6.03d Conservation in 2D: vector momentum

4 From a helicopter, a small plane is spotted 3750 m away on a bearing of \(075 ^ { \circ }\). The plane is at the same altitude as the helicopter, and is flying with constant speed \(62 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal straight line on a bearing of \(295 ^ { \circ }\). The helicopter flies with constant speed \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line, and intercepts the plane.

1. Find the bearings of the two possible directions in which the helicopter could fly.
2. Given that interception occurs in the shorter of the two possible times, find the time taken to make the interception. \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-3_668_298_260_922} A uniform lamina of mass 63 kg occupies the region bounded by the \(x\)-axis, the \(y\)-axis, and the curve \(y = 8 - x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\). The unit of length is the metre. The vertices of the lamina are \(O ( 0,0 )\), \(A ( 2,0 )\) and \(B ( 0,8 )\) (see diagram).
3. Show that the moment of inertia of this lamina about \(O B\) is \(56 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). It is given that the moment of inertia of the lamina about \(O A\) is \(1036.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }\), and the centre of mass of the lamina has coordinates \(\left( \frac { 4 } { 5 } , \frac { 24 } { 7 } \right)\). The lamina is free to rotate in a vertical plane about a fixed horizontal axis passing through \(O\) and perpendicular to the lamina. Starting with the lamina at rest with \(B\) vertically above \(O\), a couple of constant anticlockwise moment 800 Nm is applied to the lamina.
4. Show that the lamina begins to rotate anticlockwise.
5. Find the angular speed of the lamina at the instant when \(O B\) first becomes horizontal. \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-4_709_752_267_699} A smooth circular wire, with centre \(O\) and radius \(a\), is fixed in a vertical plane, and the point \(A\) is on the wire at the same horizontal level as \(O\). A small bead \(B\) of mass \(m\) can move freely on the wire. A light elastic string, with natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\), passes through a fixed ring at \(A\), and has one end fixed at \(O\) and the other end attached to \(B\). The section \(A B\) of the string is at an angle \(\theta\) above the horizontal, where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), so that \(O B\) is at an angle \(2 \theta\) to the horizontal (see diagram).
6. Taking \(O\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a ( \sqrt { 3 } + \sqrt { 3 } \cos 2 \theta + \sin 2 \theta ) .$$
7. Find the two values of \(\theta\) for which the system is in equilibrium.
8. For each position of equilibrium, determine whether it is stable or unstable. \includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-5_478_1403_267_372} A thin horizontal rail is fixed at a height of 0.6 m above horizontal ground. A non-uniform straight \(\operatorname { rod } A B\) has mass 6 kg and length 3 m ; its centre of mass \(G\) is 2 m from \(A\) and 1 m from \(B\), and its moment of inertia about a perpendicular axis through its mid-point \(M\) is \(4.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The rod is placed in a vertical plane perpendicular to the rail, with \(A\) on the ground and \(M\) in contact with the rail. It is released from rest in this position, and begins to rotate about \(M\), without slipping on the rail. When the angle between \(A B\) and the upward vertical is \(\theta\) radians, the rod has angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\), the frictional force in the direction \(A B\) is \(F \mathrm {~N}\), and the normal reaction is \(R \mathrm {~N}\) (see diagram).
9. Show that \(\omega ^ { 2 } = 4.8 - 12 \cos \theta\).
10. Find the angular acceleration of the rod in terms of \(\theta\).
11. Show that \(F = 94.8 \cos \theta - 14.4\), and find \(R\) in terms of \(\theta\).
12. Given that the coefficient of friction between the rod and the rail is 0.9 , show that the rod will slip on the rail before \(B\) hits the ground.

4 \includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-2_688_777_1382_683} From a boat \(B\), a cruiser \(C\) is observed 3500 m away on a bearing of \(040 ^ { \circ }\). The cruiser \(C\) is travelling with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line course with bearing \(110 ^ { \circ }\) (see diagram). The boat \(B\) travels with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight line course which takes it as close as possible to the cruiser \(C\).

