AP Physics C: Mechanics · Unit 1 · Topic 3 of 6 · 5 min lesson

Motion Graphs

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.

Quiz this topic · 4 questions

The big idea

On an x–t graph, slope is velocity and concavity is acceleration. On a v–t graph, slope is acceleration and area is displacement. On an a–t graph, area is the change in velocity.

1. Reading an x–t graph

A secant (line between two points) gives average velocity; a tangent gives the velocity at one instant. The curve bends down (a < 0) until the inflection point at t = 2 s, then bends up (a > 0).

Slope of a secantaverage velocity

Slope of a tangentinstantaneous velocity

Concave upa > 0

Concave downa < 0

Inflection pointa = 0

2. The whole chain

x–tslope → v; concavity → sign of a

v–tslope → a; area → Δx

a–tarea → Δv

Going down the chaintake slopes (derivatives)

Going up the chaintake areas (integrals) and add the starting value

3. Example, then try it

Example

A car’s velocity rises steadily from 0 to 12 m/s in 4.0 s, then stays at 12 m/s for 4.0 s. Find its acceleration during the first part and its total displacement.

a = slope = 12 ÷ 4.0 = 3.0 m/s².

Δx = area = ½(4.0)(12) + (4.0)(12) = 24 + 48 = 72 m.

Try it · work it on paper first

Use the graph below of x(t) = t³ − 6t² + 9t + 1. Find the average velocity from 0 to 3 s, the velocity at t = 0, when the particle is moving in the + direction while slowing down, and the distance traveled from 0 to 3 s.

Average velocity: x(3) − x(0) = 1 − 1 = 0, so 0 m/s (the flat secant).

At t = 0: v = 3t² − 12t + 9 = 9 m/s.

+ and slowing: v > 0 for t < 1 s and a = 6t − 12 < 0 for t < 2 s, so 0 < t < 1 s (rising and concave down).

Distance: it goes from x = 1 up to x(1) = 5 and back to 1: 4 + 4 = 8 m.

Answer: 0 m/s average, 9 m/s at t = 0, moving + and slowing for 0 < t < 1 s, 8 m traveled.

4. Common mistakes

Reading the height of an x–t graph as speed.Velocity is the slope, not the value.

Using the area under an x–t graph.That area means nothing useful here.

Mixing up average and instantaneous.Secant vs. tangent.

Check yourself

Part of an x–t graph slopes downward and is concave up. What is the object doing there?

Downward slope: v < 0. Concave up: a > 0. Opposite signs mean slowing down.

Ready to practice?

4 questions on Motion Graphs, each with a hint and an explanation for every wrong answer.

Quiz this topic All of Unit 1