← all cardsTopic 10 · Oscillations · State of motion & graphs
Phase relationships between x, v and a
v leads x by π/2 (max speed through equilibrium); a leads v by π/2, so a is in antiphase (π) with x — the graphical face of the minus sign. Given one graph, get the next by taking gradients, not from memory.
What this actually means
v leads x by a quarter cycle: the mass is moving fastest as it crosses zero, a quarter period before x peaks. a leads v by another quarter, which puts a a full half-cycle from x.
That antiphase between a and x is nothing more than the minus sign in a = −ω²x drawn as graphs.
Given one curve, generate the next by taking gradients, not by recalling shapes. Gradient logic survives any starting phase; memorised shapes do not.
Prove it — watch it be true
- Drop the tangent on the x–t graph: the gradient is steepest where x crosses zero, and that is where the v–t curve built from it peaks — v leads x by π/2
- Repeat the tangent construction on v–t to build a–t: another quarter-cycle shift, so a leads v by π/2
- Compare a–t with x–t: an exact flip, antiphase, the minus sign of a = −ω²x drawn as graphs