A particle $P$ of mass $0.4{\text{ }}kg$ moves in a straight line on a horizontal surface and has velocity $vm{\text{ }}{s^{ - 1}}$ at time $t\,s$. A horizontal force of magnitude $k{\text{ }}\surd v{\text{ N}}$ opposes the motion of $P$. When $t = 0$, $v = 9$ and when $t = 2$, $v = 4$.
a) Express $\frac{{dv}}{{dt}}$ in terms of $k$ and $v$, and hence show that $v = \frac{1}{4}{\left( {t - 6} \right)^2}$.
b) Find the distance travelled by $P$ in the first 3 seconds of its motion.
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