← Unit 4 Guide

Straight-Line Motion

One particle on a line, described by three functions one derivative apart. Position tells you where; its derivative, velocity, tells you how fast and which way; and its derivative, acceleration, tells you whether the motion is winding up or winding down.
Differentiate to descend the chain: position x(t) → velocity v(t) = x′(t) → acceleration a(t) = v′(t). The signs carry the meaning — the sign of v gives the direction, and comparing the signs of v and a tells you the one thing students always miss: whether the particle is speeding up or slowing down.

The Motion Scrubber

Below is a particle on its line, and the three graphs that describe it. Press play — or drag the time cursor. Watch the dot move, reverse, and race: velocity is the slope of position, acceleration is the slope of velocity, and the badge reads out the direction and whether it's speeding up or slowing down at every instant.

drag anywhere to scrub · watch the arrows — same direction means speeding up
time t0.50 s
position x(t)
velocity v(t)
acceleration a(t)
speed |v|
—
velocity arrow (v) acceleration arrow (a) arrows aligned → speeding up · opposed → slowing down

Speeding up or slowing down?

Here's the rule the whole topic hinges on: a particle speeds up when velocity and acceleration share a sign, and slows down when their signs disagree. A negative acceleration does not mean "slowing" — it only slows the particle if the particle is moving in the positive direction. The sign chart lays out every interval of our example, .

Read it in columns: on (1, 2) the particle moves left (v < 0) with negative acceleration — same sign, so it's speeding up. On (2, 3) acceleration flips positive while velocity is still negative — opposite signs, so it's slowing down toward its turn at t = 3.

The ideas everything else builds on

Descend the chain

Velocity is the derivative of position; acceleration is the derivative of velocity — so a = v′ = x″. Each step down measures the rate of the one above.

Speed is |velocity|

Velocity has a sign (direction); speed = |v| never does. A velocity of −8 and +8 are opposite directions but the same speed.

Direction from the sign of v

v > 0 moves in the positive direction, v < 0 the negative. Where v = 0 and changes sign, the particle is momentarily at rest and turns around.

Speeding up ⟺ same signs

The particle speeds up exactly when v·a > 0 (v and a agree) and slows when v·a < 0. Equivalently: speed grows when a points the same way as motion.

Distance vs. displacement

Because the particle backtracks, two "how far" questions have different answers. Displacement is where it ended up minus where it started — direction counts, so backtracking cancels. Total distance adds up every leg as a positive length — split the trip at each turn (v = 0) and sum the pieces.

Your turn

Six problems on our particle . Try each before revealing.

Quick quiz

Six fast checks. Pick an answer to see whether it's right and why — then work through all six.

Question 1 of 6
Score: 0 / 0

Common mistakes & exam tips

"Negative acceleration = slowing down"

Only if velocity is positive. If v < 0 and a < 0, the particle is speeding up (moving negative, faster). Always compare the two signs.

Confusing speed with velocity

Speed is |v| and is never negative. "Speeding up" is about |v| growing, which is the v·a > 0 test — not about the sign of v alone.

Distance = displacement

Only when the particle never turns around. If v changes sign, split at each turn and add the absolute legs for total distance; displacement is just end − start.

"At rest" means a = 0

At rest means v = 0, not a = 0. At a turning point v = 0 but acceleration is usually nonzero — that's what pulls it back.

Forgetting to check both v = 0 roots

Velocity is often a quadratic with two zeros. Both can be turning points; test the sign of v on each interval before concluding direction.

Reading position height as speed

On the position graph, speed is the steepness, not the height. The particle can be far out (high x) yet momentarily still (slope 0).

On the AP exam

You can now read a particle's whole story off its position function — direction, speed, and whether it's winding up or down. Next, two quantities change at once and their rates are chained together by a single equation: continue to Related Rates →, or head back to the Unit 4 Guide.