4. Ryan is playing a game of snooker. The horizontal table is modelled as the horizontal \(x - y\) plane with the point \(O\) as the origin and unit vectors parallel to the \(x\)-axis and the \(y\)-axis denoted by \(\mathbf { i }\) and \(\mathbf { j }\) respectively. All balls on the table have a common mass \(m \mathrm {~kg}\). The table and the four sides, called cushions, are modelled as smooth surfaces. The dimensions of the table, in metres, are as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-5_663_1138_667_482} Initially, all balls are stationary. Ryan strikes ball \(A\) so that it collides with ball \(B\). Before the collision, \(A\) has velocity \(( - \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and, after the collision, it has velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Show that the velocity of ball \(B\) after the collision is \(( - 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). After the collision with ball \(A\), ball \(B\) hits the cushion at point \(C\) before rebounding and moving towards the pocket at \(P\). The cushion is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the cushion and ball \(B\) is \(\frac { 5 } { 7 }\).
2. Calculate the velocity of ball \(B\) after impact with the cushion.
3. Find, in terms of \(m\), the magnitude of the impulse exerted on ball \(B\) by the cushion at \(C\), stating your units clearly.
4. Given that \(C\) has position vector \(( x \mathbf { i } + 1 \cdot 75 \mathbf { j } ) \mathrm { m }\),
   1. determine the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\),
   2. find the value of \(x\).
5. Describe one way in which the model used could be refined. Explain how your refinement would affect your answer to (d)(i).