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A cannon of mass $m$ is fired at speed $u$ at an angle $?$ (with $0???2??$ ) from the origin $O$, and it lands at a point away from $O$ on the same level. Assume that there is no air resistance during the flight of the cannon. Let $(x,y)=(x(t),y(t))$ be the position of the cannon from the origin $O$ at time $t$, and $g$ be the magnitude of gravitational acceleration. (a) By setting up the equations of motion of along the horizontal and vertical directions, or otherwise, find $x=x(t)$ and $y=y(t)$ at time $t$. (3 marks) (b) The length of the trajectory $s$ of a particle is given by $s:=?_{t_{initial}}x?_{2}+y??_{2}?dt=?_{x_{initial}}1+(dxdy?)_{2}?dx$ where $t_{initial}$ and $t_{final}$ are the initial and final times during travelling along the trajectory, and $x_{initial}$ and $x_{initial}$ are the initial and final $x$ positions. Using this fact, show that the length of the trajectory $s$ that the cannon travels is $s=2gu_{2}?{cos_{2}(?)[?ln(1?sin(?))+ln(sin(?)+1)]+2sin(?)}$ (4 marks) (c) By using the previous results, or otherwise, show that in order to attain maximum $s$, the angle of projection $?$ satisfies $ln[cos(?)sin(?)+1?]sin(?)=1.$ Find the numerical value of $?$. Hence, show that the corresponding $s$ at this angle is $max(s)?1.1997gu_{2}?.$ (5 marks) (d) Show that the area under the trajectory of a projectile $A$ and above the horizontal ground is $A:=?_{x_{initial}}ydx=3g_{2}2u_{4}?sin_{3}(?)cos(?).$ (4 marks) (e) By using the previous results, or otherwise, find the angle $?$ so as to achieve the maximum $A$. Hence, find the corresponding maximised area $A$ at this angle. (4 marks)

SOLUTION : a) maximum height