4. A car of mass \(M\) moves along a straight horizontal road. The total resistance to motion of the car is modelled as having constant magnitude \(R\). The engine of the car works at a constant rate \(R U\). Find the time taken for the car to accelerate from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\). \section*{5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]} The vector \(\mathbf { n } = \left( - \frac { 3 } { 5 } \mathbf { i } + \frac { 4 } { 5 } \mathbf { j } \right)\) and the vector \(\mathbf { p } = \left( - \frac { 4 } { 5 } \mathbf { i } + \frac { 3 } { 5 } \mathbf { j } \right)\) are perpendicular unit vectors.

1. Verify that \(\frac { 9 } { 5 } \mathbf { n } + \frac { 13 } { 5 } \mathbf { p } = ( \mathbf { i } + 3 \mathbf { j } )\). A smooth uniform sphere \(S\) of mass 0.5 kg is moving on a smooth horizontal plane when it collides with a fixed smooth vertical wall which is parallel to \(\mathbf { p }\). Immediately after the collision the velocity of \(S\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The coefficient of restitution between \(S\) and the wall is \(\frac { 9 } { 16 }\).
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(S\) immediately before the collision.
3. Find the energy lost in the collision. \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d6e5bd56-0a01-44a2-b439-f80cb356d46d-3_681_747_1121_679}
   \end{figure} A smooth wire \(P M Q\) is in the shape of a semicircle with centre \(O\) and radius \(a\). The wire is fixed in a vertical plane with \(P Q\) horizontal and the mid-point \(M\) of the wire vertically below \(O\). A smooth bead \(B\) of mass \(m\) is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity \(4 m g\) and natural length \(\frac { 5 } { 4 } a\). The other end of the string is attached to a fixed point \(F\) which is a distance \(a\) vertically above \(O\), as shown in Fig. 1.