# Impulse (physics)

Template:Infobox physical quantity Template:Classical mechanics

In classical mechanics, **impulse** (symbolized by **J** or **Imp**^{[1]}) is the product of a force, F, and the time, t, for which it acts. The impulse of a force acting for a given time interval is equal to the change in linear momentum produced over that interval.^{[2]} Impulse is a vector quantity since it is the result of integrating force, a vector quantity, over time. The SI unit of impulse is the newton second (N·s) or, in base units, the kilogram meter per second (kg·m/s).

A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly.

The impulse is the integral of the resultant force (*F*) with respect to time:

## Mathematical derivation in the case of an object of constant mass

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Impulse **J** produced from time *t*_{1} to *t*_{2} is defined to be^{[4]}

where **F** is the resultant force applied from *t*_{1} to *t*_{2}.

From Newton's second law, force is related to momentum **p** by

Therefore

where Δ**p** is the change in linear momentum from time *t*_{1} to *t*_{2}. This is often called the impulse-momentum theorem.^{[5]}

As a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant:

where

**F**is the resultant force applied,*t*_{1}and*t*_{2}are times when the impulse begins and ends, respectively,*m*is the mass of the object,**v**_{2}is the final velocity of the object at the end of the time interval, and**v**_{1}is the initial velocity of the object when the time interval begins.

The term "impulse" is also used to refer to a fast-acting force or impact. This type of impulse is often *idealized* so that the change in momentum produced by the force happens with no change in time. This sort of change is a step change, and is not physically possible. However, this is a useful model for computing the effects of ideal collisions (such as in game physics engines).

Impulse has the same units (in the International System of Units, kg·m/s = N·s) and dimensions (*M* *L* *T*^{−1}) as momentum.

## Variable mass

The application of Newton's second law for variable mass leads to the Tsiolkovsky rocket equation.

## See also

- Specific impulse
- Wave–particle duality defines the impulse of a wave collision. The preservation of momentum in the collision is then called phase matching. Applications include:
- Compton effect
- nonlinear optics
- Acousto-optic modulator
- Electron phonon scattering

## Notes

- ↑ Beer, F.P., E.R. Johnston, Jr., D.F. Mazurek, P.J. Cornwell, and E.R. Eisenberg. (2010).
*Vector Mechanics for Engineers; Statics and Dynamics.*9th ed. Toronto: McGraw-Hill. - ↑ Impulse of Force, Hyperphysics
- ↑ http://www.materialseducation.org/educators/mated-modules/docs/Property_Differences_in_Polymers.pdf
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ See, for example, section 9.2, page 257, of Serway (2004).

## Bibliography

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## External links

Template:Classical mechanics derived SI units

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