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

$\overline{F}=\frac{\sum _n{F}_n}{N}$

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

For a collision that occurs over a finite time interval, $\mathrm{\Delta }t$, 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 $\mathrm{\Delta }t/\delta t=N$, 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:

$\overline{F}\mathrm{\Delta }t=\frac{\sum _n{F}_n}{N}\cdot \mathrm{\Delta }t=\sum _{n=1}^N[F_n\cdot \left\{\frac{\mathrm{\Delta }t/\delta t}{N}\right\}\cdot \delta t]=\sum _{n=1}^N{F}_n\cdot \delta t\to \underset{0}{\overset{\mathrm{\Delta }t}{\int }}F(t)dt$

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