# Impulse (physics)

(Redirected from Impulse momentum theorem)
Common symbols J, Imp N · s = kg · m/s

In classical mechanics, impulse (symbolized by J or Imp[1]) is the change in linear momentum of a body. It may be defined or calculated as the product of the average force multiplied by the time over which the force is exerted.[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 quantity of an impulse is average force × time interval, or in shorthand notation:

$J = F_{average} (t_2 - t_1)$

If a force is not of constant magnitude over time, the impulse is the integral of the magnitude of the resultant force (F) with respect to time:

$J = \int F dt$

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

Impulse J produced from time t1 to t2 is defined to be[3]

$\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, dt$

where F is the resultant force applied from t1 to t2.

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

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

Therefore

\begin{align} \mathbf{J} &= \int_{t_1}^{t_2} \frac{d\mathbf{p}}{dt}\, dt \\ &= \int_{p_1}^{p_2} d\mathbf{p} \\ &= \mathbf{p_2} - \mathbf{p_1} = \Delta \mathbf{p} \end{align}

where Δp is the change in linear momentum from time t1 to t2. This is often called the impulse-momentum theorem.[4]

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:

$\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, dt = \Delta\mathbf{p} = m \mathbf{v_2} - m \mathbf{v_1}$

where

F is the resultant force applied,
t1 and t2 are times when the impulse begins and ends, respectively,
m is the mass of the object,
v2 is the final velocity of the object at the end of the time interval, and
v1 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).

The longer the club remains in contact with the ball the greater the impulse

Impulse has the same units (in the International System of Units, kg·m/s = N·s) and dimensions (MLT−1) as momentum.

## Variable mass

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

## Notes

1. ^ 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.
2. ^ Impulse of Force, Hyperphysics
3. ^ Hibbeler, Russell C. (2010). Engineering Mechanics (12th ed.). Pearson Prentice Hall. p. 222. ISBN 0-13-607791-9.
4. ^ See, for example, section 9.2, page 257, of Serway (2004).

## Bibliography

• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
• Tippler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.
1. ^ 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.
2. ^ Impulse of Force, Hyperphysics
3. ^ Hibbeler, Russell C. (2010). Engineering Mechanics (12th ed.). Pearson Prentice Hall. p. 222. ISBN 0-13-607791-9.
4. ^ See, for example, section 9.2, page 257, of Serway (2004).