kinematic equations

Under constant acceleration — the simplest non-trivial case of motion — three compact equations describe everything. Given any three of the five quantities (initial position x₀, initial velocity v₀, acceleration a, elapsed time t, final velocity v, final position x), you can solve for the rest.

The first, v = v₀ + at, says that velocity grows linearly with time. Start at v₀, add a constant amount every second, and that is your current velocity. The second, x = x₀ + v₀t + ½at², says that position is the sum of three parts: where you started, how far you would have travelled at the initial velocity, and the extra distance covered because you were accelerating. The factor of ½ is not arbitrary — it is the area of the triangular region under a linearly growing velocity graph, a geometric identity that Nicole Oresme had proven three centuries before Galileo. The third, v² = v₀² + 2a(x − x₀), is the work-energy theorem in disguise, and it has no time in it at all: given a starting speed and a distance covered under constant acceleration, you know the final speed directly, without tracking how long the journey took.

These equations are the working tools of introductory mechanics. Projectile problems, brake distances, rocket burns, falling-object drops — anything where acceleration can be treated as constant — collapse to algebraic exercises once you write them down. The moment acceleration stops being constant, the equations fail, and you need calculus proper. But inside their domain they are exact, not approximate, and they are the residue of the more general calculus of motion boiled down to pure algebra.