← all cardsTopic 10 · Oscillations · Equations & the two systems
The two maxima
● VERBATIM — examined word-for-wordEach differentiation multiplies the amplitude by ω. Speed peaks at equilibrium; acceleration peaks at the extremes, directed towards equilibrium.
v_max = ωx₀ (at x = 0) · a_max = ω²x₀ (at x = ±x₀)What this actually means
Speed peaks at equilibrium, where all the energy is kinetic. Acceleration peaks at the extremes, where the restoring force is largest, and it always points towards the centre.
The ω powers are the classic mix-up: v_max = ωx₀ but a_max = ω²x₀. Each differentiation buys exactly one more factor of ω.
The two maxima never happen at the same place or moment: max speed at the centre, max acceleration at the edges, a quarter period apart.
The trap
Swapping the ω powers — v_max = ωx₀, a_max = ω²x₀.
Prove it — watch it be true
- Read the v-axis intercept of the v–x ellipse: ±ωx₀, reached at x = 0 — v_max lives at equilibrium
- Read the ends of the a–x line at x = ±x₀: |a| is largest there at ω²x₀, pointing inward — a_max lives at the extremes
- Line up the x, v and a–t graphs: the v peak and the a peak sit a quarter cycle apart, so the two maxima never coincide