as \(p = mv\)

Below the momentum chart draw a force diagram for the object. The net force gives the direction of the impulse and \(\Delta p\).