Edexcel M4 (Mechanics 4) 2007 June

1. A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is \(e\). Immediately before the impact the direction of motion of the ball makes an angle of \(60 ^ { \circ }\) with the wall. Immediately after the impact the direction of motion of the ball makes an angle of \(30 ^ { \circ }\) with the wall.
   1. Find the fraction of the kinetic energy of the ball which is lost in the impact.
      (6)
   2. Find the value of \(e\).
      (4)
   3. A lorry of mass \(M\) moves along a straight horizontal road against a constant resistance of magnitude \(R\). The engine of the lorry works at a constant rate \(R U\), where \(U\) is a constant. At time \(t\), the lorry is moving with speed \(v\).
   4. Show that \(M v \frac { \mathrm {~d} v } { \mathrm {~d} t } = R ( U - v )\).
      (3)
   At time \(t = 0\), the lorry has speed \(\frac { 1 } { 4 } U\) and the time taken by the lorry to attain a speed of \(\frac { 1 } { 3 } U\) is \(\frac { k M U } { R }\), where \(k\) is a constant.
2. Find the exact value of \(k\).
   (7)

3. \begin{figure}[h] \end{figure} A framework consists of two uniform rods \(A B\) and \(B C\), each of mass \(m\) and length \(2 a\), joined at \(B\). The mid-points of the rods are joined by a light rod of length \(a \sqrt { } 2\), so that angle \(A B C\) is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point \(A\) and is perpendicular to the plane of the framework. The angle between the \(\operatorname { rod } A B\) and the downward vertical is denoted by \(\theta\), as shown in Fig. 1.

1. Show that the potential energy of the framework is $$- m g a ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
2. Find the value of \(\theta\) when the framework is in equilibrium, with \(B\) below the level of \(A\).
3. Determine the stability of this position of equilibrium.

4. At 12 noon, \(\operatorname { ship } A\) is 20 km from ship \(B\), on a bearing of \(300 ^ { \circ }\). Ship \(A\) is moving at a constant speed of \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(070 ^ { \circ }\). Ship \(B\) moves in a straight line with constant speed \(V \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and intercepts \(A\).

1. Find, giving your answer to 3 significant figures, the minimum possible for \(V\). It is now given that \(V = 13\).
2. Explain why there are two possible times at which ship \(B\) can intercept ship \(A\).
3. Find, giving your answer to the nearest minute, the earlier time at which ship \(B\) can intercept ship \(A\).
   (8)

5. A smooth uniform sphere \(A\) has mass \(2 m \mathrm {~kg}\) and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide. At the instant of collision the line joining the centres of the spheres is parallel to j. Immediately after the collision, the velocity of \(A\) is ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).

1. Find the velocities of the two spheres immediately before the collision.
2. Find the magnitude of the impulse in the collision.
3. Find, to the nearest degree, the angle through which the direction of motion of \(A\) is deflected by the collision.

6. A small ball is attached to one end of a spring. The ball is modelled as a particle of mass 0.1 kg and the spring is modelled as a light elastic spring \(A B\), of natural length 0.5 m and modulus of elasticity 2.45 N . The particle is attached to the end \(B\) of the spring. Initially, at time \(t = 0\), the end \(A\) is held at rest and the particle hangs at rest in equilibrium below \(A\) at the point \(E\). The end \(A\) then begins to move along the line of the spring in such a way that, at time \(t\) seconds, \(t \leq 1\), the downward displacement of \(A\) from its initial position is 2 sin \(2 t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of the particle below \(E\) is \(y\) metres.

1. Show, by referring to a simple diagram, that \(y + 0.2 = x + 2 \sin 2 t\).
2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 49 y = 98 \sin 2 t\). Given that \(y = \frac { 98 } { 45 } \sin 2 t\) is a particular integral of this differential equation,
3. find \(y\) in terms of \(t\).
4. Find the time at which the particle first comes to instantaneous rest.