A particle is projected from \(O\) at an angle of \(\theta\) with a velocity \(V\) metres per second. The particle lands at a distance of \(R\) metres from the origin at time \(T\). The equations of motion of the particle are \(\ddot{x} = 0\) and \(\ddot{y} = -g\).

i) Using calculus, derive the expressions for the position of the particle and time \(t\).
ii) Show that \(\displaystyle \tan\theta = \frac{gT^2}{2R}\).
iii) Show that \(\displaystyle R = \frac{V^2\sin2\theta}{g}\).