1 A particle \(P\) is projected with speed \(u\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac { 2 } { 3 } T\) after projection.
\includegraphics[max width=\textwidth, alt={}, center]{6dcd0997-d7a1-463c-9040-96a5e81623cf-04_362_750_258_653} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3 m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2 \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }\). Show that \(\cos \theta = \frac { 1 } { 3 }\) and find \(x\) in terms of \(a\).