Read force from a potential-energy slope
You will be able to: Differentiate U(x) and use Fₓ = −dU/dx to predict force.
How can an energy curve tell you which way an object accelerates?
Imagine an energy graph rising by 6 J per metre near your position. The associated force points toward decreasing x with magnitude 6 N. The slope, not the height of the curve, determines the force.
A useful starting point: Build spring energy from tiny stretches →
Words and symbols before equations
- Derivative dU/dx
- Local slope of the energy–position curve, measured in J/m = N.
- Conservative force Fₓ
- The negative derivative of U with respect to x.
- Tangent line
- A straight line matching the curve’s local slope at a selected point.
- Additive constant C
- A shift of the energy zero; its derivative is zero.
What this picture assumes
One-dimensional conservative model U = ax² + bx + C. The orange tangent shows local slope in energy/position units. Force is its negative derivative; no initial velocity is specified.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- At x = 1 m: U = 5 J, slope = 0 J/m and Fₓ = 0 N. Equilibrium x = 1 m; changing C leaves it and the force unchanged.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Starting from dU = −Fₓ dx, divide by dx to obtain Fₓ = −dU/dx. A rising U curve has negative force; a falling curve has positive force. This describes force, not necessarily velocity: an object can move uphill while slowing down.
For U(x) = ax² + bx + C, the power rule gives dU/dx = 2ax + b. Therefore Fₓ = −(2ax + b). Coefficient a has units J/m², b has J/m, and C has J. Units provide a useful check on differentiation.
Adding a constant to U changes its graph’s height but not its slope or force. To find speed, you also need the total energy and mass; a force curve alone does not specify an initial velocity.
A worked example, step by step
U(x) = (2 J/m²)x² − (4 J/m)x + 7 J. Find force at x = 0 m and x = 2 m, and locate zero force.
- Differentiate term by term: dU/dx = (4 J/m²)x − 4 J/m.
- Apply the minus sign: Fₓ = 4 N − (4 N/m)x.
- At x = 0, Fₓ = +4 N; at x = 2 m, Fₓ = −4 N.
- Set Fₓ = 0 to obtain x = 1 m. The constant 7 J affects none of these forces.
A high U value does not imply a large force. A steep slope does; the minus sign sets its direction.
Can an object move toward increasing U even though force points toward decreasing U?
Compare with an explanation
Yes. Its existing velocity can carry it uphill while the conservative force slows it down.
Predict. Change one thing. Explain.
Move the position through the minimum. Predict the force sign on either side. Then change only the energy offset and confirm that the force at that position stays fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At x = 1 m: U = 5 J, slope = 0 J/m and Fₓ = 0 N. Equilibrium x = 1 m; changing C leaves it and the force unchanged.
One-dimensional conservative model U = ax² + bx + C. The orange tangent shows local slope in energy/position units. Force is its negative derivative; no initial velocity is specified.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant work, system boundary, energy or calculus 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 questionU(x) = (1 J/m³)x³ − (6 J/m)x + C. Derive Fₓ, calculate F at x = 1 m, explain the acceleration direction for positive mass, and assess the role of C.
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Compare with the answer and four-point rubric
- 1 point: Differentiate: dU/dx = (3 J/m³)x² − 6 J/m.
- 1 point: Fₓ = 6 N − (3 N/m²)x², so Fₓ(1 m) = +3 N.
- 1 point: With positive mass, acceleration is in the +x direction by Fₓ = maₓ.
- 1 point: C shifts energy values but contributes no force because dC/dx = 0.
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 feature of U determines force?
The negative local slope, with units J/m = N.
RECALL 2Does force direction tell you velocity direction?
No. Velocity depends on the previous motion and initial conditions.
RECALL 3How do you get U from a known conservative force?
Integrate −Fₓ dx and choose an arbitrary additive constant.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read force from a potential-energy slope
- Fₓ = −dU/dx.
- ΔU = −∫Fₓ dx.
- U + C gives the same force as U.
Remember: A high U value does not imply a large force. A steep slope does; the minus sign sets its direction.
Conditions: One-dimensional conservative model U = ax² + bx + C. The orange tangent shows local slope in energy/position units. Force is its negative derivative; no initial velocity is specified.
Refresh Kid · AP Physics C: Mechanics Unit 3 (official Unit 3) · Objectives 3.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.3, objectives 3.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 3: Work, Energy, and Power. The unit covers Topics 3.1–3.5. Calculus connects work to force integrals, force to potential-energy derivatives, and power to the rate of energy transfer. Models distinguish object-only and multi-object systems; translational models exclude rotational energy unless explicitly noted. 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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