# Physics equations/08-Linear Momentum and Collisions

#### CALCULUS-based generalization to non-uniform force

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 ${\displaystyle {\bar {F}}\Delta t}$  as the impulse, with the understanding that ${\displaystyle {\bar {F}}}$  represents a time average. Recall that the average of a large set of numbers is the sum divided by the ${\displaystyle N}$ :

${\displaystyle {\bar {F}}={\frac {\sum _{n}F_{n}}{N}}}$

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

For a collision that occurs over a finite time interval, ${\displaystyle \Delta t}$ , we break that collision time into much smaller intervals ${\displaystyle \delta t}$ . 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 ${\displaystyle \Delta t/\delta t=N}$ , where ${\displaystyle N}$  is the number of frames of the camera. Let ${\displaystyle F_{n}}$  be the force associated with the n-th frame. The discretely defined average force associated with that camera is:

${\displaystyle {\bar {F}}\Delta t={\frac {\sum _{n}F_{n}}{N}}\cdot \Delta t=\sum _{n=1}^{N}\left[F_{n}\cdot \left\{{\frac {\Delta t/\delta t}{N}}\right\}\cdot \delta t\right]=\sum _{n=1}^{N}F_{n}\cdot \delta t\rightarrow \int _{0}^{\Delta t}F(t)\,dt}$

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: