8.1.2 v-t and a-t Equations

Remember that \displaystyle v=\frac{{dx}}{{dt}} and \displaystyle a=\frac{{dv}}{{dt}}?

If we begin with \displaystyle x={{x}_{0}}\sin \omega t

and differentiate both sides with respect to time, we obtain \displaystyle \displaystyle v=\omega {{x}_{0}}\cos \omega t

and differentiate both sides with respect to time, again we obtain \displaystyle a=-{{\omega }^{2}}{{x}_{0}}\sin \omega t

With just two steps of differentiation, we learn that

  1. In a SHM, displacement, velocity and acceleration all vary sinusoidally with time.
  2. Acceleration leads velocity by a quarter cycle, and velocity in turn leads displacement by a quarter cycle.
  3. Maximum velocity \displaystyle {{v}_{{\max }}}=\omega {{x}_{0}}
  4. Maximum acceleration \displaystyle {{a}_{{\max }}}={{\omega }^{2}}{{x}_{0}}.

Video Explanation

x-t, v-t and a-t Graphs of SHM

Concept Test

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