AP Calculus AB and BC

Speed vs Velocity

Velocity is signed, so it tells you both how fast and in which direction. Speed is the absolute value of velocity and is never negative. Speed increases exactly when velocity and acceleration have the same sign, which is not the same as acceleration being positive.

Velocity

Use when: The question asks about direction, when the particle is at rest, or when it changes direction.

Speed

Use when: The question asks how fast, or whether the particle is speeding up or slowing down.

Side by side

	Velocity	Speed
Definition	$v(t) = s'(t)$	$\lvert v(t) \rvert$
Can be negative	Yes	No
Carries direction	Yes	No
Integrates to	Displacement	Total distance travelled

The speeding-up rule is about agreement of signs, not the sign of acceleration alone. A particle with velocity $-4$ and acceleration $-2$ is speeding up, because it is moving in the negative direction faster and faster.

The integration difference follows directly. Integrating velocity gives displacement, which can be small or zero after a round trip. Integrating speed gives total distance, which counts every unit travelled.

Frequently asked questions

Does negative acceleration mean slowing down?

Not necessarily. It means slowing down only when the velocity is positive. If velocity is also negative, the particle is speeding up.

In the CED: Unit 4: Contextual Applications, Unit 8: Applications of Integration