# 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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