5 A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The point \(O\) is at a perpendicular distance of 1 m from the inclined plane. The ball is projected with velocity \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-14_478_913_571_561}

1. 1. Find the time taken by the ball to travel from \(O\) to \(A\).
   2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\).
2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\).
   Show that \(e < k\), where \(k\) is a constant to be determined.
   [0pt] [4 marks] \(6 \quad\) In this question use \(\cos 30 ^ { \circ } = \sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
   A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60 ^ { \circ }\) with the wall, as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-18_499_1036_721_593} The coefficient of restitution between \(A\) and \(B\) is \(e\).
3. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } u ( 1 + e )\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision.
4. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 + e } { 7 - e }\).
   [0pt] [7 marks]