In classical mechanics, impulse (symbolized by or Imp) is the change in momentum of an object. It is most often used to describe forces which act over short time periods, specifically in the case of impacts and collisions, for which it gets its namesake. A vector quantity, the impulse has a magnitude, which describes the amount by which the momentum changed, and a direction, which describes the direction in which the momentum changed.
For a force acting over a short time, the impulse is often idealized so that the change in momentum produced by the force is modelled as happening instantaneously. 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".
Impulse has the same units and dimensions as momentum. The SI unit of impulse is the newton-second (), and the dimensionally equivalent unit of momentum is the kilogram-metre per second (). The corresponding English engineering unit is the pound-second (), and in the British Gravitational System, the unit is the slug-foot per second ().
If, during a period of time, the initial momentum of an object is , and the final momentum is then the net impulse on the object is defined as,
By Newton's second law of motion, the rate of change of momentum of an object is equal to the resultant force acting on the object:
so the impulse delivered by a constant force acting for time is:
In the continuous time case, the impulse delivered onto a constant mass by a varying force acting from time to is defined to be the integral of the force with respect to time: From Newton's second law, force is related to momentum by Therefore,
where is the change in linear momentum from time to . This is often called the impulseâÂÂmomentum theorem (analogous to the workâÂÂenergy theorem).
As a result of the previous 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:
where
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, the impulse per unit mass. 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.
Newton's second law is often written as,
But this is only the case when there is no mass transfer into or out of . When the mass of an object is changing by way of exuding or accumulating mass, then we have that the momentum will have a time dependence in its mass aspect as well as in its velocity aspect. If you were to naively apply Newton's second law, you'd erroneously conclude,
Rather, suppose that the occurrence of the mass leaving or entering happens after every period , at which point a mass is exuded from the original mass and proceeds to travel with velocity . By NewtonâÂÂs third law, the impulse (i.e., the momentum transfer) between the two constant masses will be equal and opposite,
Thus, the impulse on by is
Suppose an external force is applied to the first object. Then, in a unit of time , it gains an impulse and then loses a mass , yielding the net impulse,
Hence, what we find is that in the infinitesimal limit, that,
Relating this back to Newton's second law, we have that the law only applies to the total momentum of the system. Using this and labeling mass that has left the system with , Yielding the same result. Thus, using either the step-wise impulse method or the infinitesimal Newtonian force method work in this case.
Suppose our previous example was for a rocket. Then, using the equation for , we have that the specific impulse will be,
In this case, we would call this quantity the exhaust velocity since it is equal to the velocity at which the material is leaving the rocket relative to the frame of the rocket. The thrust is then,