AP Calculus AB and BC

Acceleration vs Speeding Up

Q: Does positive acceleration mean speeding up?

No. If velocity is negative and acceleration positive, the particle is slowing down.

Q: What is the rule?

The particle speeds up when velocity and acceleration have the same sign, and slows down when the signs differ.

Q: What if acceleration is zero?

Then speed is momentarily unchanging. The particle is neither speeding up nor slowing down at that instant.

Positive acceleration does not mean speeding up. Acceleration is the rate of change of velocity, and it can be positive while a particle slows down. What decides speeding up is whether velocity and acceleration have the SAME sign, not whether acceleration is positive.

Acceleration

Use when: You are asked for a rate of change of velocity, or need the second derivative of position, or are checking concavity of the position graph.

Speeding up

Use when: You are asked whether the particle is gaining speed, which requires comparing two signs rather than reading one.

Side by side

	Acceleration	Speeding up
What it is	$a(t) = v'(t)$ , a signed quantity	A yes or no condition
Decided by	Its own sign	Whether $\text{sign}(v) = \text{sign}(a)$
Positive acceleration means	Velocity is increasing	Nothing on its own
Example: $v < 0$ , $a > 0$	Positive	NO, it is slowing down
Example: $v < 0$ , $a < 0$	Negative	YES, it is speeding up

The last two rows are the whole page. With velocity negative and acceleration positive, the acceleration is pushing back against the motion, so the particle slows. With both negative, the acceleration reinforces the motion and the particle gains speed while moving in the negative direction.

Why the sign rule works

Speeding up means the SIZE of velocity grows. Acceleration in the same direction as motion adds to that size; acceleration against the motion subtracts from it. The signs matching is exactly the statement that the push agrees with the travel.

Frequently asked questions

Does positive acceleration mean speeding up?

No. If velocity is negative and acceleration positive, the particle is slowing down.

What is the rule?

The particle speeds up when velocity and acceleration have the same sign, and slows down when the signs differ.

What if acceleration is zero?

Then speed is momentarily unchanging. The particle is neither speeding up nor slowing down at that instant.

In the CED: Unit 4: Contextual Applications