Oblique collision of spheres

3 \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_348_828_251_621} \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_2717_35_110_2012} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2 u\) and \(3 u\) respectively. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\theta\) with the line of centres and \(B\) 's direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\).

1. Find the value of \(\tan \theta\). \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-05_2723_33_99_21}
2. Find the total loss of kinetic energy as a result of the collision.
3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision. \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-06_776_785_255_680} The end \(A\) of a uniform rod \(A B\) of length \(6 a\) and weight \(W\) is in contact with a rough vertical wall. One end of a light inextensible string of length \(3 a\) is attached to the midpoint \(C\) of the rod. The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The rod is in equilibrium when the angle between the rod and the wall is \(\theta\), where \(\tan \theta = \frac { 3 } { 2 }\). A particle of weight \(W\) is attached to the point \(E\) on the rod, where the distance \(A E\) is equal to \(k a ( 3 < k < 6 )\) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\frac { 1 } { 3 }\). The rod is about to slip down the wall.
4. Find the value of \(k\).
   
5. Find, in terms of \(W\), the magnitude of the frictional force between the rod and the wall.

8 Two smooth circular discs, \(A\) and \(B\), have equal radii and are free to move on a smooth horizontal plane. The masses of \(A\) and \(B\) are 1 kg and \(m \mathrm {~kg}\) respectively. \(B\) is initially placed at rest with its centre at the origin, \(O\). \(A\) is projected towards \(B\) with a velocity of \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) to the negative \(y\)-axis where \(\tan \theta = \frac { 5 } { 2 }\). At the instant of collision the line joining their centres lies on the \(x\)-axis. There are two straight vertical walls on the plane. One is perpendicular to the \(x\)-axis and the other is perpendicular to the \(y\)-axis. The walls are an equal distance from \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-7_944_1241_694_242} After \(A\) and \(B\) have collided with each other, each of them goes on to collide with a wall. Each then rebounds and they collide again at the same place as their first collision, with disc \(B\) again at \(O\). The coefficient of restitution between \(A\) and \(B\) is denoted by \(e\). The coefficient of restitution between \(A\) and the wall that it collides with is also \(e\) while the coefficient of restitution between \(B\) and the wall that it collides with is \(\frac { 5 } { 9 } e\). It is assumed that any resistance to the motion of \(A\) and \(B\) may be ignored.

1. Explain why it must be the case that the collision between \(A\) and the wall that it collides with is not inelastic.
2. Show that \(\mathrm { e } = \frac { 1 } { \mathrm {~m} }\).
3. Show that \(m = \frac { 5 } { 3 }\).
4. State one limitation of the model used.

3 Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is 5 kg and the mass of \(B\) is 3 kg . At the instant before they collide,

• the velocity of \(A\) is \(4 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres,
• the velocity of \(B\) is \(6 \mathrm {~ms} ^ { - 1 }\) along the line of centres
  (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{894be707-4f7b-4647-b209-805522556196-4_531_1683_651_191}

The coefficient of restitution for collisions between the two discs is \(\frac { 3 } { 4 }\).
Determine the angle that the velocity of \(A\) makes with the line of centres after the collision. \(4 A B C D\) is a uniform lamina in the shape of a kite with \(\mathrm { BA } = \mathrm { BC } = 0.37 \mathrm {~m} , \mathrm { DA } = \mathrm { DC } = 0.91 \mathrm {~m}\) and \(\mathrm { AC } = 0.7 \mathrm {~m}\) (see diagram). The centre of mass of \(A B C D\) is \(G\).

1. Explain why \(G\) lies on \(B D\).
2. Show that the distance of \(G\) from \(B\) is 0.36 m . The lamina \(A B C D\) is freely suspended from the point \(A\).
3. Determine the acute angle that \(C D\) makes with the horizontal, stating which of \(C\) or \(D\) is higher.

6 Two smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide

• the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
• the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 6).

\begin{figure}[h] \end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision, find

1. the coefficient of restitution between \(A\) and \(B\),
2. the total loss of kinetic energy as a result of the collision.

7 Two smooth circular discs \(A\) and \(B\) of masses \(m _ { A } \mathrm {~kg}\) and \(m _ { B } \mathrm {~kg}\) respectively are moving on a horizontal plane. At the instant before they collide the velocities of \(A\) and \(B\) are as follows, as shown in the diagram below.

• The velocity of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the line of centres, where \(\tan \alpha = \frac { 4 } { 3 }\).
• The velocity of \(B\) is \(4 \mathrm {~ms} ^ { - 1 }\) at an angle of \(\frac { 1 } { 3 } \pi\) radians to the line of centres. \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-5_469_873_548_274}

The direction of motion of \(B\) after the collision is perpendicular to the line of centres.

