Impulse
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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.
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[edit] Mathematical derivation
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:
where
- p is momentum
This is often called the impulse-momentum theorem.[1]
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:
where
- F is the constant total net force applied,
- Δt 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:
can be calculated, if initial and final velocities are known,
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,
- v1 is the final velocity of the object at the end of the time interval, and
- v0 is the initial velocity of the object when the time interval begins.
[edit] See also
- Specific impulse
- Momentum
- Wave-particle duality defines an impulse for waves. The preservation of momentum at a collision is then called phase matching. Applications include:
- Compton effect
- nonlinear optics
- Acousto-optic modulator
- Umklapp scattering
- electron phonon scattering
[edit] Notes
- ^ See, for example, section 9.2, page 257, of Serway (2004).
[edit] 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.
