Impulse

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Impulse

By Wikipedia,
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http://en.wikipedia.org/wiki/Impulse

In classical mechanics, an impulse is defined as the integral of a force with respect to time. When a force is applied to a rigid body it changes the momentum of that body. A small force applied for a long time can produce the same momentum change as a large force applied briefly, because it is the product of the force and the time for which it is applied that is important.

Mathematical derivation

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

where

I is impulse (frequently marked J),

F is the force, and

dt is an infinitesimal amount of time.

t₁ and t₂ denote a time interval

A simple derivation using Newton's second law yields:

$\mathbf{I} = \int_{t_1}^{t_2} \frac{d\mathbf{p}}{dt}\, dt$

$\mathbf{I} = \int_{t_1}^{t_2} d\mathbf{p}$

$\mathbf{I} = \Delta \mathbf{p}$

where

p is momentum

This is often called the impulse-momentum theorem.

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

$\mathbf{I} = \mathbf{F}\Delta t = m \Delta \mathbf{v} = \Delta\mathbf{p}$

where

F is the constant total net force applied,

Δ t

is the time interval over which the force is applied,

m is the constant mass of the object,

Δv is the change in velocity produced by the force in the considered time interval, and

mΔv = Δ(mv) is the change in linear momentum.

It is often the case that not just one but both of these two quantities vary.

In the technical sense, impulse is a physical quantity, not an event or force. The term "impulse" is also used to refer to a fast-acting force. 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. This is a useful model for certain purposes, such as computing the effects of ideal collisions, especially in game physics engines.

Impulse has the same units and dimensions as momentum (kg m/s = N·s).

Impulse can be calculated using the equation:

$\mathbf{F}\Delta t = \Delta\ p$

$\Delta\ p$ can be calculated, if initial and final velocities are known,

$\mathbf{F}\Delta t = mv_1 - mv_0$

where

F is the constant total net force applied,

t

is the time interval over which the force is applied,

m is the constant mass of the object,

v 1

is the final velocity of the object at the end of the time interval, and

v 0

is the initial velocity of the object when the time interval begins.

Bibliography

Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed. ed.). Brooks/Cole. ISBN 0-534-40842-7.
Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed. ed.). W. H. Freeman. ISBN 0-7167-0809-4.

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Published in July 2009.

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Mathematical derivation

See also

Bibliography