Turning: velocity and inward acceleration
You will be able to: Separate tangential velocity from inward acceleration and identify the actual force causing a turn.
How can constant speed still mean acceleration?
A toy car goes around a circular track at a steady 4 m/s. Its speedometer stays steady, but its direction changes continuously. That changing velocity requires acceleration.
A useful starting point: Changing velocity →
Words and symbols before equations
- Tangent
- Direction touching the circle at one point; the direction of instantaneous velocity.
- Radius r
- Distance from the center to the path, in m.
- Centripetal
- Center-seeking: the inward component of acceleration.
- Period T / frequency f
- Seconds per revolution / revolutions per second (Hz).
What this picture assumes
Uniform circular motion, mass 2 kg. Scrub phase to inspect an instant; this is not elapsed time. Velocity and acceleration use separate arrow scales.
Connect the picture to the physics
Uniform circular motion has acceleration magnitude v²/r directed toward the center. The velocity is tangent, perpendicular to that inward direction. Without the inward interaction, an object leaves along its instantaneous tangent.
Centripetal force is a name for the inward net force, not an additional interaction. Friction, tension, gravity or combinations of real forces can supply it. Write their inward components and set the sum equal to mv²/r.
If speed also changes, tangential acceleration exists as well. It is perpendicular to the inward acceleration, so the total magnitude is √(a_c²+a_t²). For uniform motion, one revolution covers 2πr, so T=2πr/v and f=1/T.
A worked example, step by step
A 2 kg cart moves at 4 m/s on a circle of radius 2 m. Find inward acceleration and net inward force, then compare them if speed doubles.
- a_c = 4²/2 = 8 m/s² inward.
- F_inward = 2×8 = 16 N. Name the actual source, such as a guide rail, in a physical force diagram.
- Doubling speed gives a_c=8²/2=32 m/s² and force 64 N.
- Both grow by a factor four because speed is squared.
Do not draw a separate centripetal-force arrow in addition to the real forces whose sum supplies it.
If inward acceleration is 3 m/s² and tangential acceleration is 4 m/s², what is total acceleration?
Compare with an explanation
5 m/s² from √(3²+4²), because the components are perpendicular.
Predict. Change one thing. Explain.
Keep radius and mass fixed. Double speed and compare inward force. Scrub the orbital phase to see the tangent velocity and inward acceleration change direction together.
Inward acceleration 8 m/s²; inward net force 16 N; period 3.14 s; frequency 0.32 Hz. Radius scale 25 drawing units/m. Velocity scale 10 per m/s; acceleration scale 2 per m/s².
Uniform circular motion, mass 2 kg. Scrub phase to inspect an instant; this is not elapsed time. Velocity and acceleration use separate arrow scales.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use a force or motion 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 0.5 kg object moves uniformly at 6 m/s in a circle of radius 3 m. (a) Find a_c, (b) find net inward force, (c) calculate period, and (d) draw velocity and acceleration at the rightmost point for counterclockwise motion.
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Compare with the answer and four-point rubric
- 1 point: 12 m/s² inward.
- 1 point: 6 N inward.
- 1 point: T=2π(3)/6=π s≈3.14 s.
- 1 point: Velocity up, acceleration left toward the center.
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 1Why acceleration at constant speed?
Velocity direction changes.
RECALL 2Is centripetal an extra force type?
No; it describes the inward net force requirement.
RECALL 3What changes if the object speeds up on the circle?
It also has tangential acceleration.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Turning: velocity and inward acceleration
- a_c=v²/r; ΣF_inward=mv²/r.
- Uniform circular motion: T=2πr/v; f=1/T.
- Changing speed: combine inward and tangential acceleration as perpendicular components.
Remember: Do not draw a separate centripetal-force arrow in addition to the real forces whose sum supplies it.
Conditions: Uniform circular motion, mass 2 kg. Scrub phase to inspect an instant; this is not elapsed time. Velocity and acceleration use separate arrow scales.
Refresh Kid · Unit 2 · Objectives 2.9.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.9, objectives 2.9.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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