Point a cannon at the horizon, light the fuse, and you can say — before the ball has moved a centimetre — exactly where it will be one second, two seconds, ten seconds later. Not roughly: exactly, to as many decimal places as you care to compute. The same short set of rules that governs the cannonball steers the Moon around the Earth, the Earth around the Sun, a hockey puck across the ice, and the water sloshing in a braking bus. This astonishing reach — one tiny rulebook, the entire mechanical world — is what we mean by Newtonian mechanics, and this page is about the rulebook itself: the three laws Isaac Newton laid down in 1687 as the axioms of the whole theory.

An axiom is a starting assumption you do not prove — you adopt it, and then you check whether the world it predicts is the world you see. Newton's three laws are exactly that: not theorems derived from something deeper, but the bedrock we build on. Everything else in classical mechanics — energy, momentum, orbits, oscillations, the whole Lagrangian reformulation — is unpacked from these three sentences plus a rule for the forces. Our job here is to state them cleanly, to read the second law as what it really is (a differential equation whose solution is the motion), and to be honest about the one piece of fine print that Newton half-hid: the laws only hold in a special kind of viewpoint called an inertial frame.

The three laws, stated as axioms

Here they are, in the order Newton gave them. Read the third one twice — it is the one everybody thinks they remember and almost everybody misquotes.

Notice how little each law says on its own, and how much they say together. The first law tells you what "no force" looks like. The second tells you what a force does — it makes the second law the engine of the whole subject. The third guarantees that forces come in balanced pairs, which is what makes momentum a conserved bookkeeping quantity for a closed system.

F = ma is a differential equation

The second law is usually taught as an arithmetic recipe: multiply mass by acceleration, out comes force. That is true, but it badly undersells what the equation is for. Acceleration is the second derivative of position with respect to time, \mathbf{a} = \ddot{\mathbf{x}} = \dfrac{d^2\mathbf{x}}{dt^2}. So the moment you write down what the force actually is — and the force generally depends on where the body is, how fast it is going, and possibly the clock — the second law becomes a differential equation for the position:

This is the sentence that makes Newtonian mechanics a predictive theory rather than a collection of facts. The unknown is not a number — it is the whole trajectory \mathbf{x}(t), the function that says where the body is at every instant. You feed in the force law on the right, you feed in the initial conditions — the position \mathbf{x}(0) and velocity \dot{\mathbf{x}}(0) at one starting moment — and the equation hands you the entire past and future of the motion. That is why the cannonball's landing point is fixed the instant it leaves the barrel.

A worked case makes the machinery concrete. Suppose the force is a constant F along one axis — a falling stone (with F = mg), or a charged bead in a uniform field. In one dimension the equation is m\,\ddot{x} = F, so the acceleration is the constant a = F/m. Integrate once for the velocity, once more for the position:

There it is: two integrations, two constants (x_0 and v_0), and a trajectory that is a parabola in time. Every projectile problem you have ever solved is this one solution, dressed up. Different forces give different equations — a spring's F = -kx gives m\ddot{x} = -kx, whose solutions are sines and cosines — but the program never changes: write m\ddot{\mathbf{x}} = \mathbf{F}, then solve.

Watch the solution move

Because a constant force gives x(t) = x_0 + v_0 t + \tfrac12 a t^2, the trajectory is a parabola whose shape you can dial in. The chart below plots that solution against time (taking x_0 = 0). Turn the acceleration slider up and the parabola curves harder — that is the second law's right-hand side steering the left-hand side. Set a = 0 and you get a straight line: with no net force the motion is uniform, which is precisely the first law falling out as the special case F = 0 of the general solution. The initial-velocity slider tilts the curve at t = 0 without changing its curvature — that is the second constant of integration at work.

The picture is worth pausing on. The same differential equation, with the same force, produces a whole family of motions — one for each choice of starting velocity. The force law does not pick the trajectory by itself; the force law plus the initial conditions does. That split — dynamics on the right, initial data supplied separately — is the shape of every problem in the subject.

Bookkeeping the forces: the free-body diagram

To actually use \mathbf{F} = m\mathbf{a} you need the total force — the vector sum of every push and pull acting on the body. The tool for that sum is the free-body diagram: strip the body down to a single point, then draw every force on it as an arrow. Reveal the figure one arrow at a time to build the picture for a box being shoved along the ground.

Once the arrows are drawn, the second law is just arithmetic per direction. Vertically the box does not accelerate, so the up and down arrows must cancel: N = W = mg. Horizontally the net force is F - f, so the acceleration is

No new physics — just the second law applied to the sum of the arrows. This is the everyday loop of Newtonian problem-solving: draw the forces, add them, set the total equal to m\mathbf{a}, solve.

The fine print: which observer? Inertial frames

Now the piece that first-year courses often rush past. The first law says a force-free body moves in a straight line at constant speed — but as measured by whom? Motion is only ever described relative to a frame of reference: a choice of origin, axes, and clock. And the laws are not true in every frame. A frame in which Newton's first law holds — a force-free body really does drift straight and steady — is called an inertial frame. That is the whole definition: an inertial frame is one in which the law of inertia is true.

Sit in a smoothly cruising train and roll a ball across the floor: it goes straight, exactly as it would on the platform. Both frames are inertial, and — this is Galilean relativity — no mechanics experiment done inside the sealed carriage can tell you whether you are moving or at rest. All frames drifting at constant velocity relative to one inertial frame are themselves inertial, and the laws look identical in every one of them. There is no privileged "truly at rest" frame; uniform motion is undetectable from the inside.

But now let the driver brake hard. The ball on the floor — with no horizontal force touching it — suddenly rolls forward all by itself. In the braking carriage, the first law appears to fail: something accelerated with nothing pushing it. The braking frame is non-inertial (it is accelerating), and inside it Newton's laws in their bare form no longer hold. What went wrong is not the physics but the viewpoint: from the platform (an inertial frame) the ball simply kept moving at constant velocity while the train decelerated out from under it — the first law was fine all along.

If you insist on doing mechanics in an accelerating frame, you can — but you must add fictitious forces (also called inertial or pseudo-forces): extra terms, equal to -m\mathbf{a}_{\text{frame}}, invented to account for the frame's own acceleration. The forward lurch of the ball, the "push" that pins you to your seat when a car accelerates, the outward "centrifugal" tug on a roundabout — all are fictitious forces, artefacts of working in a non-inertial frame. They have no third-law partner and no physical agent pushing; they are the price of describing motion from an accelerating deck.