In a game of “Airwars” one player has to attempt to shoot down another’s toy aircraft in mid-air using a foam missile.  

In a particular game a player launches their aircraft from the origin with velocity $(3i+18.7j)m{s}^{-1}$.  At the same instant their opponent launches their missile with velocity $(−5i+18.7j)m{s}^{-1}$  from the point with coordinates $(24,0),$ where the coordinates are expressed in metres. 

The flight paths of both the aircraft and the missile occur in the same vertical plane, and i and j  and  are respectively the unit vectors in the positive horizontal and vertical directions (where in the vertical direction upwards is taken to be positive). 

Modelling the motion of both the model aircraft and the model missile as projectiles moving under gravity alone, find the coordinates at which the missile hits the aircraft and how long both had been airborne prior to colliding.

For a particular projectile, $tan \alpha =\frac{3}{4}$, $U=10m{s}^{-1}$ and the particle is projected from the point $(3,8).$  

Find an expression for the trajectory of the particle, giving your answer in the form

 $y=\frac{a{x}^{2}+bx+c}{128}$

where the constants $a$, $b$ and $c$ are expressed in terms of $g$.

Find the distance between the particle and the origin after two seconds of motion, giving your answer to three significant figures.

This question has 3 part(s).