Read position, velocity and acceleration graphs
You will be able to: Connect graph height, slope and signed area without confusing their meanings.
How can three different graphs describe the same motion?
A cart starts at x = 0 with v = −4 m/s and a = +2 m/s². It first moves left, pauses at 2 s, then moves right. The three motion graphs tell the same story in different ways.
A useful starting point: Accumulation and initial values →
Words and symbols before equations
- Graph ordinate
- The vertical-axis value; read its label and units.
- Slope
- Vertical change divided by horizontal change; a tangent gives the local slope.
- Signed area
- Area above the time axis is positive, below is negative.
- Turning point
- A reversal in direction, identified by a change in the sign of velocity.
What this picture assumes
x₀ = 0 m, v₀ = −4 m/s, a = +2 m/s². The reversal is at 2 s. Signed velocity area gives displacement; absolute area gives distance.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- t = 3 s: x = -3 m, v = 2 m/s, a = 2 m/s². Displacement -3 m; distance 5 m; velocity change 6 m/s.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For this cart, v(t) = −4 + 2t and x(t) = −4t + t² in SI units. The position graph curves upward because its slope is increasing. It can curve upward while its slope is still negative: moving left and accelerating right can happen together.
From 0 to 2 s, the triangular area below the velocity axis is −4 m. From 2 to 4 s the area above is +4 m. Net displacement is zero, but total distance is 8 m. Areas cancel for displacement, not for distance.
At 2 s the position graph is horizontal, the velocity graph crosses zero and acceleration remains +2 m/s². A graph’s height and slope are distinct measurements. Motion dots at equal time intervals become closer before the turn and farther apart afterward.
| Starting graph | Slope means | Signed area means |
|---|---|---|
| Position vs time | Velocity | Not displacement |
| Velocity vs time | Acceleration | Displacement |
| Acceleration vs time | Rate of change of acceleration | Change in velocity |
A worked example, step by step
For v(t) = −4 + 2t in m/s with t in s, find displacement and distance from 0 to 3 s.
- Find the reversal: −4 + 2t = 0 gives t = 2 s.
- From 0 to 2 s, signed triangular area = −(1/2)(2 s)(4 m/s) = −4 m.
- From 2 to 3 s, area = +(1/2)(1 s)(2 m/s) = +1 m.
- Displacement is −3 m; distance is 4 + 1 = 5 m.
Area under a position–time graph is not displacement. The axes determine the physical meaning.
Can x be negative while v is positive?
Compare with an explanation
Yes. An object left of the origin can be moving toward positive x.
Predict. Change one thing. Explain.
Move time across 2 s. Compare position, velocity and acceleration at the turning point. Then compare signed area and absolute area up to that time.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t = 3 s: x = -3 m, v = 2 m/s, a = 2 m/s². Displacement -3 m; distance 5 m; velocity change 6 m/s.
x₀ = 0 m, v₀ = −4 m/s, a = +2 m/s². The reversal is at 2 s. Signed velocity area gives displacement; absolute area gives distance.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant position, velocity, acceleration or integral relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA cart has v(t) = −6 + 2t in SI units from 0 to 5 s. Locate the reversal and calculate displacement, distance and acceleration.
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Compare with the answer and four-point rubric
- 1 point: The reversal is at t = 3 s.
- 1 point: Signed areas are −9 m and +4 m, so displacement is −5 m.
- 1 point: Distance is 9 + 4 = 13 m.
- 1 point: The slope dv/dt is +2 m/s² throughout.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does the slope of v(t) mean?
Acceleration.
RECALL 2What does negative velocity indicate?
Motion toward the negative coordinate direction.
RECALL 3Is v = 0 enough to prove a reversal?
No. Check whether velocity changes sign across that instant.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read position, velocity and acceleration graphs
- Slope of x(t) → v(t); slope of v(t) → a(t).
- Signed area under v(t) → Δx; under a(t) → Δv.
- For distance, add magnitudes of areas after splitting at velocity sign changes.
Remember: Area under a position–time graph is not displacement. The axes determine the physical meaning.
Conditions: x₀ = 0 m, v₀ = −4 m/s, a = +2 m/s². The reversal is at 2 s. Signed velocity area gives displacement; absolute area gives distance.
Refresh Kid · AP Physics C: Mechanics Unit 1 (official Unit 1) · Objectives 1.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.3, objectives 1.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is Mechanics Unit 1: Kinematics. Quantitative motion problems stay in one or two dimensions; the optional spatial view clarifies vector notation. Students should study calculus alongside this course. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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