Impulse (physics)

Impulse J produced from time t₁tot₂ is defined to be^[2] $${\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t,}$$  where F is the resultant force applied from t₁tot₂.

From Newton's second law, force is related to momentum pby$${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}.}$$ 

Therefore, $${\displaystyle {\begin{aligned}\mathbf {J} &=\int _{t_{1}}^{t_{2}}{\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\mathrm {d} t\\&=\int _{\mathbf {p} _{1}}^{\mathbf {p} _{2}}\mathrm {d} \mathbf {p} \\&=\mathbf {p} _{2}-\mathbf {p} _{1}=\Delta \mathbf {p} ,\end{aligned}}}$$  where Δp is the change in linear momentum from time t₁tot₂. This is often called the impulse-momentum theorem^[3] (analogous to the work-energy theorem).

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: $${\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t=\Delta \mathbf {p} =m\mathbf {v_{2}} -m\mathbf {v_{1}} ,}$$ 

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.

Impulse has the same units and dimensions (MLT⁻¹) as momentum. In the International System of Units, these are kg⋅m/s = N⋅s. In English engineering units, they are slug⋅ft/s = lbf⋅s.

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 videogame physics engines). Additionally, in rocketry, the term "total impulse" is commonly used and is considered synonymous with the term "impulse".

The application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse. This fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse (or nozzle exhaust velocity) and the vehicle's propellant-mass ratio.

• 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

• Dirac delta function, mathematical abstraction of a pure impulse
• One-way wave equation

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

• Dynamics