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Velocity and Acceleration in Simple Harmonic Motion

It is instructive to learn how velocity and acceleration in an SHM vary with time. We know that displacement x(t) is given by

X(t) = A sin (ωt + φ )
The velocity V and acceleration a are given by



We notice that when the displacement is maximum (+A or-A) the velocity V = 0, because now the oscillator has to return and the velocity must change its direction. But when x is maximum (+A or –A) the acceleration is also maximum (-ω 2 A and + ω 2 A respectively) and is directed opposite to the displacement. When x = 0, i.e. when sin (ωt + φ) = 0, velocity V is maximum (Aω or - Aω ) and acceleration is zero. In the given figure we have plotted separately the x versus t, V versus t and the a versus t curves for an SHM, taking, for simplicity, φ = 0.

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