Newton's Laws of Motion

Nothing moves, stops, or turns without a force

Junior level — plain language, no maths

Newton boiled all of motion down to three rules, and the first is the least obvious: things keep doing what they're already doing. A ball on a smooth floor rolls until something - friction, a wall, your foot - stops it. A hockey puck on ice glides for ages. We only think objects naturally slow down because on Earth there's always friction quietly pushing back. Take friction away and motion simply continues. This reluctance to change is called inertia.

The second rule is the one you feel every day: push something and it speeds up; push harder and it speeds up faster; push something heavier and it barely budges. That's \(F = ma\) put into words - the force you apply equals the mass times the acceleration you get. A shopping trolley proves it perfectly: empty, a gentle shove sends it rolling; loaded with bricks, the very same shove hardly moves it.

The third rule is the surprising one: every push comes with an equal push back. Lean on a wall and the wall pushes on you just as hard, which is exactly why you don't topple through it. A rocket hurls hot gas downward and the gas hurls the rocket up. A swimmer shoves water backward and the water shoves the swimmer forward. Forces always come in pairs, pointing opposite ways - you can never touch without being touched back.

The three laws, made quantitative

Student level — the core equations

Newton's three laws, published in the Principia in 1687, are the foundation of all mechanics. First: an object stays at rest, or moving in a straight line at constant speed, unless a net external force acts on it - the principle of inertia. Second, the workhorse: the net force equals mass times acceleration, \(\mathbf{F}_{\text{net}} = m\mathbf{a}\), a vector equation you apply one axis at a time. Third: for every force there is an equal and opposite one, \(\mathbf{F}_{AB} = -\mathbf{F}_{BA}\), and crucially the two act on different bodies.

The trick to using them is the free-body diagram: draw the object, mark every force on it - gravity \(mg\) pulling down, the normal force \(N\) from a surface pushing up, any applied push, and friction - then add them as vectors. Friction resists motion up to a limit \(f \le \mu N\); below that limit a stationary object simply won't move, because the forces balance and \(\mathbf{a} = 0\). Push past the limit and it accelerates at \(a = F_{\text{net}}/m\).

Two consequences are worth burning into memory. Acceleration is inversely proportional to mass - double the mass and the same force delivers half the acceleration. And weight is not the same as mass: weight is merely gravity's force on you, \(W = mg\). Fly to the Moon and your mass is unchanged, yet your weight drops sixfold, because \(g\) does.

From three laws to momentum, frames, and their limits

Scholar level — full mathematical depth

01The second law is really about momentum

Newton did not write \(F = ma\); he wrote \(\mathbf{F} = \dfrac{d\mathbf{p}}{dt}\), the rate of change of momentum \(\mathbf{p} = m\mathbf{v}\). For constant mass this collapses to \(m\mathbf{a}\), but the momentum form is the honest one: it handles rockets and other variable-mass systems, where mass is being ejected, and it is the version that survives - suitably modified - into relativity.

02The third law is conservation of momentum

Because internal forces cancel in equal-and-opposite pairs, the total momentum of an isolated system never changes: \(\sum \mathbf{p} = \text{const}\). This is arguably deeper than the force statement itself. By Noether's theorem it follows from the translational symmetry of space - the fact that physics is the same here as over there - and it holds even where the naive action-reaction picture strains, such as when electromagnetic fields quietly carry momentum of their own.

03Inertial frames and Galilean relativity

The laws hold only in inertial (non-accelerating) frames. Step into an accelerating frame and you must invent fictitious forces - centrifugal, Coriolis - to keep \(F = ma\) bookkeeping straight. Yet all inertial frames are equivalent: no mechanical experiment can single out a state of absolute rest. That principle of relativity, already present in Galileo and Newton, is the seed Einstein grew into special relativity.

04Where Newton breaks down

Newtonian mechanics is a spectacularly accurate limit - low speeds, weak gravity, macroscopic sizes - not the final law. Approach the speed of light and momentum becomes \(\mathbf{p} = \gamma m\mathbf{v}\), so force no longer runs parallel to acceleration. Shrink to atomic scales and the sharp trajectory dissolves into quantum probability. Turn up gravity and the "force" is revealed as the curvature of spacetime. The three laws are the first, superbly useful chapter of dynamics - and knowing exactly where they fail is part of understanding why they work so well everywhere else.