1. Show that the bearing of the course of \(B\) is \(073 ^ { \circ }\), correct to the nearest degree.
2. Find the magnitude and the bearing of the velocity of \(C\) relative to \(B\).
3. Find the shortest distance between \(B\) and \(C\) in the subsequent motion.

6 Two ships \(P\) and \(Q\) are moving on straight courses with constant speeds. At one instant \(Q\) is 80 km from \(P\) on a bearing of \(220 ^ { \circ }\). Three hours later, \(Q\) is 36 km due south of \(P\).

1. Show that the velocity of \(Q\) relative to \(P\) is \(19.1 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(063.8 ^ { \circ }\) (both correct to 3 significant figures).
2. Find the shortest distance between the two ships in the subsequent motion. Given that the speed of \(P\) is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and \(Q\) is travelling in the direction with bearing \(105 ^ { \circ }\), find
3. the bearing of the direction in which \(P\) is travelling,
4. the speed of \(Q\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-2_460_388_1160_826} A ship \(S\) is travelling with constant velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(120 ^ { \circ }\). A patrol boat \(B\) observes the ship when \(S\) is due north of \(B\). The patrol boat \(B\) then moves with constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line (see diagram).

1. Given that \(V = 18\), find the bearing of the course of \(B\) such that \(B\) intercepts \(S\).
2. Given instead that \(V = 9\), find the bearing of the course of \(B\) such that \(B\) passes as close as possible to \(S\).
3. Find the smallest value of \(V\) for which it is possible for \(B\) to intercept \(S\).

4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.
Immediately before the collision, A is moving with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) with the line of centres, where \(\sin \alpha = 0.6\), and B is moving with speed \(0.3 \mathrm {~ms} ^ { - 1 }\) at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 . \begin{figure}[h] \end{figure}

1. Find the speeds of A and B immediately after the collision.
2. Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
3. Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
4. How does your answer to part (b) change if the contact between B and the wall is not smooth?

5 Two small uniform discs, A of mass \(2 m \mathrm {~kg}\) and B of mass \(3 m \mathrm {~kg}\), slide on a smooth horizontal surface and collide obliquely with smooth contact. Immediately before the collision, A is moving towards B along the line of centres with speed \(2 \mathrm {~ms} ^ { - 1 }\) and B is moving towards A with speed \(\sqrt { 3 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-5_366_976_539_244}

1. Explain how you know that the motion of A will be along the line of centres after the collision.
2. - Determine the maximum possible speed of A after the collision.
   When the speed of B after the collision is a minimum, the loss of kinetic energy in the collision is 1.4625 J .
3. Determine the value of \(m\).

4 Two small smooth discs, A of mass 0.5 kg and B of mass 0.4 kg , collide while sliding on a smooth horizontal plane. Immediately before the collision A and B are moving towards each other, A with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Before the collision the direction of motion of A makes an angle \(\alpha\) with the line of centres, where \(\tan \alpha = 0.75\), and the direction of motion of B makes an angle of \(60 ^ { \circ }\) with the line of centres, as shown in Fig. 4. \begin{figure}[h] \end{figure} After the collision, one of the discs moves in a direction perpendicular to the line of centres, and the other disc moves in a direction making an angle \(\beta\) with the line of centres.

1. Explain why the disc which moves perpendicular to the line of centres must be A .
2. Determine the value of \(\beta\).
3. Determine the kinetic energy lost in the collision.
4. Determine the value of the coefficient of restitution between A and B .

4 Two uniform circular discs with the same radius, A of mass 1 kg and B of mass 5.25 kg , slide on a smooth horizontal surface and collide obliquely with smooth contact. Fig. 4 gives information about the velocities of the discs just before and just after the collision.

