Impulse (physics)

In classical mechanics, impulse (symbolized by Template:Math or Imp) is the change in momentum of an object. It is most often used to describe forces which act over short time periods, specifically in the case of impacts and collisions, for which it gets its namesake. Impulse is a vector quantity, meaning it has both a magnitude, which describes the amount by which the momentum changed, and a direction, which describes the direction in which the momentum changed.

For a force acting over a short time, the impulse is often idealized so that the change in momentum produced by the force is modelled as happening instantaneously. 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".

Impulse has the same units and dimensions Template:Nowrap as momentum. The SI unit of impulse is the newton-second (Template:Nowrap), and the dimensionally equivalent unit of momentum is the kilogram-metre per second (Template:Nowrap). The corresponding English engineering unit is the pound-second (Template:Nowrap), and in the British Gravitational System, the unit is the slug-foot per second (Template:Nowrap).

Definition

If, during a period of time, the initial momentum of an object is Template:Mvarb, and the final momentum is Template:Mvarb then the net impulse Template:Mvarb on the object is defined as, $$𝐉={𝐩}_{\mathrm{f}}-{𝐩}_{\mathrm{i}}.$$

By Newton's second law of motion, the rate of change of momentum of an object is equal to the resultant force Template:Math acting on the object: $$𝐅=\frac{{𝐩}_{\mathrm{f}}-{𝐩}_{\mathrm{i}}}{\mathrm{\Delta }t},$$ so the impulse Template:Math delivered by a constant force Template:Math acting for time Template:Math is: $$𝐉=𝐅\mathrm{\Delta }t.$$

Integral definition

File:Happy vs. Sad Ball.webm In the continuous time case, the impulse delivered onto a constant mass by a varying force acting from time Template:Math to Template:Math is defined to be the integral of the force Template:Math with respect to timeTemplate:Sfn: $$𝐉={\int }_{t_1}^{t_2}𝐅\mathrm{d}t.$$From Newton's second law, force is related to momentum Template:Math by $$𝐅=\frac{\mathrm{d}𝐩}{\mathrm{d}t}.$$Therefore, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Class "Wikibase\Client\WikibaseClient" not found"): {\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 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).

Constant mass

As a result of the previous 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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Class "Wikibase\Client\WikibaseClient" not found"): {\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

• 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.

Variable mass

Template:Further 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, the impulse per unit mass. 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.

Application: Newton's second law

Newton's second law is often written as $$𝐅=\frac{\mathrm{d}𝐩}{\mathrm{d}t}.$$ But this is the case only when there is no mass transfer into or out of Template:Mvarb. When the mass of an object changes by exuding or accumulating mass, the momentum has a time dependence in its mass aspect as well as in its velocity aspect. If Newton's second law is applied naively, the erroneous conclusion is $$𝐅=m\frac{\mathrm{d}𝐯}{\mathrm{d}t}+𝐯\frac{\mathrm{d}m}{\mathrm{d}t}\mathrm{(}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{)}.$$ Rather, suppose that the occurrence of the mass leaving or entering happens after every period Template:Math, at which point a mass Template:Math is exuded from the original mass Template:Mvar and proceeds to travel with velocity Template:Mvar. By Newton's third law, the impulse (i.e., the momentum transfer) between the two constant masses is equal and opposite, $${𝐉}_m=-{𝐉}_{\mathrm{\Delta }m}.$$ Thus, the impulse on Template:Mvar by Template:Math is $${𝐉}_m=\mathrm{\Delta }m(𝐯-𝐮).$$ Suppose an external force Template:Mvarb is applied to the first object. Then, in a unit of time Template:Math, it gains an impulse Template:Math and then loses a mass Template:Math, yielding the net impulse, $$𝐉={𝐉}_{\mathrm{e}\mathrm{x}\mathrm{t}}+{𝐉}_m={𝐅}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\Delta }t+\mathrm{\Delta }m(𝐯-𝐮).$$ Hence, what we find is that in the infinitesimal limit, that, $${𝐅}_{\mathrm{e}\mathrm{x}\mathrm{t}}=m𝐚-\frac{\mathrm{d}m}{\mathrm{d}t}(𝐮-𝐯).$$ Relating this back to Newton's second law, we have that the law applies only to the Template:Em Template:Mvarb of the system. Using this and labeling mass that has left the system with Template:Var, $$\begin{array}{cc}{𝐅}_{\mathrm{e}\mathrm{x}\mathrm{t}}& =\frac{\mathrm{d}𝐏}{\mathrm{d}t}\\ & =\frac{\mathrm{d}𝐩}{\mathrm{d}t}+\frac{\mathrm{d}{𝐩}_{\mathrm{e}\mathrm{x}}}{\mathrm{d}t}\\ & =m𝐚+𝐯\frac{\mathrm{d}m}{\mathrm{d}t}+{𝐯}_{\mathrm{e}\mathrm{x}}\frac{\mathrm{d}{m}_{\mathrm{e}\mathrm{x}}}{\mathrm{d}t}\\ & =m𝐚-\frac{\mathrm{d}m}{\mathrm{d}t}(𝐮-𝐯),\end{array}$$yielding the same result. Thus, using either the step-wise impulse method or the infinitesimal Newtonian force method work in this case.

Application: specific impulse

Suppose our previous example was for a rocket. Then, using the equation for Template:Mvarb, we have that the specific impulse Template:Math is $$I_{\mathrm{s}\mathrm{p}}=𝐯-𝐮.$$ In this case, we call this quantity the exhaust velocity since it is equal to the velocity at which the material is leaving the rocket relative to the frame of the rocket. The thrust Template:Mvarb is then $𝐓=-\frac{\mathrm{d}m}{\mathrm{d}t}{I}_{\mathrm{s}\mathrm{p}}=\frac{\mathrm{d}m}{\mathrm{d}t}(𝐯-𝐮).$

See also

• Wave–particle duality defines the impulse of a wave interaction. 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

References

• Template:Cite book
• Template:Cite book

External links

• Dynamics Template:Webarchive