Coordinate displacement, velocity and acceleration phases

1. Frame the question

For release from the positive endpoint at t=0, write x=A cos(ωt). The velocity is v=−Aω sin(ωt) and acceleration is a=−Aω² cos(ωt)=−ω²x. You can interpret these without differentiating: velocity is the slope of the position graph, and acceleration tells how that slope changes.

2. Build the mechanism

At the positive turning point, x=A and v=0 but acceleration is maximally negative. One-quarter cycle later, the mass crosses equilibrium moving negatively at maximum speed; acceleration is zero at that instant. At half a cycle it reaches the negative turning point, again with v=0 but now maximum positive acceleration. At three-quarters of a cycle it crosses equilibrium in the positive direction.

Position and acceleration have opposite signs except where both are zero. Velocity is one-quarter cycle out of phase with position and cannot be inferred from position alone. At a given interior position the mass passes twice per cycle with opposite velocities. A graph’s vertical scale and units matter: displacement, velocity and acceleration amplitudes are A, Aω and Aω², not one common numerical quantity.

Other initial conditions require a phase shift, x=A cos(ωt+φ). For example, starting at equilibrium moving positively corresponds to a different phase than endpoint release. Shifting the time origin changes the phase label but does not change period, amplitude or the restoring equation. At an equilibrium crossing, zero instantaneous acceleration does not mean zero acceleration over a finite surrounding interval.

3. Work an example

Let A=0.20 m and ω=2 rad/s, so T=π s. At T/4, x=0, v=−0.40 m/s and a=0. At T/2, x=−0.20 m, v=0 and a=+0.80 m/s². The mass turns around at the second instant because acceleration is nonzero.

4. Predict and investigate

At the negative endpoint, is velocity negative, zero or positive?

Capture phase fractions 0, 0.25, 0.5 and 0.75 for A=0.2 m. Predict signs before reading the ledger. Sketch separate labeled x–t, v–t and a–t graphs, and explain the slope relationships rather than copying a single curve into all three.

5. Transfer to a new case

Try a new practice case

For A=0.3 m and ω=4 rad/s, find acceleration at the negative endpoint.

Your numerical result (m/s²) 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

How can the same x=0 occur with either positive or negative velocity?

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 7.3. Mechanism reference: OpenStax University Physics 1, simple harmonic motion and pendulums. No official assessment items reproduced.