6 A particle is projected with speed \(v \mathrm {~ms} ^ { - 1 }\) from a point \(O\) on horizontal ground. The angle of projection is \(\theta ^ { \circ }\) above the horizontal. At time \(t\) seconds after the instant of projection the horizontal displacement of the particle from \(O\) is \(x \mathrm {~m}\) and the upward vertical displacement from \(O\) is \(y \mathrm {~m}\).

1. Show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$ A stone is thrown from the top of a vertical cliff 100 m high. The initial speed of the stone is \(16 \mathrm {~ms} ^ { - 1 }\) and the angle of projection is \(\theta ^ { \circ }\) to the horizontal. The stone hits the sea 40 m from the foot of the cliff.
2. Find the two possible values of \(\theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-3_623_995_1475_536} A uniform ladder \(A B\) of weight \(W \mathrm {~N}\) and length 4 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). A small object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at a point \(C\), which is 1 m from \(A\). The coefficient of friction between the ladder and the ground is \(\mu\) and the system is in limiting equilibrium.