Impulse (physics)

In classical mechanics, impulse (symbolized by Template:Math or Imp) is the change in momentum of an object. If the initial momentum of an object is Template:Math, and a subsequent momentum is Template:Math, the object has received an impulse Template:Math:

$$\mathbf{𝐉}={\mathbf{𝐩}}_2-{\mathbf{𝐩}}_1.$$

Momentum is a vector quantity, so impulse is also a vector quantity: $$\sum \mathbf{𝐅}\times \mathrm{\Delta }t=\mathrm{\Delta }\mathbf{𝐩}.$$ Newton’s second law of motion states that the rate of change of momentum of an object is equal to the resultant force Template:Mvar acting on the object: $$\mathbf{𝐅}=\frac{{\mathbf{𝐩}}_2-{\mathbf{𝐩}}_1}{\mathrm{\Delta }t},$$

so the impulse Template:Mvar delivered by a steady force Template:Mvar acting for time Template:Math is: $$\mathbf{𝐉}=\mathbf{𝐅}\mathrm{\Delta }t.$$

The impulse delivered by a varying force acting from time Template:Mvar to Template:Mvar is the integral of the force Template:Mvar with respect to time: $$\mathbf{𝐉}=\underset{a}{\overset{b}{\int }}\mathbf{𝐅}\mathrm{d}t.$$

The SI unit of impulse is the newton second (N⋅s), and the dimensionally equivalent unit of momentum is the kilogram metre per second (kg⋅m/s). The corresponding English engineering unit is the pound-second (lbf⋅s), and in the British Gravitational System, the unit is the slug-foot per second (slug⋅ft/s).

Mathematical derivation in the case of an object of constant mass

File:Happy vs. Sad Ball.webm

The impulse delivered by the "sad" ball is Template:Math, where Template:Math is the speed upon impact. To the extent that it bounces back with speed Template:Math, the "happy" ball delivers an impulse of Template:Math.

Impulse Template:Math produced from time Template:Math to Template:Math is defined to beTemplate:Sfn where Template:Math is the resultant force applied from Template:Math to Template:Math.

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

Therefore, where Template:Math is the change in linear momentum from time Template:Math to Template:Math. This is often called the impulse-momentum theorem (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:

where

• Template:Math is the resultant force applied,
• Template:Math and Template:Math are times when the impulse begins and ends, respectively,
• Template:Mvar is the mass of the object,
• Template:Math is the final velocity of the object at the end of the time interval, and
• Template:Math 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".

Variable mass

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.

See also

• 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

External links

• Dynamics