Problem

A particle $P$ is projected with speed $u\,\mathrm{m\,s^{-1}}$ at an angle $\theta$, where $\tan\theta=2$, above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a time $t\,\mathrm{s}$ are denoted by $x\,\mathrm{m}$ and $y\,\mathrm{m}$ respectively.

(a) Use the equation of the trajectory given in the list of formulae (MF 19) to show that

$$y=2x-\frac{25x^2}{u^2}.$$

(b) In the subsequent motion, $P$ passes through the point with coordinates $(8,12)$. The particle then hits a fixed vertical barrier $7\,\mathrm{m}$ high that is at a horizontal distance of $D\,\mathrm{m}$ from the point of projection.

Find the set of possible values of $D$.