$(49\mathrm{.}1)$ A merry$-$go$-$round starts at rest. From $t=0s$ to $t=12s$ it accelerates at a constant rate $=\frac{}{48}ra\frac{d}{s^2}.$ From $t=12s$ to $t=132s$ the merry$-$go$-$round rotates at constant angular speed. After $t=132s$ the merry$-$go$-$round slows uniformly and comes to a halt at $t=150s.$

$($a$)$ Consider the first regime $($from $t=012s).$

$($i$)$ Determine the angular speed of the merry$-$go$-$round at the end of this regime.

$($ii$)$ Determine the angle that the merry$-$go$-$round travelled during this regime in radians and in revolutions.

$($b$)$ Consider the second regime $($from $t=12132s).$ Determine the angle that the merry$-$go$-$round travelled during this regime in radians and in revolutions.

$($c$)$ Consider the third regime $($from $t=132150s).$

$($i$)$ Determine the $[$average$]$ angular acceleration of the merry$-$go$-$round during this regime.

$($ii$)$ Determine the angle that the merry$-$go$-$round travelled during this regime in radians and in revolutions.

$($d$)($i$)$ Determine the total angular displacement of the merry$-$go$-$round throughout the entire two minute and thirty second ride in radians and in revolutions.

$($ii$)$ Determine the average angular speed of the merry$-$go$-$round during the entire ride in radians per second and revolutions per minute.