Force & Impulse: A Summary of Equations & Concepts

[Greg Bernhardt]

Definition/Summary

Force = impulse per time: $\mathbf{F}\ =\ d\mathbf{I}/dt$.

For constant force, impulse = force times time: $\mathbf{I}\ =\ \mathbf{F}\,\Delta t$ (by comparison, work done = force "dot" distance: $W\ =\ \mathbf{F}\cdot \Delta\mathbf{s}$).

For varying force, impulse is the integral of force over time: $\mathbf{I}\ =\ \int\mathbf{F}\,dt$ (and work done is the integral of force over distance: $W\ =\ \int\mathbf{F}\cdot d\mathbf{s}$).

Newton's second law (force = rate of change of momentum: $\mathbf{F}\ =\ d(m\mathbf{v})/dt$) integrated over time becomes:

impulse = total change of momentum: $\mathbf{I}\ =\ \int d(m\mathbf{v})/dt\,dt\ =\ \int d(m\mathbf{v})\ =\ \Delta(m\mathbf{v})$.

Impulse is a vector, with the same dimensions as momentum: $ML/T$, and is measured in units of Newton seconds ($N.s\text{, or }kg\,m\,s^{-1}$).

Equations

$$\mathbf{F}\ =\ \frac{d\mathbf{I}}{dt}$$

$$\mathbf{I}\ =\ \int\mathbf{F}\,dt$$

Impulse-momentum theorem:

$$\mathbf{I}\ =\ \int\frac{d(m\mathbf{v})}{dt}\,dt\ =\ \int d(m\mathbf{v})\ =\ \Delta(m\mathbf{v})$$

Extended explanation

Impulse is sometimes easier to measure:

When, for example, a bat hits a ball, it is is contact with the ball for a substantial time, and the force changes considerably during that time.

It is not usually practical to measure the instantaneous force, and it is not usually helpful to try to integrate it even if it is known, so instead the "total" force applied is measured, and that is the impulse.

Specific impulse:

Specific impulse is impulse per mass of propellant (fuel).

It equals change in momentum per mass of propellant, and so measures the efficiency of rocket and jet engines.

See http://en.wikipedia.org/wiki/Specific_impulse.

(This is rocket science! )​

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[AaronS]

Thanks for the great explanation! This is really helpful for understanding the concept of impulse. It's also interesting to learn about specific impulse and how it relates to rocket science.