$($AP$,$ CH$2)$ A particle's position is given by the equation: $x(t)=a{t}^3-b{t}^2+c,$ where x is in meters and t is in second. The constants $a,b$ and c are given as$,a=1\mathrm{.}0m{s}^{-3},b=15m{s}^{-2},$ and $c=4\mathrm{.}0m.$

$($a$)$ Determine the velocity of particle as a function of time.

$($b$)$ Determine the acceleration of the particle as a function of time.

$($c$)$ Find the acceleration of the particle at $t=5\mathrm{.}0s.$