What use are Newton's "Laws" of Mechanics? Even a glib answer to that question can easily fill a 1-year course, if you really want to know. My purpose here is merely to offer some hints of how people learned to apply Newton's Laws to different types of Mechanics problems, began to notice that they were repeating certain calculations over and over in certain wide classes of problems, and eventually thought of cute shortcuts that then came to have a life of their own. That is, in the sense of Michael Polanyi's The Tacit Dimension, a number of new paradigms emerged from the technology of practical application of Newton's Mechanics.
The mathematical process of emergence generally works like this: we take the SECOND LAW and transform it using a formal mathematical identity operation such as "Do the same thing to both sides of an equation and you get a new equation that is equally valid." Then we think up names for the quantities on both sides of the new equation and presto! we have a new paradigm. I will show three important example of this process, not necessarily the way they first were "discovered," but in such a way as to illustrate how such things can be done. But first we will need a few new mathematical tools.