Motion Along a Line

AP Calculus · Axiom Academy

Analyzing position, velocity, and acceleration using derivatives Position, Velocity, and Acceleration When a particle moves along a line, its motion is described by three related functions: Position: s(t) — where the particle is at time t Velocity: v(t) = s'(t) — the rate of change of position Acceleration: a(t) = v'(t) = s''(t) — the rate of change of velocity Speed: |v(t)| — the magnitude of velocity (always non-negative) A particle moves along the x -axis with position s(t) = t^3 - 6t^2 + 9t for . Velocity: v(t) = 3t^2 - 12t + 9 = 3(t-1)(t-3) When is the particle at rest? v(t) = 0 at t = 1 and t = 3 When is it moving right? v(t) > 0 on At t = 0.5 : v > 0 , a = -9 < 0 → opposite signs → slowing down At t = 2 : v < 0 , a = 0 → neither At t = 4 : v > 0 , a = 12 > 0 → same signs → speeding up Velocity = derivative of position; Acceleration = derivative of velocity Particle changes direction when v(t) changes sign Speeding up: v and a have the same sign Slowing down: v and a have opposite signs Total distance ≠ displacement (total distance uses |v(t)| )

This is the written version of the interactive lesson above. See the full AP Calculus course.