$$\begin{gathered} A goose is flying horizontally along at a speed of S m/s and at an altitude of H m when it passes a goose \\ hunter below lying at (0,0) with a goose rifle. Simul\tan eously, the hunter shoots a bullet up at an angle \theta from \\ the horizontal with a velocity V m/s hoping to shoot the goose. Assume the acceleration due to gravity is g m/s^2. \\ \left(i\right) Explain why x = St describes the dis\tan ce the goose has flown after t seconds. You may assume the equations \\ of motion x =Vt \cos \theta and y =Vt \sin \theta - \frac{g{t}^2}{2} describe the horizontal and vertical dis\tan ces of the bullet after t seconds. \\ Assume that the bullet hits the goose. \\ \left(ii\right) Explain why S =V \cos \theta . \\ \left(iii\right) Hence show that 2 \tan 2 H = - \frac{g{t}^2}{2} + St \theta . \\ \left(iv\right) Hence show that there are two possible occasions when the bullet can shoot the goose if S^2 {\tan }^2\theta > 2gH . \end{gathered}$$