Problem

A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. The initial velocity of $P$ is $25\,\mathrm{m\,s^{-1}}$ at an angle $\theta$ above the horizontal, where $\tan\theta=\frac43$. At point $A$, the direction of motion of $P$ makes an angle of $45^\circ$ with the downward vertical through $A$.

(a) By differentiating the equation of the trajectory or otherwise, find the coordinates of $A$.

At point $A$, the particle strikes a fixed smooth barrier, rebounds, and lands on the horizontal plane. The barrier is inclined at an angle of $45^\circ$ to the horizontal.

(b) Find the speed of $P$ immediately before it collides with the barrier.

(c) Given that the coefficient of restitution between the barrier and the particle is $\frac19$, find the horizontal distance travelled by $P$ after it strikes the barrier.