View source for Impulse (physics)

{{short description|Integral of a comparatively larger force over a short time interval}}
{{Infobox physical quantity
|name = Impulse
|image = Armedforces jeffery tee shot.jpg
|caption = A large force applied for a very short duration, such as a golf shot, is often described as the club giving the ball an ''impulse''.
|unit = [[newton-second]] ([[Newton (unit)|N]]⋅[[second|s]])
|otherunits = [[kilogram|kg]]⋅[[metre|m]]/[[second|s]] in SI base units, [[pound (force)|lbf]]⋅[[second|s]]
|symbols = '''J''', '''Imp'''
|dimension = wikidata
|conserved = Yes
|derivations =
}}
{{Classical mechanics|cTopic=Fundamental concepts}}

In [[classical mechanics]], '''impulse''' (symbolized by {{math|'''J'''}} or '''Imp''') is the change in [[momentum]] of an object. If the initial momentum of an object is {{math|'''p'''<sub>1</sub>}}, and a subsequent momentum is {{math|'''p'''<sub>2</sub>}}, the object has received an impulse {{math|'''J'''}}:

<math display=block>\mathbf{J}=\mathbf{p}_2 - \mathbf{p}_1.</math>

[[Momentum]] is a [[Vector (physics)|vector]] quantity, so impulse is also a vector quantity: <math display="block">\sum \mathbf{F} \times \Delta t = \Delta \mathbf{p}.</math><ref>{{Cite book |title=Basic Physics: A Self-Teaching Guide |publisher=John Wiley & Sons |year=2020 |isbn=9781119629900 |pages=34 |language=en}}</ref> 
[[Newton’s second law of motion]] states that the rate of change of momentum of an object is equal to the resultant force {{mvar|F}} acting on the object:
<math display="block">\mathbf{F}=\frac{\mathbf{p}_2 - \mathbf{p}_1}{\Delta t},</math>

so the impulse {{mvar|J}} delivered by a steady [[force]] {{mvar|F}} acting for time {{math|Δ''t''}} is:
<math display="block">\mathbf{J}=\mathbf{F} \Delta t.</math>

The impulse delivered by a varying force is the [[integral]] of the force {{mvar|F}} with respect to time:
<math display="block">\mathbf{J}= \int\mathbf{F} \, \mathrm{d}t.</math>

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

==Mathematical derivation in the case of an object of constant mass==
[[File:Happy vs. Sad Ball.webm|thumbnail|The impulse delivered by the "sad" ball is {{math|''mv''<sub>0</sub>}}, where {{math|''v''<sub>0</sub>}} is the speed upon impact.  To the extent that it bounces back with speed {{math|''v''<sub>0</sub>}}, the "happy" ball delivers an impulse of {{math|1=''m''&Delta;''v'' = 2''mv''<sub>0</sub>}}.<ref>[http://materialseducation.org/educators/matedu-modules/docs/Property_Differences_in_Polymers.pdf Property Differences In Polymers: Happy/Sad Balls]</ref>]]
Impulse {{math|'''J'''}} produced from time {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}} is defined to be{{sfn|Serway|Jewett|2004|loc=chpt. 9.2 Impulse and Momentum}}
<math display=block qid=Q837940>\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, \mathrm{d}t,</math>
where {{math|'''F'''}} is the resultant force applied from {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}}.

From [[Newton's laws of motion#Newton's second law|Newton's second law]], force is related to [[momentum]] {{math|'''p'''}} by
<math display=block>\mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}.</math>

Therefore,
<math display=block qid=Q837940>\begin{align}
 \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{align}</math>
where {{math|Δ'''p'''}} is the change in linear momentum from time {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}}. 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:
<math display=block qid=Q837940>\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, \mathrm{d}t =  \Delta\mathbf{p} = m \mathbf{v_2} - m \mathbf{v_1},</math>

where
*{{math|'''F'''}} is the resultant force applied,
*{{math|''t''<sub>1</sub>}} and {{math|''t''<sub>2</sub>}} are times when the impulse begins and ends, respectively,
*{{mvar|m}} is the mass of the object,
*{{math|'''v'''<sub>2</sub>}} is the final velocity of the object at the end of the time interval, and
*{{math|'''v'''<sub>1</sub>}} is the initial velocity of the object when the time interval begins.

Impulse has the same units and dimensions {{nowrap|(MLT<sup>&minus;1</sup>)}} as momentum. In the [[International System of Units]], these are {{nowrap|1=[[kilogram|kg]]⋅[[meter per second|m/s]] =}} {{nowrap|[[newton (units)|N]]⋅[[second|s]]}}. In [[English engineering units]], they are {{nowrap|1=[[Slug (unit)|slug]]⋅[[foot per second|ft/s]] =}} {{nowrap|[[pound (force)|lbf]]⋅[[second|s]]}}.

The term "impulse" is also used to refer to a fast-acting force or [[Impact (mechanics)|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 function|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 engine]]s).  Additionally, in rocketry, the term "total impulse" is commonly used and is considered synonymous with the term "impulse".

==Variable mass==
{{Further|Specific impulse}}
The application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for [[jet propulsion|jet]]- or [[rocket]]-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of [[rocket propellant|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 [[Nonlinear optics#Phase matching|phase matching]]. Applications include:
**[[Compton effect]]
**[[Nonlinear optics]]
**[[Acousto-optic modulator]]
**Electron [[phonon]] scattering
* [[Dirac delta function]], mathematical abstraction of a pure impulse

==Notes==
{{reflist}}

==References==
*{{cite book | last=Serway | first=Raymond A. | last2=Jewett | first2=John W. |title=Physics for Scientists and Engineers |url=https://archive.org/details/physicssciengv2p00serw |url-access=registration |edition=6th|publisher=Brooks/Cole |year=2004 |isbn=0-534-40842-7}}
*{{cite book |author=Tipler, Paul |title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics |edition=5th|publisher=W. H. Freeman |year=2004 |isbn=0-7167-0809-4}}

==External links==
* [http://www.rwc.uc.edu/koehler/biophys/2c.html Dynamics]

{{Classical mechanics derived SI units}}

[[Category:Classical mechanics]]
[[Category:Vector physical quantities]]
[[Category:Mechanical quantities]]

[[de:Impuls#Kraftstoß]]