A particle $P$ is projected horizontally from point $O$ along a rough horizontal surface. The coefficient of friction between $P$ and the surface is $0.2$. A horizontal force of magnitude $0.06t$ N, acting away from $O$, is exerted on $P$, where $t$ is the time after projection. $P$ is at rest when $t = 4$.
(i)[2]
The particle starts moving once more when $t = 8$. Show that the mass of $P$ is $0.24\,\text{kg}$.
(ii)[5]
Show that, for $0 \leq t \leq 4$, $\frac{dv}{dt} = 0.25t - 2$, and find the speed at which $P$ is projected.
(iii)[4]
Find the distance from $O$ where $P$ is brought to rest.
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