The kinematic equations

Most textbooks hand you the kinematic equations as a list to memorize. You're going to read them off a picture instead — because when acceleration is constant, the v–t graph is a straight line, and everything about straight lines is easy.

The setup

Constant acceleration a, starting velocity v₀. The velocity line is:

$v = v_0 + at$

That's not a new fact — it's just "slope × run": start at v₀ and add a for every second. First equation done.

Displacement is the area

How far did you go by time t? The area under the velocity line — and that area splits into two friendly shapes:

The rectangle is the distance you'd cover just coasting at v₀. The triangle — base t, height at — is the extra distance the acceleration adds:

$x = v_0 t + \tfrac{1}{2} a t^2$

The famous ½ isn't mysterious. It's the area of a triangle.

The shortcut without time

Sometimes you know speeds and distance but not the time ("how long a runway does a plane need to reach takeoff speed?"). Solve v = v₀ + at for t, plug it into the area formula, and t cancels:

$v^2 = v_0^2 + 2a\,\Delta x$

Worth checking the algebra once by hand — after that, it's yours.

The toolkit

Three equations, one straight line, zero memorization: the first is the line's slope, the second is the area under it, the third is the two combined with t eliminated.

Why this matters

Constant acceleration isn't a toy case — it's gravity. Everything in the next module (free fall, projectiles) is these three equations with a = g pointed down.