E.4 Kinematics in Calculus

Calculus IS motion. The FORWARD chain differentiates displacement s(t) once for velocity v(t) = ds/dt and again for acceleration a(t) = dv/dt = d²s/dt². The REVERSE chain integrates acceleration back to velocity (+ C₁ pinned by v(0)) and then to displacement (+ C₂ pinned by s(0)). Signed displacement ∫v dt under-counts the path when v changes sign — the absolute total distance is ∫|v| dt, found by splitting the interval at the zeros of v.