5 A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(u\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The particle \(P\) moves freely under gravity and passes through the point with coordinates \(\left( 3 a , \frac { 4 } { 5 } a \right)\) relative to horizontal and vertical axes through \(O\) in the plane of the motion.

1. Use the equation of the trajectory to show that \(u ^ { 2 } = 25 a g\).
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-09_2725_35_99_20} At the instant when \(P\) is moving horizontally, a particle \(Q\) is projected from \(O\) with speed \(V\) at an angle \(\alpha\) above the horizontal. The particles \(P\) and \(Q\) reach the ground at the same point and at the same time.
2. Express \(V ^ { 2 }\) in the form \(k a g\), where \(k\) is a rational number.
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-10_506_803_255_630} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and the string makes an angle \(\theta\) with the downward vertical through \(O\). The particle \(P\) is projected at right angles to the string with speed \(\frac { 1 } { 3 } \sqrt { 10 a g }\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point \(A\) which is vertically below \(O\). The circular path and the point \(A\) are in the same vertical plane. After the string strikes the peg, the particle \(P\) begins to move in a vertical circle with centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\) the string becomes slack (see diagram). The distance of \(A\) below \(O\) is \(\frac { 5 } { 9 } a\).
3. Find the value of \(\cos \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-11_2725_35_99_20}
4. Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg.
   \(7 \quad\) A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1 m v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\).
5. Find an expression for \(v\) in terms of \(t\).
   The displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\) is \(x \mathrm {~m}\).
6. Find an expression for \(v ^ { 2 }\) in terms of \(x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-14_2716_31_106_2016}