At time \(t \mathrm{~s}\), a particle \(P\) is projected with speed \(40 \mathrm{~ms}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H \mathrm{~m}\) and the corresponding time is \(T \mathrm{~s}\).
(a) Obtain expressions for \(H\) and \(T\) in terms of \(\theta\).

During the time between \(t=T\) and \(t=3, P\) descends a distance \(\frac{1}{4} H\).
(b) Find the value of \(\theta\).

(c) Find the speed of \(P\) when \(t=3\).