3.102: Velocity–time graph and rectilinear motion

A vehicle travels along a straight, horizontal road. At time $t = 0$ seconds, the vehicle, travelling at a velocity of $w\,\mathrm{ms}^{-1}$, passes point $O$. The vehicle travels at this constant velocity for $12$ seconds. It then slows down, with constant deceleration, for $10$ seconds until it reaches a velocity of $(w - 14)\,\mathrm{ms}^{-1}$. It continues to travel at this velocity for $28$ seconds until it reaches point $A$, $458\,\mathrm{m}$ from $O$.

Find the value of $w$.

A particle moves in a straight line. The velocity, $v\,\mathrm{ms}^{-1}$, at time $t$ seconds ($t \geqslant 0$) is $v = (t - 4)(t - 5)$.

Find the value of $t$ for which the acceleration is $0\,\mathrm{ms}^{-2}$.

Find the set of values of $t$ for which the velocity of the particle is negative.

Find the distance travelled by the particle in the first $5$ seconds of its motion.