Impulse and Momentum

This page defines impulse and describes its relationship to momentum.

The Main Idea

Impulse is a vector quantity describing both the nature and duration of a force. It is defined as the time integral of the net force vector: [math]\displaystyle{ \vec{J} = \int \vec{F}_{net}dt }[/math]. For constant forces, this simplifies to the product of the force vector and the time interval over which it is applied: [math]\displaystyle{ \vec{J} = \vec{F}_{net} \Delta t }[/math]. Impulse is represented by the letter [math]\displaystyle{ \vec{J} }[/math]. The most commonly used metric unit for impulse is the Newton*second.

People are interested in impulse primarily because of its relationship to momentum, as described by the impulse-momentum theorem. The theorem states that if an impulse is exerted on a system, the change in that system's momentum caused by the force is equal to the impulse: [math]\displaystyle{ \Delta \vec{p} = \vec{J} }[/math]. This works out dimensionally because the units for impulse are equivalent to the units for momentum. For example, the Newton*second is equivalent to the kilogram*meter/second because a Newton is defined as a kilogram*meter/second^2.

A Mathematical Model

The impulse-momentum theorem is a consequence of the momentum principle (see Newton's Second Law: the Momentum Principle). Below is its derivation:

[math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]

can be arranged to [math]\displaystyle{ d\vec{p} = \vec{F}_{net}dt }[/math].

Integrating both sides yields [math]\displaystyle{ \int d\vec{p} = \int \vec{F}_{net}dt }[/math]

which simplifies to [math]\displaystyle{ \Delta \vec{p} = \int \vec{F}_{net}dt = \vec{J} }[/math].

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

Be sure to show all steps in your solution and include diagrams whenever possible