• The line XY passes through the centres of the discs at the moment of collision
• The components parallel and perpendicular to XY of the velocities of A are shown
• Before the collision, B is at rest and after it is moving at \(2 \mathrm {~ms} ^ { - 1 }\) in the direction XY

\begin{figure}[h] \end{figure} The coefficient of restitution between the two discs is \(\frac { 2 } { 3 }\).

1. Find the values of \(U\) and \(u\).
2. What information in the question tells you that \(v = V\) ? The speed of disc A before the collision is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the speed of disc A after the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_398_396_397_475} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_399_332_399_945} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_305_326_493_1354} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} Fig. 5.1 shows a vertical light elastic spring. It is fixed to a horizontal table at one end. Fig 5.2 shows the spring with a particle of mass \(m \mathrm {~kg}\) attached to it at the other end. The system is in equilibrium when the spring is compressed by a distance \(h \mathrm {~m}\).

5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .

11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.

1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
3. 1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
   2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?

5. Two smooth spheres \(A\) and \(B\), of equal radii, are moving on a smooth horizontal plane when they collide. Immediately after the collision sphere \(A\) has velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, their line of centres is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the spheres is \(\frac { 2 } { 5 }\). Sphere \(A\) has mass 4 kg and sphere \(B\) has mass 2 kg .

1. Find the velocity of \(A\) and the velocity of \(B\) immediately before the collision. After the collision, sphere \(A\) continues to move with velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) until it collides with a smooth vertical wall. The impulse exerted by the wall on \(A\) is \(32 \mathbf { j }\) Ns.
2. State whether the wall is parallel to the vector \(\mathbf { i }\) or to the vector \(\mathbf { j }\). Give a reason for your answer.
3. Find the speed of \(A\) after the collision with the wall.
4. Calculate the loss of kinetic energy caused by the collision of sphere \(A\) with the wall.

5. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Two smooth spheres \(P\) and \(Q\), of equal radii, are moving on a smooth horizontal surface. The mass of \(P\) is 2 kg and the mass of \(Q\) is 6 kg . The velocity of \(P\) is \(( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). At a particular instant, \(Q\) is positioned 12 m east and 48 m south of \(P\).

1. Prove that \(P\) and \(Q\) will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\). Immediately after the collision, sphere \(Q\) has speed \(5 \mathrm {~ms} ^ { - 1 }\).
2. Determine the coefficient of restitution between the spheres and hence calculate the velocity of \(P\) immediately after the collision.
3. Find the magnitude of the impulse required to stop sphere \(P\) after the collision.

5. A particle \(A\), of mass \(m \mathrm {~kg}\), has position vector \(11 \mathbf { i } + 6 \mathbf { j }\) and a velocity \(2 \mathbf { i } + 7 \mathbf { j }\). At the same moment, second particle \(B\), of mass \(2 m \mathrm {~kg}\), has position vector \(7 \mathbf { i } + 10 \mathbf { j }\) and a velocity \(5 \mathbf { i } + 4 \mathbf { j }\).

1. If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
2. Determine the impulse exerted by \(A\) on \(B\).
3. Calculate the loss of kinetic energy caused by the collision.

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere \(A\) has mass 0.2 kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\) The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\)

1. Find the velocity of \(A\) immediately after the collision.
2. Find the magnitude of the impulse received by \(A\) in the collision.
3. Find, to the nearest degree, the size of the angle through which the direction of motion of \(A\) is deflected as a result of the collision.

1. A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.2 kg .

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The kinetic energy of \(Q\) immediately after the collision is half the kinetic energy of \(Q\) immediately before the collision.

1. Find
   1. the velocity of \(P\) immediately after the collision,
   2. the velocity of \(Q\) immediately after the collision,
   3. the coefficient of restitution between \(P\) and \(Q\),
      carefully justifying your answers.
2. Find the size of the angle through which the direction of motion of \(P\) is deflected by the collision.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.5 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( u \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(u\) is a positive constant, and the velocity of \(Q\) is \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant when the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\).
The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 } { 5 }\) As a result of the collision, the direction of motion of \(P\) is deflected through an angle of \(90 ^ { \circ }\) and the direction of motion of \(Q\) is deflected through an angle of \(\alpha ^ { \circ }\)

1. Find the value of \(u\)
2. Find the value of \(\alpha\)
3. State how you have used the fact that \(P\) and \(Q\) have equal radii.

