$O$ and $A$ are fixed points on a horizontal surface, with $OA = 0.5{\text{ }}m$. A particle $P$ of mass $0.2{\text{ }}kg$ is projected horizontally with speed $3{\text{ }}m{\text{ }}{s^{ - 1}}$ from $A$ in the direction $OA$ and moves in a straight line (see diagram). At time $t\,s$ after projection, the velocity of $P$ is $vm{\text{ }}{s^{ - 1}}$ and its displacement from $O$ is $x{\text{ m}}$.

The coefficient of friction between the surface and $P$ is $0.5$, and a force of magnitude $\frac{{0.4}}{{{x^2}}}N$ acts on $P$ in the direction $PO$.

a) Show that, while the particle is in motion, $v\frac{{dv}}{{dx}} =  - \left( {5 + \frac{2}{{{x^2}}}} \right)$.

b) Calculate the distance travelled by $P$ before it comes to rest, and show that $P$ does not subsequently move.