Impulse
Encyclopedia
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 equal to the change of momentum.
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.
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:
where
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 engine
s).
Impulse has the same units (in the International System of Units
, and dimensions as momentum.
Impulse can be calculated using the equation
where
; 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:
where
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, an impulse (abbreviated I or J) is defined as the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of a force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
with respect to time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
. When a force is applied to a rigid body it changes the momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
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
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of the force and the time for which it is applied that is important. The impulse is equal to the change of momentum.
Mathematical derivation in the case of an object of constant mass
Impulse I produced from time t1 to t2 is defined to bewhere 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.
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:
where
- F is the constant total net force applied,
- Δt is the time interval over which the force is applied,
- m is the constant mass of the object,
- Δv is the change in velocity produced by the force in the considered time interval, and
- Δp is the change in linear momentum.
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
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...
, and is not physically possible. This is a useful model for computing the effects of ideal collisions (such as in game physics engine
Physics engine
A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
s).
Impulse has the same units (in the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
, and dimensions 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.
Variable mass
When a system expels mass in one direction, the force the expelled mass applies to the system is called thrustThrust
Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....
; the force the system applies to the mass being expelled is of equal magnitude but opposite direction.
Consider for example a rocket
Rocket
A rocket is a missile, spacecraft, aircraft or other vehicle which obtains thrust from a rocket engine. In all rockets, the exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction...
. 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:
- 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 infinitesimalInfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
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
- 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 impulseSpecific impulseSpecific impulse is a way to describe the efficiency of rocket and jet engines. It represents the derivative of the impulse with respect to amount of propellant used, i.e., the thrust divided by the amount of propellant used per unit time. If the "amount" of propellant is given in terms of mass ,...
- Wave–particle dualityWave–particle dualityWave–particle duality postulates that all particles exhibit both wave and particle properties. A central concept of quantum mechanics, this duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the behavior of quantum-scale objects...
defines an impulse for waves. The preservation of momentum at a collision is then called phase matching. Applications include:- Compton effect
- nonlinear opticsNonlinear opticsNonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
- Acoustic-optic modulator
- Unflappable scattering
- electron phononPhononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
scattering