A particle moves along the x axis. It is initially at the position $0\mathrm{.}130$ m$,$ moving with velocity $0\mathrm{.}250$ m$/$s and acceleration $-0\mathrm{.}320$ m$/$s$2.$ Suppose it moves with constant acceleration for $4\mathrm{.}20$ s$.$

$($a$)$ Find the position of the particle after this time.

$($b$)$ Find its velocity at the end of this time interval.

Next, assume it moves with simple harmonic motion for $4\mathrm{.}20$ s and x $=0$ is its equilibrium position. $($Assume that the velocity and acceleration is the same as in parts $($a$)$ and $($b$).)$

$($c$)$ Find its position.

$($d$)$ Find its velocity at the end of this time interval.