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

Impulse I produced from time t₁ to t₂ is defined to be

where F is the force applied dt denotes an infinitesimal amount of time.

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

Therefore

where Δp is the change in momentum from time t₁ to t₂. 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 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.

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.

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

Notes

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. 

External links and references

• Dynamics