Follow position, velocity and acceleration through one cycle
You will be able to: Describe the motion at turning points, equilibrium crossings and intermediate positions.
How can a block be momentarily stopped but still accelerating?
Release a spring block from its rightmost position. At that instant its speed is zero, but the spring pull is largest. It starts moving left, reaches its greatest speed at the center, and then slows toward the leftmost point.
A useful starting point: Measure gravity with pendulum timing →
Words and symbols before equations
- Turning point
- An endpoint x=±A where velocity becomes zero and reverses.
- Velocity v
- Signed rate of position change, in m/s.
- Acceleration a
- Signed rate of velocity change, in m/s².
- Quarter cycle
- One-fourth of a full period, T/4.
What this picture assumes
Horizontal ideal spring: m=1 kg, k=4 N/m, A=0.3 m. Release from +A at rest. T=π s. Velocity and acceleration arrows use separate labeled scales; the slider advances time through one ideal cycle.
Connect the picture to the physics
Start at x=+A at rest. The force and acceleration point left with maximum magnitude. After T/4 the block passes x=0 moving left at maximum speed. There its net spring force and acceleration are zero at that instant.
At T/2 it reaches x=−A, where velocity is zero and acceleration is largest rightward. At 3T/4 it crosses equilibrium rightward at maximum speed. At T it returns to +A and the same starting state. The block does not pause for a finite time at a turning point; its nonzero acceleration reverses the velocity.
At any nonzero position, acceleration points toward equilibrium. Velocity can point either way there, depending on which part of the cycle the block is in. Opposite signs of v and a mean slowing; matching signs mean speeding up. These horizontal-spring statements do not erase a pendulum’s radial acceleration.
| Position | Speed | Acceleration |
|---|---|---|
| x=+A | Zero | Largest magnitude, left |
| x=0 | Largest magnitude | Zero |
| x=−A | Zero | Largest magnitude, right |
A worked example, step by step
A horizontal spring block has T=4 s and is released from x=+0.20 m at t=0. Describe x, velocity direction and acceleration direction at t=1, 2 and 3 s.
- At 1 s=T/4: x=0, moving left at maximum speed, a=0.
- At 2 s=T/2: x=−0.20 m, v=0, acceleration right with maximum magnitude.
- At 3 s=3T/4: x=0, moving right at maximum speed, a=0.
- At the same position x=0, velocity directions differ; a position alone does not specify the entire state.
At a turning point v=0 but a is not zero. At equilibrium a=0 but speed is greatest in a nonzero-amplitude ideal spring oscillation.
At x>0 with v>0, is the block speeding up?
Compare with an explanation
No. Acceleration is negative, toward equilibrium, so it slows while moving outward.
Predict. Change one thing. Explain.
Move through a cycle in quarter-cycle steps, then inspect intermediate states. Predict each sign before moving the control. Velocity and acceleration arrows have separate labeled scales and must not be compared as the same quantity.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At t=0 s (0 of a cycle): x=0.3 m; v=0 m/s; a=-1.2 m/s². Turning point: zero velocity and maximum acceleration magnitude.
Horizontal ideal spring: m=1 kg, k=4 N/m, A=0.3 m. Release from +A at rest. T=π s. Velocity and acceleration arrows use separate labeled scales; the slider advances time through one ideal cycle.
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 spring block is released at x=+A at t=0. Its period is 8 s. (a) Describe its state at 2 s. (b) Describe its state at 4 s. (c) State when it first returns to +A. (d) Explain why zero velocity at 4 s does not mean zero acceleration.
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Compare with the answer and four-point rubric
- 1 point: At equilibrium, moving left at maximum speed, a=0.
- 1 point: At −A, v=0 and acceleration maximum rightward.
- 1 point: 8 s.
- 1 point: The spring is maximally displaced, so the restoring force and acceleration are nonzero.
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 1Where is speed greatest?
At equilibrium for ideal nonzero-amplitude SHM.
RECALL 2Where is acceleration magnitude greatest?
At the turning points.
RECALL 3Does x alone tell the velocity direction?
Usually not; the same position is passed in two directions.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Follow position, velocity and acceleration through one cycle
- Turning points: x=±A, v=0, |a| maximum.
- Equilibrium: x=0, |v| maximum, a=0 (horizontal ideal spring).
Remember: At a turning point v=0 but a is not zero. At equilibrium a=0 but speed is greatest in a nonzero-amplitude ideal spring oscillation.
Conditions: Horizontal ideal spring: m=1 kg, k=4 N/m, A=0.3 m. Release from +A at rest. T=π s. Velocity and acceleration arrows use separate labeled scales; the slider advances time through one ideal cycle.
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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