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

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