← all cardsTopic 8 · 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
- Find the v–t peak: it lines up with x crossing zero
- Find the a–t peak: it lines up with x at an extreme
- Read the two peak values: they differ by exactly one factor of ω