4. \begin{figure}[h] \end{figure} Two smooth uniform spheres, \(A\) and \(B\), have equal radii. The mass of \(A\) is \(3 m\) and the mass of \(B\) is \(4 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, \(A\) is moving with speed \(3 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres and \(B\) is moving with speed \(2 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(90 ^ { \circ }\) by the collision, as shown in Figure 3.

1. Find the size of the angle through which the direction of motion of \(A\) is turned as a result of the collision.
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision.

5. \begin{figure}[h] \end{figure} A smooth uniform sphere \(S\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(S\) collides obliquely with another uniform sphere of mass \(M\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision the direction of motion of \(S\) makes an angle \(\alpha\), where \(0 < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(S\) makes an angle \(\beta\) with the line joining the centres of the spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that \(\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }\) Given that \(m = e M\),
2. show that the directions of motion of the two spheres immediately after the collision are perpendicular.

7. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(A\) collides obliquely with a smooth uniform sphere \(B\) of mass \(3 m\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of \(A\) is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
2. Show that \(e > \frac { 1 } { 3 }\)
3. Show that \(0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

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 \(3 m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide obliquely.
Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The coefficient of restitution between the spheres is \(\frac { 1 } { 4 }\)

1. Find the velocity of \(B\) immediately after the collision.
2. Find, to the nearest degree, the size of the angle through which the direction of motion of \(B\) is deflected as a result of the collision.

2 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. Particle \(A\) has mass 2 kg and velocity \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). Particle \(B\) has mass 3 kg and velocity \(\left[ \begin{array} { r } - 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). The two particles collide, and they coalesce during the collision.

1. Find the velocity of the combined particles after the collision.
2. Find the speed of the combined particles after the collision.

5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the coefficient of restitution between the spheres.
3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
4. Find the mass of \(B\).

7 Two snooker balls, one white and one red, have equal mass. The balls are on a horizontal table \(A B C D\) The white ball is struck so that it moves at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to \(A B\) The white ball hits a stationary red ball.
After the collision, the white ball moves at a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(30 ^ { \circ }\) to \(A B\) After the collision, the red ball moves at a speed \(v \mathrm {~ms} ^ { - 1 }\) and at an angle \(\theta\) to \(C D\) When the collision takes place, the white ball is the same distance from \(A B\) as the distance the red ball is from CD The diagram below shows the table and the velocities of the balls after the collision. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-08_595_1370_1121_335} Not to scale After the collision, the white ball hits \(A B\) and the red ball hits \(C D\) Model the balls as particles that do not experience any air resistance.
7

1. Explain why the two balls hit the sides of the table at the same time.
   7
2. Show that \(\theta = 17.0 ^ { \circ }\) correct to one decimal place.
   7
3. \(\quad\) Find \(v\) 7
4. Determine which ball travels the greater distance after the collision and before hitting the side of the table. Fully justify your answer.
   7
5. State one possible refinement to the model that you have used. \(8 \quad\) In this question use \(g\) as \(9.8 \mathrm {~ms} ^ { - 2 }\) A rope is used to pull a crate, of mass 60 kg , along a rough horizontal surface.
   The coefficient of friction between the crate and the surface is 0.4 The crate is at rest when the rope starts to pull on it.
   The tension in the rope is 240 N and the rope makes an angle of \(30 ^ { \circ }\) to the horizontal.
   When the crate has moved 5 metres, the rope becomes detached from the crate.

4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).

1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(I = \frac { \sqrt { 3 } m v } { 2 ( 1 + m ) }\).
2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.

1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.

Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Calculate the coefficient of restitution between \(A\) and \(B\).
3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
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