Explain by drawing graphs of displacement $x(t) \to t$,velocity $v(t) \to t$ and acceleration $a(t) \to t$ of $SHM$ for initial phase zero.

(N/A) For a particle in Simple Harmonic Motion $(SHM)$ with initial phase zero,the displacement is given by $x(t) = A \cos(\omega t)$.
The velocity is the time derivative of displacement: $v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t)$.
The acceleration is the time derivative of velocity: $a(t) = \frac{dv}{dt} = -A\omega^2 \cos(\omega t)$.
These equations show that all three quantities vary periodically with time with the same period $T = \frac{2\pi}{\omega}$.
$1$. Displacement $x(t)$ varies between $-A$ and $+A$.
$2$. Velocity $v(t)$ varies between $-A\omega$ and $+A\omega$.
$3$. Acceleration $a(t)$ varies between $-A\omega^2$ and $+A\omega^2$.
Phase relationships:
- Velocity leads displacement by a phase of $\frac{\pi}{2}$.
- Acceleration leads velocity by a phase of $\frac{\pi}{2}$,and leads displacement by a phase of $\pi$.