Oblique collision of spheres

5
\includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-3_362_841_264_651} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.

4
\includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-2_332_995_1375_575} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 3 kg respectively. They are moving on a horizontal surface, and they collide. Immediately before the collision, \(A\) is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the line of centres, where \(\sin \alpha = 0.8\), and \(B\) is moving along the line of centres with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.
[0pt] [10]

3
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-2_387_561_1055_794} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 0.8 kg and 2.0 kg respectively. The spheres are on a horizontal surface. \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres when it collides with \(B\), which is stationary (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the speed and direction of motion of \(A\) immediately after the collision.

2
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_460_725_731_708} Two uniform rods, \(A B\) and \(B C\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is 1.5 m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is 3 m (see diagram). The weight of \(A B\) is 80 N and the frictional force acting on \(A B\) at \(A\) is 14 N .

1. Write down the horizontal component of the force acting on \(A B\) at \(B\) and show that the vertical component of this force is 33 N upwards.
2. Given that the force acting on \(B C\) at \(C\) has magnitude 50 N , find the weight of \(B C\).
   \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_421_949_1793_598} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \mathrm {~ms} ^ { - 1 }\). The spheres are moving in opposite directions, each at \(60 ^ { \circ }\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
3. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres.
4. Find the coefficient of restitution between the spheres.

4 One end of a light inextensible string of length 2 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(O P\) makes an angle of 0.15 radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(O P\) makes an angle of \(\theta\) radians with the downward vertical.

1. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. Using the simple harmonic approximation,
2. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = - 0.1\),
3. find the angular speed of \(O P\) and the linear speed of \(P\) when \(t = 0.5\).
   \includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-3_606_1006_973_568} Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram).
4. Given that \(\sin \beta = 0.96\) and \(\frac { v _ { B } } { u _ { B } } = 1.2\), find the value of \(\sin \gamma\).
5. Given also that, before the collision, the component of \(A\) 's velocity parallel to the line of centres is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the values of \(u _ { B }\) and \(v _ { B }\).
6. Find the coefficient of restitution between the spheres.
7. Given that the kinetic energy of \(A\) immediately before the collision is \(6.5 m \mathrm {~J}\), where \(m \mathrm {~kg}\) is the mass of \(A\), find the value of \(v _ { A }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 2 kg and \(m \mathrm {~kg}\) respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving towards \(B\) at an angle of \(\alpha\) to the line of centres, where \(\cos \alpha = 0.6\). \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(A\) along the line of centres (see diagram). As a result of the collision, \(A\) 's loss of kinetic energy is \(7.56 \mathrm {~J} , B\) 's direction of motion is reversed and \(B\) 's speed after the collision is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) after the collision,
2. the component of \(A\) 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,
3. the value of \(m\),
4. the coefficient of restitution between \(A\) and \(B\).
   \(7 S _ { A }\) and \(S _ { B }\) are light elastic strings. \(S _ { A }\) has natural length 2 m and modulus of elasticity \(120 \mathrm {~N} ; S _ { B }\) has natural length 3 m and modulus of elasticity 180 N . A particle \(P\) of mass 0.8 kg is attached to one end of each of the strings. The other ends of \(S _ { A }\) and \(S _ { B }\) are attached to fixed points \(A\) and \(B\) respectively, on a smooth horizontal table. The distance \(A B\) is \(6 \mathrm {~m} . P\) is released from rest at the point of the line segment \(A B\) which is 2.9 m from \(A\).
5. For the subsequent motion, show that the total elastic potential energy of the strings is the same when \(A P = 2.1 \mathrm {~m}\) and when \(A P = 2.9 \mathrm {~m}\). Deduce that neither string becomes slack.
6. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
7. State, giving a reason, the type of motion of \(P\) and find the time taken between successive occasions when \(P\) is instantaneously at rest. For the instant 0.6 seconds after \(P\) is released, find
8. the distance travelled by \(P\),
9. the speed of \(P\).

