Английская Википедия:Impulse (physics)
Шаблон:Short description Шаблон:Infobox physical quantity Шаблон:Classical mechanics
In classical mechanics, impulse (symbolized by Шаблон:Math or Imp) is the change in momentum of an object. If the initial momentum of an object is Шаблон:Math, and a subsequent momentum is Шаблон:Math, the object has received an impulse Шаблон:Math:
<math display=block>\mathbf{J}=\mathbf{p}_2 - \mathbf{p}_1.</math>
Momentum is a vector quantity, so impulse is also a vector quantity.
Newton’s second law of motion states that the rate of change of momentum of an object is equal to the resultant force Шаблон:Mvar acting on the object: <math display=block>\mathbf{F}=\frac{\mathbf{p}_2 - \mathbf{p}_1}{\Delta t},</math>
so the impulse Шаблон:Mvar delivered by a steady force Шаблон:Mvar acting for time Δt is: <math display=block>\mathbf{J}=\mathbf{F} \Delta t.</math>
The impulse delivered by a varying force is the integral of the force Шаблон:Mvar with respect to time: <math display=block>\mathbf{J}= \int\mathbf{F} \, \mathrm{d}t.</math>
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
Impulse Шаблон:Math produced from time Шаблон:Math to Шаблон:Math is defined to be[1] <math display=block qid=Q837940>\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, \mathrm{d}t,</math> where Шаблон:Math is the resultant force applied from Шаблон:Math to Шаблон:Math.
From Newton's second law, force is related to momentum Шаблон:Math 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 Шаблон:Math is the change in linear momentum from time Шаблон:Math to Шаблон:Math. This is often called the impulse-momentum theorem[2] (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
- Шаблон:Math is the resultant force applied,
- Шаблон:Math and Шаблон:Math are times when the impulse begins and ends, respectively,
- Шаблон:Mvar is the mass of the object,
- Шаблон:Math is the final velocity of the object at the end of the time interval, and
- Шаблон:Math is the initial velocity of the object when the time interval begins.
Impulse has the same units and dimensions Шаблон:Nowrap as momentum. In the International System of Units, these are Шаблон:Nowrap Шаблон:Nowrap. In English engineering units, they are Шаблон:Nowrap Шаблон:Nowrap.
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
Шаблон: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. 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
- One-way wave equation
Notes
Bibliography
External links
Шаблон:Classical mechanics derived SI units
- ↑ Шаблон:Cite book
- ↑ See, for example, section 9.2, page 257, of Serway (2004).
