Mechanics & Motion

For centuries the motion of the planets was a mystery. Kepler found the pattern — planets trace ellipses, sweeping out equal areas in equal times — but not the reason. Newton's brilliant idea was that planets are constantly falling toward the Sun while moving sideways so fast they keep missing it. He found one universal rule: everything pulls on everything else with a force that weakens as the inverse square of the distance. That single law explains falling apples, the ocean tides, and the orbits of distant galaxies.

The world is always changing — how do we describe change precisely? The ancient Greeks tied themselves in knots over this. The answer required a new branch of mathematics: calculus. If a car is speeding up, what is its speed right now? Look at the tiny distance it covers in a tiny interval of time and take the ratio. Calculus is the machinery for doing this rigorously. It was not an abstract game — it was invented as the natural language for the physics of motion.

Newton gave us a program for predicting the future. His second law, F = ma, is really a rule about change: it does not tell you where something is, but how its velocity will change in the next instant. If you know where an object is, how it is moving, and the forces on it, you can compute its position a moment later — then repeat, step by step, to trace its whole future path. It is a complete recipe for the motion of the universe, given the starting conditions and the forces.

Newton's third law says that for every action there is an equal and opposite reaction: push on me and I push back just as hard. A wonderful consequence is that the total momentum — mass times velocity — of an isolated system never changes, because all the internal pushes cancel. This lets us analyze a car crash or a rocket launch without knowing the messy details of the forces: the total momentum before must equal the total momentum after.

The laws of physics do not care whether you run an experiment here or there, facing north or facing east — they must be the same regardless of how you orient your coordinates. To capture this we invented the vector: not just three numbers, but three numbers that transform together in a special way when you rotate your axes, just like the coordinates of a point. Writing laws as vector equations, like F = ma, guarantees they respect this deep symmetry of space.

What is a force? Saying it equals ma is only a definition; the real content of Newton's laws is that forces have simple, independent properties. The truly fundamental forces — gravity and the electromagnetic force — have beautifully simple laws. Most forces we feel every day, like friction or the pull of a spring, are not fundamental at all. They are the enormously complicated, large-scale result of countless tiny electrical forces between atoms.

When a force moves an object, we say it does work, transferring energy. For some forces, like gravity, the work to move from A to B does not depend on the path — like climbing a hill, only the change in altitude matters, not the trail. For these 'conservative' forces we can define a potential energy: the energy stored in the system. Kinetic energy plus potential energy then stays constant.

The power of potential energy is that it lets us use conservation of energy directly. For a conservative force — gravity, or an ideal spring — the sum of kinetic and potential energy is a constant, a result that follows straight from Newton's laws. For nonconservative forces like friction, energy is not conserved in the same simple way; it leaks away as heat. The clean test for a conservative force: the work done around any closed loop is zero, so you can never get free energy by going around in a circle.