We derived the 1-dimensional conservation laws of Newtonian mechanics, then established general solution methods for problems where the force has special form 1) F = F(t) or 2) F = F(v).
Conservation Laws 1) make solving for classical motions technically easier, and 2) reveal underlying symmetry structure that physical motions must obey. This second point will become much more beautiful and transparent when we study Lagrangian mechanics. We stated laws in 3 dimensions for
We then began discussing the problem of solving Newton's second law (NII), in the case where the mass is constant. Derived from NII: momentum conservation, the work-energy theorem, and energy conservation for conservative forces F=F(r).
Then considered methods to solve NII in 1-d, for different possible forms of force laws:
1) F = F(t). Here it makes sense to just integrate NII twice with respect to t to get motion x(t). We were careful about integrating over dummy variables to get to the endpoint t of the motion.
2) F=F(v). Derived from NII an integral equation, relating the time period of the motion to an integral over velocities. Gives t(v), which can be inverted to get v(t), which can then be integrated with respect to t to get the motion x(t).
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KB