Newton's Laws in Calculus Form and Free-Body Diagrams

The three laws, stated the way Physics C uses them

First law. There exist reference frames — inertial frames — in which a body with zero net force keeps a constant velocity vector. The first law is not a special case of the second; it is the statement that defines when the second law may be written at all.

Second law. The exam version is the momentum form:

For constant mass this reduces to $\sum\vec F=m\vec a$, but you should read the general form first: it is what makes a time-dependent force $F(t)$ a calculus problem rather than an algebra problem. Once $a(t)=F(t)/m$ is known, velocity and position follow by integration, with the constants fixed by the initial conditions.

Third law. $\vec F_{AB}=-\vec F_{BA}$: equal magnitude, opposite direction, same type of interaction, and — the part that is tested — acting on different bodies. A third-law pair therefore never appears on the same free-body diagram.

Free-body diagram procedure. Isolate one body; draw only forces exerted on it by other objects; choose axes that make the acceleration lie along one axis; write $\sum F_x$ and $\sum F_y$ separately. Contact forces (normal, friction, tension) are constraint forces: their magnitudes are unknowns to be solved for, not formulas to be recalled.