Use energy to reconstruct speed and turning points

1. Frame the question

For a horizontal ideal spring, choose spring potential zero at equilibrium. Then U=½kx² and total energy E=½mv²+½kx². At either turning point x=±A, velocity is zero, so E=½kA². In the absence of dissipation or external work, this same total applies at every other position.

2. Build the mechanism

Subtracting potential from total gives K=½k(A²−x²) and v²=(k/m)(A²−x²). This determines speed magnitude, not signed velocity. The allowed positions satisfy |x|≤A; asking for a real speed outside that range contradicts the assigned energy rather than revealing a negative kinetic energy.

At equilibrium all this mechanical energy is kinetic and speed is greatest. At the endpoints it is all potential. At x=A/2, potential is one-quarter of total and kinetic is three-quarters: energy depends on displacement squared, so halfway in position is not halfway in energy. Equal energy shares occur at |x|=A/√2.

A potential-energy-versus-position graph is an upward parabola. The fixed total-energy line intersects it at the two turning points, and their vertical separation gives kinetic energy. This is not a time graph: the oscillator spends more time near turning points where it moves slowly. For a vertical spring the shifted effective potential must include gravity consistently rather than dropping gravitational energy silently.

3. Work an example

For k=20 N/m, m=2 kg and A=0.20 m, E=0.40 J. At x=0.10 m, U=0.10 J and K=0.30 J, giving speed √0.30≈0.548 m/s. Either velocity sign is possible, depending on whether the mass is moving toward or away from the positive endpoint.

4. Predict and investigate

If amplitude doubles, what happens to total mechanical energy and maximum speed?

Compare position fractions 0, 0.5 and 1 for one amplitude; reconstruct the total from each pair of energy entries. Then double amplitude while keeping the same position fraction and predict the energy and speed factors before capture.

5. Transfer to a new case

Try a new practice case

A spring has k=50 N/m and amplitude 0.4 m. Find its total mechanical energy.

Your numerical result (J) 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. Explain the boundary

Why does an energy ledger alone not tell whether the oscillator is approaching a turning point?

These are bounded virtual models and supplied numerical cases, not independent measurements or an observed physical investigation. Do not construct spring, pendulum or collision apparatus from these instructions. Practical evidence requires an appropriate supervised setting.

Sources

Original explanations, worked cases and numerical models. Framework checked 30 October 2026: AP Physics 1 Course and Exam Description, topics 7.4. Mechanism reference: OpenStax University Physics 1, simple harmonic motion and pendulums. No official assessment items reproduced.