Edexcel M4 (Mechanics 4) 2013 June

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]

Boat \(A\) is moving with velocity ( \(3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and boat \(B\) is moving with velocity \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find

1. the magnitude of the velocity of \(A\) relative to \(B\),
2. the direction of the velocity of \(A\) relative to \(B\), giving your answer as a bearing.

2. \begin{figure}[h] \end{figure} A smooth fixed plane is inclined at an angle \(\alpha\) to the horizontal. A smooth ball \(B\) falls vertically and hits the plane. Immediately before the impact the speed of \(B\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Immediately after the impact the direction of motion of \(B\) is horizontal. The coefficient of restitution between \(B\) and the plane is \(\frac { 1 } { 3 }\). Find the size of angle \(\alpha\).

1. A smooth uniform sphere \(A\), of mass \(5 m\) and radius \(r\), is at rest on a smooth horizontal plane. A second smooth uniform sphere \(B\), of mass \(3 m\) and radius \(r\), is moving in a straight line on the plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes \(A\). Immediately before the impact the direction of motion of \(B\) makes an angle of \(60 ^ { \circ }\) with the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(30 ^ { \circ }\) by the impact.

Find

1. the speed of \(B\) immediately after the impact,
2. the coefficient of restitution between the spheres.

1. At 10 a.m. two walkers \(A\) and \(B\) are 4 km apart with \(A\) due north of \(B\). Walker \(A\) is moving due east at a constant speed of \(6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Walker \(B\) is moving with constant speed \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and walks in the straight line which allows him to pass as close as possible to \(A\).

Find

1. the direction of motion of \(B\), giving your answer as a bearing,
2. the least distance between \(A\) and \(B\),
3. the time when the distance between \(A\) and \(B\) is least.

5. A van of mass 1200 kg travels along a straight horizontal road against a resistance to motion which is proportional to the speed of the van. The engine of the van is working at a constant rate of 40 kW . The van starts from rest at time \(t = 0\). At time \(t\) seconds, the speed of the van is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed of the van is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that $$75 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2500 - v ^ { 2 }$$
2. Find \(v\) in terms of \(t\).

6. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(4 m\) and length \(4 l\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). A particle of mass \(k m\), where \(k < 7\), is attached to the rod at \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity 4 mg , is attached to the point \(D\) of the rod, where \(A D = 3 l\). The other end of the string is attached to a fixed point \(E\) which is vertically above \(A\), where \(A E = 3 l\), as shown in Figure 2. The angle between the rod and the upward vertical is \(2 \theta\), where \(\arcsin \left( \frac { 1 } { 6 } \right) < \theta \leqslant \frac { \pi } { 2 }\).

1. Show that, while the string is stretched, the potential energy of the system is $$8 m g l \left\{ ( 7 - k ) \sin ^ { 2 } \theta - 3 \sin \theta \right\} + \text { constant }$$ There is a position of equilibrium with \(\theta \leqslant \frac { \pi } { 6 }\).
2. Show that \(k \leqslant 4\) Given that \(k = 4\),
3. show that this position of equilibrium is stable.

7. A particle \(P\) of mass 0.5 kg is attached to the end \(A\) of a light elastic spring \(A B\), of natural length 0.6 m and modulus of elasticity 2.7 N . At time \(t = 0\) the end \(B\) of the spring is held at rest and \(P\) hangs at rest at the point \(C\) which is vertically below \(B\). The end \(B\) is then moved along the line of the spring so that, at time \(t\) seconds, the downwards displacement of \(B\) from its initial position is \(4 \sin 2 t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of \(P\) below \(C\) is \(y\) metres.

1. Show that $$y + \frac { 49 } { 45 } = x + 4 \sin 2 t$$
2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 9 y = 36 \sin 2 t$$ Given that \(y = \frac { 36 } { 5 } \sin 2 t\) is a particular integral of this differential equation,
3. find \(y\) in terms of \(t\),
4. find the speed of \(P\) when \(t = \frac { 1 } { 3 } \pi\).