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.
With a bit of algebra, we can turn this into a Riemann sum.
For a collision that occurs over a finite time interval, , we break that collision time into much smaller intervals . 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 , where is the number of frames of the camera. Let be the force associated with the n-th frame. The discretely defined average force associated with that camera is:
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: