AP Calculus AB and BC glossary

Speeding up and slowing down

A particle is speeding up when its velocity and acceleration have the same sign, and slowing down when they have opposite signs. Speeding up means the speed is increasing, not that the motion is in the positive direction, so a particle can speed up while moving backward.

The rule comes from looking at speed, which is $|v(t)|$ . Its size grows exactly when velocity and acceleration point the same way, so the real question is never whether acceleration is positive but whether $v(t)$ and $a(t)$ share a sign.

A particle with $v = -3$ and $a = -2$ is speeding up, since both are negative and it moves ever faster in the negative direction. Change to $a = +2$ and it is now slowing down, even though acceleration just became positive.

The mistake

Deciding by the sign of acceleration alone. Positive acceleration does not mean speeding up. Negative velocity with positive acceleration is slowing down, while negative velocity with negative acceleration is speeding up. Always compare the two signs.

Appears in: Unit 4: Contextual Applications