Physics equations/08-Linear Momentum and Collisions

Q:oneDcollision

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