AP Calculus AB and BC

Displacement vs Total Distance Travelled

Q: How do I know where velocity changes sign?

Set v(t) = 0 and solve, then test the intervals between those times. Those are the moments the particle changes direction.

Displacement is the net change in position, found by integrating velocity, and it can be zero even after a long trip. Total distance travelled integrates the absolute value of velocity, so movement in both directions adds up.

Displacement

Use when: The question asks for net change in position, or where the particle ends up relative to where it started.

Total distance

Use when: The question asks how far the particle travelled, or uses the phrase total distance.

Side by side

	Displacement	Total distance
Integral	$\int_a^b v(t)\,dt$	$\int_a^b \lvert v(t) \rvert\,dt$
Can be zero	Yes	Only if the particle never moves
Can be negative	Yes	No
Needs sign analysis	No	Yes, split where $v$ changes sign

If a particle moves forward 5 units and back 3, the displacement is 2 and the distance travelled is 8. Whenever the velocity keeps one sign throughout the interval the two agree up to sign, which is why the distinction only bites when the particle turns around.

To compute total distance by hand, find where the velocity changes sign, split the integral at those times, and add the absolute values of the pieces. On the calculator-active section you can integrate the absolute value directly.

Position from displacement

Displacement is a change, not a location. To get the final position, add it to the starting position: $s(b) = s(a) + \int_a^b v(t)\,dt$ .

Frequently asked questions

How do I know where velocity changes sign?

Set $v(t) = 0$ and solve, then test the intervals between those times. Those are the moments the particle changes direction.

In the CED: Unit 8: Applications of Integration