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

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

{\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:

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