Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m . Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac{35}{2} \mathrm{~ms}^{-1}\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha=\frac{4}{3}\). Particle \(Q\) is projected with speed \(u \mathrm{~ms}^{-1}\) at an angle \(\beta\) above the horizontal, where \(\tan \beta=\frac{1}{2}\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T \mathrm{~s}\) after the projection of particle \(Q\).
(a) Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4 u T=21 \sqrt{5}(T+1)\).
(b) Find the value of \(T\).
(c) Find the horizontal and vertical displacements of the particles from \(O\) when they collide.