Momentum and Impulse

In the lesson on Newton’s laws, I mentioned that momentum is defined as , where represents momentum. We used this definition in defining force, since , but momentum is a very useful quantity in the study of mechanics for more reasons than that. Most significantly, momentum is conserved in all cases where the net force is zero. (If you’re interested in why, read the Wikipedia article on translational invariance, noting that momentum and position are Pontryagin duals per the Heisenberg uncertainty principle — and if you’re completely confused at this point, that’s no problem.) We measure momentum in units of , occasionally known as Waechtlers (make sure to include the umlaut mark). I do not advise using that unit on the AP test, though for some inexplicable reason it receives full credit in Mr. Waechtler’s class.

The law of the conservation of momentum is particularly useful in finding the results of collisions, which we will discuss in more detail in the next lesson. In general, when we work with the momentum of a system of particles, we define , making sure to compute vector sums of .

Keep in mind that all of our study of mechanics so far has been regarding objects as point particles; that is, we have concerned ourselves only with the movement of the objects’ centers of mass—more on this to come shortly.

An equally useful quantity is the change in momentum, or impulse, notated . We derive it as follows: , so ; then, integrating, . Thus . Be sure to note that we are integrating force with respect to time, not with respect to a path as we did for . The practical use of this is that if we are presented with a graph of , we can find by interpreting the definite integral as the area enclosed by the graph and the x-axis.