Problem

A particle $P$ of mass $m\,\mathrm{kg}$ moving along a rough horizontal table has displacement $x\,\mathrm{m}$ from a fixed point $O$ on the table and velocity $v\,\mathrm{m\,s^{-1}}$ at time $t\,\mathrm{s}$. The particle $P$ is subject to a resistive force of magnitude $mgkv\,\mathrm{N}$, where $k$ is a positive constant, and a frictional force of magnitude $\mu mg$. The particle $P$ is initially at $O$ with speed $U\,\mathrm{m\,s^{-1}}$.

(a) Show that

$$t=\frac1{gk}\ln\bigg(\frac{kU+\mu}{kv+\mu}\bigg).$$

It is given that $U=10$, $k=0.04$ and $\mu=0.2$.

(b) Find the distance $P$ moves before coming to rest.

(c) Find the average speed of $P$ over the period it is moving.