Edexcel M4 (Mechanics 4) 2005 June

1. A small smooth ball of mass \(\frac { 1 } { 2 } \mathrm {~kg}\) is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). Immediately before striking the plane the ball has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between ball and plane is \(\frac { 1 } { 2 }\). Find
   1. the speed, to 3 significant figures, of the ball immediately after the impact,
   2. the magnitude of the impulse received by the ball as it strikes the plane.
   3. A cyclist \(P\) is cycling due north at a constant speed of \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At 12 noon another cyclist \(Q\) is due west of \(P\). The speed of \(Q\) is constant at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find the course which \(Q\) should set in order to pass as close to \(P\) as possible, giving your answer as a bearing.
      (5)
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6895ccda-84b8-45a5-9524-e5bfc37a2fee-2_437_1232_1174_443}
   \end{figure} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \(( 1 - e ) \tan \beta = 2 \tan \alpha\).

4. A lorry of mass \(M\) is moving along a straight horizontal road. The engine produces a constant driving force of magnitude \(F\). The total resistance to motion is modelled as having magnitude \(k v ^ { 2 }\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),

1. show that \(V = \sqrt { \frac { F } { k } }\). Given instead that the lorry starts from rest,
2. show that the distance travelled by the lorry in attaining a speed of \(\frac { 1 } { 2 } V\) is $$\frac { M } { 2 k } \ln \left( \frac { 4 } { 3 } \right)$$

1. A non-uniform rod \(B C\) has mass \(m\) and length \(3 l\). The centre of mass of the rod is at distance \(l\) from \(B\). The rod can turn freely about a fixed smooth horizontal axis through \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\frac { m g } { 6 }\), is attached to \(C\). The other end of the string is attached to a point \(P\) which is at a height \(3 l\) vertically above \(B\).
   1. Show that, while the string is stretched, the potential energy of the system is
   $$m g l \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant, }$$ where \(\theta\) is the angle between the string and the downward vertical and \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\).
2. Find the values of \(\theta\) for which the system is in equilibrium with the string stretched.

6. A ship \(A\) has maximum speed \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t = 0 , A\) is 70 km due west of \(B\) which is moving at a constant speed of \(36 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). Ship \(A\) moves on a straight course at a constant speed and intercepts \(B\). The course of \(A\) makes an angle \(\theta\) with due north.

1. Show that \(- \arctan \frac { 4 } { 3 } \leq \theta \leq \arctan \frac { 4 } { 3 }\).
2. Find the least time for \(A\) to intercept \(B\).
   (5)

7. A light elastic string, of natural length \(a\) and modulus of elasticity \(5 m a \omega ^ { 2 }\), lies unstretched along a straight line on a smooth horizontal plane. A particle of mass \(m\) is attached to one end of the spring. At time \(t = 0\), the other end of the spring starts to move with constant speed \(U\) along the line of the spring and away from the particle. As the particle moves along the plane it is subject to a resistance of magnitude \(2 m \omega v\), where \(v\) is its speed. At time \(t\), the extension of the spring is \(x\) and the displacement of the particle from its initial position is \(y\). Show that

1. \(x + y = U t\),
2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 \omega ^ { 2 } x = 2 \omega U\).
3. Find \(x\) in terms of \(\omega , U\) and \(t\).