The relationships below at the left show how the maximum velocity and acceleration of an object undergoing simple harmonic motion depend on the amplitude of the motion, the mass of the oscillating object and the force constant. The panel at the right describes how these relationships can be derived.
|  | Displacement as a function of time. Equation 1: x = X cos ωt Velocity is the derivative of distance with respect to time: Equation 2: vx = – Xω sin ωt Velocity is a maximum when the displacement (x) is 0. The line just at the left can be dragged to see how the graphs line up. x (from equation 1) is 0 when ωt = ±π/2. Substitutiing ωt = π/2 in equation 2 gives: vmax = X ω
Acceleration is the derivative of velocity with respect to time: | |||||||||||||||
| | Equation 3: ax = –Xω2 cos (ωt) Acceleration is a maximum when x is X From equation 1, x = X when ωt = 0 Substituting ωt = 0 into Equation 3 gives amax = Xω2
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