Impulse
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Impulse (physics)
In classical mechanics, an impulse (abbreviated I or J) is defined as the integral of a force with respect to time. When a force is applied to a rigid body it changes the momentum of that body. A small force applied for a long time can produce the same momentum change as a large force applied briefly, because it is the product of the force and the time for which it is applied that is important. The impulse is always equal to the change of momentum. Impulse is measured in Ns (Newton seconds).
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Mathematical derivation in the case of an object of constant mass
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This section may be too technical for most readers to understand. Please help improve this article to make it understandable to non-experts, without removing the technical details. The talk page may contain suggestions. (January 2012) |
Impulse I produced from time t1 to t2 is defined to be[1]
where F is the force applied from 
to 
.
From Newton's second law, force is related to momentum p by
Therefore
where Δp is the change in momentum from time t1 to t2. This is often called the impulse-momentum theorem.[2]
As a result, an impulse may also be regarded as the change in momentum of an object to which a force is applied. The impulse may be expressed in a simpler form when both the force and the mass are constant:
It is often the case that not just one but both of these two quantities vary.
In the technical sense, impulse is a physical quantity, not an event or force. The term "impulse" is also used to refer to a fast-acting force. 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. This is a useful model for computing the effects of ideal collisions (such as in game physics engines).
Impulse has the same units (in the International System of Units, kg·m/s = N·s) and dimensions (M L T−1) as momentum.
Impulse can be calculated using the equation
where
- F is the constant total net force applied,
- t is the time interval over which the force is applied,
- m is the constant mass of the object,
- v1 is the final velocity of the object at the end of the time interval, and
- v0 is the initial velocity of the object when the time interval begins.
Examples of Impulse
Impulse is:
The gun powder accelerating the bullet in a rifle,
The bow string accelerating the arrow,
The rocket engines accelerating the shuttle,
The fist accelerating the punching bag.
In general impulse is the force applied to a given object with a given mass for a given period of time. This will decide the amount of acceleration, final velocity and the kinetic energy stored in that object. If you know momentum and the mass of the object you do not need acceleration to calculate final velocity. Momentum is conserved. If an object slows down a certain amount of energy is transferred from the object to the environment. This transfer could be in the form of acceleration or deceleration of another object or heat due to friction or collision.
Variable mass
When a system expels mass in one direction, the force the expelled mass applies to the system is called thrust; the force the system applies to the mass being expelled is of equal magnitude but opposite direction.
Consider for example a rocket. The momentum of the rocket (including the remaining fuel) changes due to two effects: one is the applied thrust, the other one is the reduction of mass:[3]
- dp = d(mv) = mdv + (dm)v = (dt)F + (dm)v = dI + (dm)v = (dm)ve + (dm)v = (dm)(ve + v)
where
- p is the momentum of the rocket including the remaining fuel
- dp is the infinitesimal change of the momentum of the rocket including the remaining fuel; it is the negative of the momentum of the mass being expelled, just after expulsion (the total momentum does not change)
- m is the mass of the rocket including the remaining fuel (it decreases when mass is expelled)
- dm is the infinitesimal change of the mass of the rocket including the remaining fuel, so the negative of the mass being expelled[4]
- v is the velocity of the rocket
- ve is the velocity of the just expelled mass relative to the rocket (effective exhaust velocity), hence:
- ve + v is the velocity of the just expelled mass
- F is the thrust
- dI is the infinitesimal impulse exerted on the rocket
See also
- Specific impulse
- Wave–particle duality defines an impulse for waves. The preservation of momentum at a collision is then called phase matching. Applications include:
- Compton effect
- nonlinear optics
- Acoustic-optic modulator
- Unflappable scattering
- electron phonon scattering
Notes
- ^ Hibbeler, Russell C. (2010), EngineEring Mechanics, 12th edition, Pearson Prentice Hall, p. 222, ISBN 0-13-607791-9
- ^ See, for example, section 9.2, page 257, of Serway (2004).
- ^ Scalar multiplication is written here in the regular notation, with the scalar first. Alternatively the vector is sometimes written first (like above), thus avoiding the need of parentheses in some cases.
- ^ With this definition dm is non-positive. Alternatively, in this context it can be convenient to define dm as a non-negative number: as the infinitesimal mass being expelled, so the negative of the infinitesimal change of the mass of the rocket including the remaining fuel
Bibliography
- Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed. ed.). Brooks/Cole. ISBN 0-534-40842-7.
- Tippler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed. ed.). W. H. Freeman. ISBN 0-7167-0809-4.
External links
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