The Constant-Acceleration Equations

Position, velocity, and acceleration are linked by derivatives: v is the derivative of x, a is the derivative of v. Run that backward with integrals and, if a is constant, you get three clean formulas: v = v0 + at, x = x0 + v0t + (1/2)at², and v² = v0² + 2a(x - x0). They feel like separate rules, but they are all one fact (constant a) unpacked three different ways, each handy for a different missing variable.

The second one comes straight from integration. Integrate a (a constant) once with respect to time: v(t) = v0 + at. Integrate again: x(t) = x0 + v0t + (1/2)at^2. The extra constants of integration are just the initial velocity and initial position, pinned down by the conditions at t = 0. The third equation, v² = v0² + 2a(x - x0), comes from combining the first two and eliminating t algebraically — useful exactly when you don't know or don't care about the time.

These equations are a special case living inside the general derivative and integral relationships from earlier in the unit. The moment acceleration starts changing with time, the integrals a·t and (1/2)at² are wrong, because they assumed a was a number, not a function. That's the bridge to the next idea: when a(t) actually varies, you must go back to integrating a(t) directly, which is where calculus-based kinematics really earns its keep over the algebra-only version.