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
Often in computational simulations of particles, a momentum variable is assigned to each particle. Such simulations usually occur in "time steps," or iterations of a loop representing a time interval. In each time step, the particles' momenta are updated according to the impulse-momentum theorem. Usually in this kind of simulation, the time steps are assumed to be small enough that the force does not significantly vary over the time step, allowing the force and time interval to be multiplied to yield the impulse. By adding this small impulse each time step, the sum approximates the time integral of force over the course of the simulation, like a Riemann sum.
The following is an example of a line of vPython responsible for updating the momentum of a particle according to the impulse-momentum theorem:
p = p + fnet*deltat
The simulation below is an example of a program that uses the momentum principle. It simulates the motion of a cart, represented by a rectangle, being blown by a gust of wind:
https://trinket.io/glowscript/ce43925647
For more information, see iterative prediction.
Examples
Simple
A 2kg particle is travelling in a northerly direction at 5m/s.