Problem

A particle $P$ of mass $m\,\mathrm{kg}$ is projected vertically upwards from a point $O$ with an initial speed of $20\,\mathrm{m\,s^{-1}}$ and moves under gravity. There is a resistive force of magnitude $0.025mv^2\,\mathrm{N}$, where $v\,\mathrm{m\,s^{-1}}$ is the speed of $P$ at time $t\,\mathrm{s}$ after projection. The displacement of $P$ from $O$ is $x\,\mathrm{m}$ at time $t\,\mathrm{s}$ after projection.

(a) Find an expression for $v$ in terms of $x$, while $P$ is moving upwards.

(b) Find an expression for $v$ in terms of $t$, while $P$ is moving upwards.