Read turning points and force directions from potential energy

1. Orient: Read turning points and force directions from potential energy

For a conservative one-particle model with total mechanical energy E, kinetic energy at position x is K=E−U(x). A position is accessible only when E≥U there. Where E=U, speed is zero; such positions can be turning points. Where E<U, a negative number is an energy deficit, not a physical negative kinetic energy.

2. Explain and derive

On a potential-energy graph, force points toward decreasing potential energy. Over a straight graph segment, Fx=−ΔU/Δx. An upward slope gives a negative force; a downward slope gives a positive force. This follows from Wforce=−ΔU together with W=FxΔx over the segment. The negative slope relation can be reasoned about without using calculus.

A local potential minimum is stable for small displacements because forces on either side point back toward it. A local maximum is unstable because small departures lead toward lower potential away from the peak. A flat region has no force from that potential. Equilibrium and zero speed are different ideas: a turning point need not be a force-balance point.

The model U=2|x−2| J is a deliberately V-shaped well. Each side has constant slope magnitude 2 J/m and force magnitude 2 N toward x=2. Its sharp cusp has no unique derivative exactly at the bottom; it stands in for a smoother well. Energy still gives speed magnitude, but not direction, and adding the same constant to both U and E changes neither accessible positions nor speeds.

3. Work an example

With E=4 J, turning points satisfy 2|x−2|=4, so x=0 and 4 m. At x=1 m, U=2 J and K=2 J. For mass 2 kg, speed is√2≈1.41 m/s. The object could be moving either right or left there. At the minimum x=2, K=4 J and speed 2 m/s.

4. Predict and investigate

Predict first: If you raise both the potential-energy curve and total-energy line by 10 J, should any turning position move?

Compare total energies 2,4 and 8 J. Locate turning points from the intersection condition before reading them. At E=2 J, test x=0 to produce a forbidden position; explain why the model withholds speed. Use graph slopes to identify force directions on both sides of the minimum. Use the lesson controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

A 2 kg object has total energy 10 J at a position with potential energy 6 J. Find speed magnitude.

Your numerical result (m/s) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

Sketch a smooth hill instead of a well. Explain why its top can be an equilibrium point but not a stable one.

Retain a first attempt, signed ledger, units, system boundary, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic outputs are not physical measurements.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 3 topic mapping: 3.3; 3.4. Checked 30 October 2026; this is a content cross-reference, not AP authorization.