2 A particle \(Q\) of mass 0.2 kg is projected horizontally with velocity \(4 \mathrm {~ms} ^ { - 1 }\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(Q\) is \(x \mathrm {~m}\) from \(A\) and is moving away from \(A\) with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of \(3 \cos 2 t \mathrm {~N}\) acting on \(Q\) in the positive \(x\)-direction.

1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies.
2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac { 3 } { 2 } \pi\).
   \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_549_1237_1724_415} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac { 4 } { 5 }\), and \(B\) is moving with speed \(3 \frac { 1 } { 4 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac { 5 } { 13 }\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).