Give examples of a one-dimensional motion where:
$(a)$ The particle moving along positive $x$-direction comes to rest periodically and moves forward.
$(b)$ The particle moving along positive $x$-direction comes to rest periodically and moves backward.

(N/A) To represent periodic motion,we use trigonometric functions like $\sin t$ or $\cos t$.
$(a)$ Consider the motion $x(t) = t - \sin t$.
Velocity $v(t) = \frac{dx}{dt} = 1 - \cos t$.
At $t = 2n\pi$,$v = 1 - 1 = 0$,so the particle comes to rest. Since $v \ge 0$ for all $t$,the particle always moves in the positive $x$-direction.
$(b)$ Consider the motion $x(t) = \sin t$.
Velocity $v(t) = \frac{dx}{dt} = \cos t$.
At $t = \frac{\pi}{2}, \frac{3\pi}{2}, \dots$,the velocity $v = 0$,so the particle comes to rest. Since $\cos t$ alternates between positive and negative values,the particle moves forward and backward periodically.