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

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