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What is the difference between impulse and momentum?

The question says it all...I know the second of of them is mass * velocity, but what is the first one for, and when is it used? Also, what are its units, and is there a symbol for it?

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Given a system of particles, the impulse exerted on the system during a time interval $[t_a, t_b]$ is defined as $$ \mathbf J(t_1, t_2) = \int_{t_1}^{t_2} dt\,\mathbf F(t) $$ where $\mathbf F$ is the net external force on the system. Since one can show that the net external force on a system is given by Newton's second law by $$ \mathbf F(t) = \dot{\mathbf P}(t) $$ where $\mathbf P$ is the total momentum of the system, one has $$ \mathbf J(t_1, t_2) = \mathbf P(t_2) - \mathbf P(t_1) $$ In other words, the impulse is equal to the change in momentum of the system. The dimensions of these quantities are the same, namely mass times velocity.

You can think of impulse as kind of the "net effect" that a force has in changing the state of motion of a system. Here is an example to illustrate what I mean.

Imagine you're pushing a shopping cart. Let's say you push the cart with a constant force for a short period of time versus a long period of time. When you push it for a short period of time, then the integral of the force with respect to time will be smaller than when you push it for a long period of time, and the result will be that the cart's momentum will not change as much. However, if you were to push the cart for a short period of time, but if you were to push it very hard, then you could make up for the short period of time for which the force acts and still get the cart going fast.

The Upshot: The impulse takes into consideration both the effect of the force on the system, and the duration of time for which the force acts.

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Thanks a bunch, the example with the shopping cart put the concept into perspective. –  brolyspikes Feb 22 '13 at 5:26
    
@brolyspikes Glad to be of service. –  joshphysics Feb 22 '13 at 5:32
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Impulse is defined as the change in momentum and shown as "J", or delta p. It has the same units as momentum (mass times velocity). It is calculated by multiplying F times T (force into time), which is useful for situations where there is a quick change in motion.

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Impulse is a change in momentum. Naturally it has the same units. It's useful for whacks and thuds and bounces and other more or less sudden changes in motion.

Think of a golf ball struck by a club. Zero momentum initially. Whatever momentum it has after, as it begins to fly through the air. We might not know, or want to bother with, the exact instantaneous changing amount of force of the club on the ball. It happens quickly, and it's just the integral $\int^{after}_{before}F(t){dt}$ that matters to knowing how fast the ball is launched. We call this impulse, and it is equal to $p_{after}-p_{before}$.

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