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Impulse (physics)

From Wikipedia, the free encyclopedia
Integral of a comparatively larger force over a short time interval
Impulse
A large force applied for a very short duration, such as a golf shot, is often described as the club giving the ball animpulse.
Common symbols
J,Imp
SI unitnewton-second (N⋅s)
Other units
kg⋅m/s in SI base units,lbf⋅s
Conserved?Yes
DimensionLMT−1{\displaystyle {\mathsf {L}}{\mathsf {M}}{\mathsf {T}}^{-1}}
Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Inclassical mechanics,impulse (symbolized byJ orImp) is the change inmomentum of an object. If the initial momentum of an object isp1, and a subsequent momentum isp2, the object has received an impulseJ:J=p2−p1.{\displaystyle \mathbf {J} =\mathbf {p} _{2}-\mathbf {p} _{1}.}

Momentum is avector quantity, so impulse is also a vector quantity:[1]∑F×Δt=Δp.{\displaystyle \sum \mathbf {F} \times \Delta t=\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 forceF acting on the object:F=p2−p1Δt,{\displaystyle \mathbf {F} ={\frac {\mathbf {p} _{2}-\mathbf {p} _{1}}{\Delta t}},}so the impulseJ delivered by a steadyforceF acting for timeΔt is:J=FΔt.{\displaystyle \mathbf {J} =\mathbf {F} \Delta t.}

The impulse delivered by a varying force acting from timea tob is theintegral of the forceF with respect to time:J=∫abFdt.{\displaystyle \mathbf {J} =\int _{a}^{b}\mathbf {F} \,\mathrm {d} t.}

TheSI unit of impulse is thenewton-second (N⋅s), and thedimensionally equivalent unit of momentum is the kilogram-metre per second (kg⋅m/s). The correspondingEnglish engineering unit is thepound-second (lbf⋅s), and in theBritish Gravitational System, the unit is theslug-foot per second (slug⋅ft/s).

Mathematical derivation in the case of an object of constant mass

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The impulse delivered by the "sad" ball ismv0, wherev0 is the speed upon impact. To the extent that it bounces back with speedv0, the "happy" ball delivers an impulse ofmΔv = 2mv0.[2]

ImpulseJ produced from timet1 tot2 is defined to be[3]J=∫t1t2Fdt,{\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t,}whereF is the resultant force applied fromt1 tot2.

FromNewton's second law, force is related tomomentump byF=dpdt.{\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}.}

Therefore,J=∫t1t2dpdtdt=∫p1p2dp=p2−p1=Δp,{\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}}}whereΔp is the change in linear momentum from timet1 tot2. This is often called the impulse–momentum theorem (analogous to thework–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:J=∫t1t2Fdt=Δp=mv2−mv1,{\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 andt2 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 theInternational System of Units, these arekg⋅m/s =N⋅s. InEnglish engineering units, they areslug⋅ft/s =lbf⋅s.

The term "impulse" is also used to refer to a short-acting force orimpact. This type of impulse is oftenidealized so that the change in momentum produced by the force is modelled as happening instantaneously. This sort of change is astep change, and is not physically possible. However, this is a useful model for computing the effects of ideal collisions (such as in videogamephysics engines). Additionally, in rocketry, the term "total impulse" is commonly used and is considered synonymous with the term "impulse".

Variable mass

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Further information:Specific impulse

The application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools forjet- orrocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit ofpropellant expended, to create a performance parameter,specific impulse. This fact can be used to derive theTsiolkovsky 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

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Notes

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  1. ^Basic Physics: A Self-Teaching Guide. John Wiley & Sons. 2020. p. 34.ISBN 9781119629900.
  2. ^Property Differences In Polymers: Happy/Sad Balls
  3. ^Serway & Jewett 2004, chpt. 9.2 Impulse and Momentum.

References

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External links

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Linear/translational quantitiesAngular/rotational quantities
Dimensions1LL2Dimensions1θθ2
Ttime:t
s		absement:A
m s		Ttime:t
s
1distance:d,position:r,s,x,displacement
m		area:A
m2		1angle:θ,angular displacement:θ
rad		solid angle:Ω
rad2, sr
T−1frequency:f
s−1,Hz		speed:v,velocity:v
m s−1		kinematic viscosity:ν,
specific angular momentum: h
m2 s−1		T−1frequency:f,rotational speed:n,rotational velocity:n
s−1,Hz		angular speed:ω,angular velocity:ω
rad s−1
T−2acceleration:a
m s−2		T−2rotational acceleration
s−2		angular acceleration:α
rad s−2
T−3jerk:j
m s−3		T−3angular jerk:ζ
rad s−3
Mmass:m
kg		weighted position:M ⟨x⟩ = ∑mxmoment of inertia: I
kg m2		ML
MT−1Mass flow rate:m˙{\displaystyle {\dot {m}}}
kg s−1		momentum:p,impulse:J
kg m s−1,N s		action:𝒮,actergy:ℵ
kg m2 s−1,J s		MLT−1angular momentum:L,angular impulse:ΔL
kg m rad s−1
MT−2force:F,weight:Fg
kg m s−2,N		energy:E,work:W,Lagrangian:L
kg m2 s−2,J		MLT−2torque:τ,moment:M
kg m rad s−2,N m
MT−3yank:Y
kg m s−3, N s−1		power:P
kg m2 s−3, W		MLT−3rotatum:P
kg m rad s−3, N m s−1
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