Classical Mechanics

Position, velocity and acceleration are one function and its first two derivatives, which means calculus already contains the whole of kinematics before any physics is added.

A vector is a bundle of independent one dimensional problems, which makes a projectile two copies of the previous lesson and turns steady circular motion into an acceleration of $v^2/r$ towards the centre.

What the three laws actually claim: that inertial frames exist, that force and mass are defined together by $\overrightarrow{F}=m\overrightarrow{a}$, and that forces always come in pairs acting on different bodies.

The catalogue of everyday forces and the free body method that turns a picture into equations, including friction as an empirical inequality and drag as a differential equation with a terminal speed.

Turning the centripetal acceleration $v^2/r$ into a demand on real forces, which fixes banking angles, loop speeds and the fastest safe corner, and exposes centrifugal force as a bookkeeping entry rather than a push.

Integrating force over distance rather than time produces the work energy theorem, which gives speed as a function of position without ever solving for when.

Some forces do work that depends only on the endpoints, which lets the work be stored as a function of position and makes the total energy a constant of the motion.

Impulse is force integrated over time, momentum is conserved whenever no external force acts, and a collision needs both that conservation and a statement about energy before it is determined.

Every system of particles has one point that obeys Newton's second law with the external forces alone, which justifies eight lessons of point particles and leads to the rocket equation.

The motion a body has about its own centre of mass needs its own variables, and its own measure of inertia, which depends on where the mass sits rather than how much of it there is.

Torque changes angular momentum exactly as force changes momentum, which explains the spinning skater, settles the race down an incline, and yields Kepler's second law before gravity is even introduced.

Every potential energy curve is a parabola near its minimum, so every stable system oscillates like a spring if disturbed gently, and the approximation can be made exact enough to keep time.

Kepler's third law forces gravity to fall off as the inverse square, the Moon test confirms it, the shell theorem makes a planet a point, and Cavendish's torsion balance turns the law into a mass for the Earth.

What an inverse square force does to a body: conic section paths, the sign of the total energy deciding bound from unbound, all three of Kepler's laws, and the 43 arcseconds where the theory fails.