Read the same oscillation on three graphs
You will be able to: Relate graph slopes, signs, peaks and zero crossings without confusing their units.
How do position, velocity and acceleration graphs fit together?
A block’s position graph is flat at its rightmost turning point. That means zero velocity, not zero acceleration. A moment later the position decreases, the velocity is negative, and the spring still accelerates the block leftward.
A useful starting point: Follow position, velocity and acceleration through one cycle →
Words and symbols before equations
- Slope of x–t
- Position change per time; gives velocity.
- Slope of v–t
- Velocity change per time; gives acceleration.
- Phase relationship
- How the repeating patterns line up in time.
- Signed area under v–t
- Net displacement over the chosen time interval.
What this picture assumes
Same ideal oscillator on all three plots: m=1 kg, k=4 N/m, A=0.3 m, release from +A. T=π s. Horizontal time axes match; vertical units and ranges differ. Teal markers select the same instant.
Connect the picture to the physics
For a smooth position graph, local slope gives velocity. A peak or trough has zero slope, so v=0 there. Where x crosses equilibrium most steeply, |v| is greatest. The sign of that slope gives direction.
Acceleration is the slope of v–t and also satisfies a=−(k/m)x. The a–t graph therefore has the opposite sign to x–t at every nonzero displacement. Position and acceleration reach extreme magnitudes together; velocity reaches extreme magnitudes one quarter-period later or earlier.
Each graph has its own vertical units and scale. Comparing numerical heights across x, v and a as if they were the same quantity is meaningless. Over one full cycle, signed velocity area is zero because the block returns to its initial position; the distance traveled is 4A, not zero.
A worked example, step by step
A block starts at x=+0.30 m at rest with T=4 s. Sketch x–t, v–t and a–t qualitatively over one full cycle.
- Mark time ticks at 0, 1, 2, 3 and 4 s on every graph.
- x values are +0.30, 0, −0.30, 0, +0.30 m, joined by a smooth sinusoidal curve.
- v is zero at 0, 2 and 4 s; it is most negative at 1 s and most positive at 3 s.
- a has the opposite sign to x: most negative at 0 and 4 s, zero at 1 and 3 s, most positive at 2 s. Give each vertical axis its own units.
A zero on one graph is not a zero on every graph. Use slope and the restoring-force relationship to connect them.
At the most negative velocity, what is the acceleration?
Compare with an explanation
Zero at that instant: the velocity graph has a minimum with zero slope, and the block is at equilibrium.
Predict. Change one thing. Explain.
Move the time marker along the three synchronized plots. Choose a position peak, an equilibrium crossing and an intermediate point; explain the other two graph values at each chosen time.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Teal markers: t=0.785 s; x=0 m, v=-0.6 m/s, a=0 m/s². All time axes cover one period, π s. Each vertical axis has its own units and scale.
Same ideal oscillator on all three plots: m=1 kg, k=4 N/m, A=0.3 m, release from +A. T=π s. Horizontal time axes match; vertical units and ranges differ. Teal markers select the same instant.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy 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 block has amplitude 0.10 m and starts at +A with period 2 s. (a) State the first time x=0. (b) Give the signs of v and a at that time. (c) State the net velocity area over 2 s. (d) Find total distance in 2 s and explain why it differs from that area.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: t=T/4=0.50 s.
- 1 point: v is negative and a=0.
- 1 point: Zero net area, hence zero net displacement.
- 1 point: Distance=4A=0.40 m; distance adds travel magnitudes while signed area allows cancellation.
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 gives v on an x–t graph?
The local slope.
RECALL 2How do signs of a and x compare in SHM?
They are opposite except at x=0, where both are zero.
RECALL 3Why can zero net displacement coexist with nonzero distance?
Forward and backward displacements cancel, but distances add.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read the same oscillation on three graphs
- Slope of x–t=v; slope of v–t=a.
- a=−(k/m)x, so x and a have opposite signs.
- One full cycle: net displacement=0; distance=4A.
Remember: A zero on one graph is not a zero on every graph. Use slope and the restoring-force relationship to connect them.
Conditions: Same ideal oscillator on all three plots: m=1 kg, k=4 N/m, A=0.3 m, release from +A. T=π s. Horizontal time axes match; vertical units and ranges differ. Teal markers select the same instant.
Refresh Kid · Unit 7 · Objectives 7.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 7.3, objectives 7.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. 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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