4 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.2 kg . Immediately before the collision \(A\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving away from \(A\) with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_422_844_431_612} The coefficient of restitution between the spheres is 0.8 . Find

1. the velocity of \(A\) immediately after the collision,
2. the angle turned through by the direction of motion of \(B\) as a result of the collision.

3
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-2_403_951_1247_559} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.4 kg . Immediately before the collision \(A\) is moving with speed \(2.8 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision \(A\) is stationary. Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the angle turned through by the direction of motion of \(B\) as a result of the collision. \section*{\(\mathrm { OCR } ^ { \text {勾 } }\)}

5
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_337_944_255_557} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(45 ^ { \circ }\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that \(a \cos \alpha = \frac { 5 } { 6 } \sqrt { 2 }\) and find \(b \cos \beta\).
2. Find the values of \(a\) and \(\alpha\).

2 A particle \(Q\) of mass 0.2 kg is projected horizontally with velocity \(4 \mathrm {~ms} ^ { - 1 }\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(Q\) is \(x \mathrm {~m}\) from \(A\) and is moving away from \(A\) with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of \(3 \cos 2 t \mathrm {~N}\) acting on \(Q\) in the positive \(x\)-direction.

1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies.
2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac { 3 } { 2 } \pi\).
   \includegraphics[max width=\textwidth, alt={}, center]{c0f31235-80aa-4838-844f-b706de55e7cd-2_549_1237_1724_415} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac { 4 } { 5 }\), and \(B\) is moving with speed \(3 \frac { 1 } { 4 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac { 5 } { 13 }\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-2_296_798_461_694} A sphere \(A\) of mass \(m\), moving on a horizontal surface, collides with another sphere \(B\) of mass \(2 m\), which is at rest on the surface. The spheres are smooth and uniform, and have equal radius. Immediately before the collision, \(A\) has velocity \(u\) at an angle \(\theta ^ { \circ }\) to the line of centres of the spheres (see diagram). Immediately after the collision, the spheres move in directions that are perpendicular to each other.

1. Find the coefficient of restitution between the spheres.
2. Given that the spheres have equal speeds after the collision, find \(\theta\).

5. Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(k m\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60 ^ { \circ }\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 3 u } { 4 ( k + 1 ) }\).
   (6) Immediately after the collision the direction of motion of \(A\) makes an angle arctan \(( 2 \sqrt { 3 } )\) with the direction of motion of \(B\).
2. Show that \(k = \frac { 1 } { 2 }\).
3. Find the loss of kinetic energy due to the collision.
   (4) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fe647e21-9c4f-4035-b28f-b12a00087692-4_515_1077_301_502}
   \end{figure} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(A B\) is \(O\). The wire is fixed in a vertical plane and hangs below \(A B\) which is horizontal. A small ring \(R\), of mass \(m \sqrt { 2 }\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac { 3 m } { 2 }\). The particles hang vertically under gravity, as shown in Figure 1.
4. Show that, when the radius \(O R\) makes an angle \(2 \theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt { } 2 m g a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
5. Find the values of \(\theta\) for which the system is in equilibrium.
6. Determine the stability of the position of equilibrium for which \(\theta > 0\).

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 2 kg and sphere \(B\) has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to \(\mathbf { i }\). Before the collision \(A\) has velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision it has velocity \(( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Before the collision the velocity of \(B\) makes an angle \(\alpha\) with the line of centres, as shown in Fig. 1, where \(\tan \alpha = 2\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) before the collision.
(9)

6. \begin{figure}[h] \end{figure} A small smooth uniform sphere \(S\) is at rest on a smooth horizontal floor at a distance \(d\) from a straight vertical wall. An identical sphere \(T\) is projected along the floor with speed \(U\) towards \(S\) and in a direction which is perpendicular to the wall. At the instant when \(T\) strikes \(S\) the line joining their centres makes an angle \(\alpha\) with the wall, as shown in Fig. 3. Each sphere is modelled as having negligible diameter in comparison with \(d\). The coefficient of restitution between the spheres is \(e\).

1. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the line of centres, are \(\frac { 1 } { 2 } U ( 1 - e ) \sin \alpha\) and \(U \cos \alpha\) respectively.
2. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the wall, are \(\frac { 1 } { 2 } U ( 1 + e ) \cos \alpha \sin \alpha\) and \(\frac { 1 } { 2 } U \left[ 2 - ( 1 + e ) \sin ^ { 2 } \alpha \right]\) respectively. The spheres \(S\) and \(T\) strike the wall at the points \(A\) and \(B\) respectively.
   Given that \(e = \frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\),
3. find, in terms of \(d\), the distance \(A B\). \section*{END}

5. A train of mass \(m\) is moving along a straight horizontal railway line. A time \(t\), the train is moving with speed \(v\) and the resistance to motion has magnitude \(k v\), where \(k\) is a constant. The engine of the train is working at a constant rate \(P\).

1. Show that, when \(v > 0 , \quad m v \frac { \mathrm {~d} v } { \mathrm {~d} t } + k v ^ { 2 } = P\). When \(t = 0\), the speed of the train is \(\frac { 1 } { 3 } \sqrt { \left( \frac { P } { k } \right) }\).
2. Find, in terms of \(m\) and \(k\), the time taken for the train to double its initial speed.
   (8) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-4_638_285_315_897}
   \end{figure} Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2 m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of \(45 ^ { \circ }\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).
3. Find the magnitude of the impulse which acts on \(A\) in the collision. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-4_481_737_1610_792}
   \end{figure} The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.
4. Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall.
   (5)

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.

5. Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
3. Find the coefficient of restitution between \(A\) and \(B\).
   \section*{June 2009}

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.

3. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius \(r\), have masses \(3 m\) and \(2 m\) respectively. The spheres are moving on a smooth horizontal plane when they collide. Immediately before the collision they are moving with speeds \(u\) and \(2 u\) respectively. The centres of the spheres are moving towards each other along parallel paths at a distance \(1.6 r\) apart, as shown in Figure 2. The coefficient of restitution between the two spheres is \(\frac { 1 } { 6 }\).
Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision.

4. A smooth uniform sphere \(S\) is moving on a smooth horizontal plane when it collides obliquely with an identical sphere \(T\) which is at rest on the plane. Immediately before the collision \(S\) is moving with speed \(U\) in a direction which makes an angle of \(60 ^ { \circ }\) with the line joining the centres of the spheres. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\) and \(U\) where necessary,
   1. the speed and direction of motion of \(S\) immediately after the collision,
   2. the speed and direction of motion of \(T\) immediately after the collision. The angle through which the direction of motion of \(S\) is deflected is \(\delta ^ { \circ }\).
2. Find
   1. the value of \(e\) for which \(\delta\) takes the largest possible value,
   2. the value of \(\delta\) in this case.

5. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3 u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\frac { 1 } { 5 }\). It is given that \(\cos \alpha = \frac { 1 } { 3 }\) and \(\cos \beta = \frac { 2 } { 3 }\) and that \(\alpha\) and \(\beta\) are both acute angles.

1. Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\).
2. Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy.

3. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) with equal radii have masses \(m\) and \(2 m\) respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, \(A\) has speed \(3 u\) with its direction of motion at an angle \(\theta\) to the line of centres, and \(B\) has speed \(u\) with its direction of motion at an angle \(\theta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 1 } { 8 }\) Immediately after the collision, the speed of \(A\) is twice the speed of \(B\).
Find the size of the angle \(\theta\).

1. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a second smooth uniform sphere \(B\), which is at rest on the plane. The sphere \(B\) has mass \(4 m\) and the same radius as \(A\). Immediately before the collision the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres of the spheres, as shown in Figure 1. The direction of motion of \(A\) is turned through an angle of \(90 ^ { \circ }\) by the collision and the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\) Find the value of \(\tan \alpha\).
1.