# Physics equations/08-Linear Momentum and Collisions

#### CALCULUS-based generalization to non-uniform force[edit | edit source]

Here we use the *Riemann sum* to clarify what happens when the force is not constant.

If the force is not constant, we can still use as the *impulse*, with the understanding that represents a time average. Recall that the average of a large set of numbers is the sum divided by the :

With a bit of algebra, we can turn this into a Riemann sum.

For a collision that occurs over a finite time interval, , we break that collision time into much smaller intervals . The former might be the collision time between a golf ball and the club, while the latter would be the time interval of an ultra high-speed camera. Note that , where is the number of frames of the camera. Let be the force associated with the n-th frame. The discretely defined average force associated with that camera is:

**Footnote:** This conversion from discrete to continuous math is easy to grasp, although the details are difficult to master: Other examples of this method include:

and**Descrete**expection values.**continuous**- The
, the**Discrete Fourier transform**, and the**Fourier series****Fourier transform**