Once you know the position of the oscillator for all times, you can work out the velocity and acceleration functions.

  x(t) = A cos (wt + f)


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The velocity is the time derivative of the position so:
   v(t) = -A w sin (wt + f)
The change from cos to sin means that the velocity is 90^o out of phase with the displacement-when x = 0 the velocity is a maximum and when x is a maximum v = 0. This is seen nicely in the page on the connection between SHM and circular motion.


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The acceleration is the time derivative of the velocity so:
   a(t) = -A w² cos (wt + f)
Notice also from the preceding that:
a(t) = -w²x   The acceleration is exactly out of phase with the displacement. This can also be seen in the page on the connection between SHM and circular motion

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