# (Solved): A cannon of mass m is fired at speed u at an angle (with 02 ) from the origin O, a ...

A cannon of mass is fired at speed at an angle (with ) from the origin , and it lands at a point away from on the same level. Assume that there is no air resistance during the flight of the cannon. Let be the position of the cannon from the origin at time , and be the magnitude of gravitational acceleration. (a) By setting up the equations of motion of along the horizontal and vertical directions, or otherwise, find and at time . (3 marks) (b) The length of the trajectory of a particle is given by where and are the initial and final times during travelling along the trajectory, and and are the initial and final positions. Using this fact, show that the length of the trajectory that the cannon travels is (4 marks) (c) By using the previous results, or otherwise, show that in order to attain maximum , the angle of projection satisfies Find the numerical value of . Hence, show that the corresponding at this angle is (5 marks) (d) Show that the area under the trajectory of a projectile and above the horizontal ground is (4 marks) (e) By using the previous results, or otherwise, find the angle so as to achieve the maximum . Hence, find the corresponding maximised area at this angle. (4 marks)

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