Position, velocity, and acceleration
In calculus, velocity is the derivative of position and acceleration is the derivative of velocity. Working backward, integrating acceleration gives velocity, and integrating velocity gives position.
Position, velocity, and acceleration describe straight-line motion, and calculus links them in a chain of derivatives. If s(t) gives an object's position at time t, then velocity is the rate of change of position, v(t) = s′(t), and acceleration is the rate of change of velocity, a(t) = v′(t) = s″(t). Differentiating moves down the chain; integrating moves back up.
Velocity carries a sign: positive velocity means motion in the positive direction, negative velocity the opposite. Speed is the absolute value of velocity, |v(t)|. A particle is speeding up when velocity and acceleration share the same sign, and slowing down when their signs differ — a subtlety exams love, since a particle with negative velocity and negative acceleration is speeding up.
Average rates use the familiar difference quotient: average velocity over [a, b] is displacement divided by time, (s(b) − s(a)) / (b − a), which is generally different from the average of instantaneous velocities. On the integral side, the integral of v(t) over [a, b] gives displacement (net change in position), while the integral of |v(t)| gives total distance traveled — displacement can be zero even when the particle traveled a long way and came back.
Straight-line (rectilinear) motion is a signature AP Calculus AB topic, appearing in both the derivative-applications and integral-applications units. Expect free-response questions asking when a particle changes direction, whether it is speeding up, and how far it traveled over an interval.
Key takeaways
- Velocity is the derivative of position; acceleration is the derivative of velocity.
- Integrating acceleration recovers velocity, and integrating velocity recovers position (up to initial conditions).
- Speed is |v(t)|; a particle speeds up when velocity and acceleration have the same sign.
- Average velocity over [a, b] is (s(b) − s(a)) / (b − a) — displacement over elapsed time.
- The integral of v(t) gives displacement, while the integral of |v(t)| gives total distance traveled.