A particle $P$ travels in a straight line. It passes through the point $O$ of the line with velocity $5{\text{ }}m{\text{ }}{s^{ - 1}}$ at time $t = 0$, where $t$ is in seconds. P’s velocity after leaving $O$ is given by
$\left( {0.002{t^3} - 0.12{t^2} + 1.8t + 5} \right){\text{ }}m{\text{ }}{s^{ - 1}}$
The velocity of $P$ is increasing when $0 \lt t \lt {T_1}$ and when $t \gt {T_2}$, and the velocity of $P$ is decreasing when ${T_1} \lt t \lt {T_2}$.
a) Find the values of ${T_1}$ and ${T_2}$ and the distance $OP$ when $t = {T_2}$.
b) Find the velocity of $P$ when $t = {T_2}$ and sketch the velocity-time graph for the motion of $P$.
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