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.

Explanation

When we are writing an equation belonging to periodic nature it include sine or cosine function.
(a) The particle will be moveing along positive x-direction only if
`x(t) =1 - sin t`
Velocity `v (t) = (dx(t))/(dt) =1 - cos t`
Acceleration `a (t) = (dv)/(dt) = sin t `
When `t =0, x (t) =0`
When `t = pi , x (t) = pi gt 0`
When `t =0, x (t) = 2pi gt 0`
(b) Equation can be represented by
`x (t) = sin t `
`therefore v = (d)/(dt) x (t) = cos t `
As displacement and velocity includes sin t and cos t hence these equations represent periodic.