Impulse

The momentum transferred during a collision that takes a time $T$ is

$$\vec{I} = \Delta \vec{p} = \int_0^T \vec{F}(t)dt$$ (4.64)

This transferred momentum is called the ``impulse''. The function $\vec{F}(t)$ may be very complicated; we frequently don't care much about its details but only wish to know the change in momentum it produces, which is why the notion of impulse was invented. Impulse can be easily measured independent of the details of the collision.

Sometimes (such as when we develop the kinetic theory of gases later!) we will want to know what the average force exerted during a collision (that takes a known time $T$) is. By definition, it is

$$\vec{F}_{\rm avg} = \frac{1}{T}\int_0^T \vec{F}(t)dt = \frac{\vec{I}}{T}.$$ (4.65)

The average force is thus the transferred momentum divided by the time it takes to transfer it, which comes very close to its definition in calculus.