1. Show that \(\frac { 3 } { 2 } \leqslant \frac { m _ { B } } { m _ { A } } \leqslant 4\).
2. Given that \(\mathrm { m } _ { \mathrm { A } } = 2\) and \(\mathrm { m } _ { \mathrm { B } } = 6\), find the total loss of kinetic energy as a result of the collision.

6 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a smooth horizontal surface when they collide. \(A\) has mass 2.5 kg and \(B\) has mass 3 kg . Immediately before the collision \(A\) and \(B\) each has speed \(u \mathrm {~ms} ^ { - 1 }\) and each moves in a direction at an angle \(\theta\) to their line of centres, as indicated in Fig. 1. Immediately after the collision \(A\) has speed \(v _ { 1 } \mathrm {~ms} ^ { - 1 }\) and moves in a direction at an angle \(\alpha\) to the line of centres, and \(B\) has speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a direction at an angle \(\beta\) to the line of centres as indicated in Fig. 2. The coefficient of restitution between \(A\) and \(B\) is \(e\). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(\tan \beta = \frac { 11 \tan \theta } { 10 e - 1 }\).
2. Given that after impact sphere \(A\) moves at an angle of \(50 ^ { \circ }\) to the line of centres and \(B\) moves perpendicular to the line of centres, find \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_817_848_374_210} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_819_953_376_1062} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 32\) and the curve \(y = \mathrm { e } ^ { 0.8 x }\) for \(0 \leq x \leq \ln 32\), is occupied by a uniform lamina (see Fig. 3).
3. Show that the \(x\)-coordinate of the centre of mass of the lamina is given by \(\frac { 16 } { 3 } \ln 2 - \frac { 5 } { 4 }\).
4. Calculate the \(y\)-coordinate of the centre of mass of the lamina.
5. The region bounded by the \(x\)-axis, the line \(x = 16\) and the curve \(y = 1.25 \ln x\) for \(1 \leq x \leq 16\), is occupied by a second uniform lamina (see Fig. 4). By using your answer to part (i) find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of this second lamina. {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-05_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).

1. Find the speed of \(B\) immediately after the collision.
2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.

6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).

1. Find the speed of \(B\) immediately after the collision.
2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.

6 A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) makes an angle of \(30 ^ { \circ }\) with the line of the centres of the two balls, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-4_362_1632_621_242} The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Given that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\), show that the speed of \(B\) immediately after the collision is $$\frac { \sqrt { 3 } } { 4 } u ( 1 + e )$$
2. Find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision.
3. Given that \(e = \frac { 2 } { 3 }\), find the angle that the velocity of \(A\) makes with the line of centres immediately after the collision. Give your answer to the nearest degree.
   (3 marks)

4 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(m\) and \(2 m\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) when it collides with the sphere \(B\) which has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } )\). After the collision, the velocity of \(B\) is \(( \mathbf { i } + \mathbf { j } )\).

1. Find the velocity of \(A\) immediately after the collision.
2. Find the angle between the velocities of \(A\) and \(B\) immediately after the collision.
3. Find the impulse exerted by \(B\) on \(A\).
4. State, as a vector, the direction of the line of centres of \(A\) and \(B\) when they collide.
   (1 mark)

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 1 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to the unit vector \(\mathbf { i }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-16_456_1052_721_550}

1. Briefly state why the components of the velocities of \(A\) and \(B\) parallel to the unit vector \(\mathbf { j }\) are not changed by the collision.
2. The coefficient of restitution between the spheres is 0.5 . Find the velocities of \(A\) and \(B\) immediately after the collision. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-17_2484_1709_223_153} \(7 \quad\) A ball is projected from a point \(O\) on a smooth plane which is inclined at an angle of \(35 ^ { \circ }\) above the horizontal. The ball is projected with velocity \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the plane, as shown in the diagram. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane at the point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-18_321_838_605_577}
3. Find the components of the velocity of the ball, parallel and perpendicular to the plane, as it strikes the inclined plane at \(A\).
4. On striking the plane at \(A\), the ball rebounds. The coefficient of restitution between the plane and the ball is \(\frac { 4 } { 5 }\). Show that the ball next strikes the plane at a point lower down than \(A\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-19_2484_1709_223_153}

7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(4 m\) and \(3 m\) respectively. The sphere \(A\) is moving on a smooth horizontal surface and collides with the sphere \(B\), which is stationary on the same surface. Just before the collision, \(A\) is moving with speed \(u\) at an angle of \(30 ^ { \circ }\) to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_449_927_1244_456} The coefficient of restitution between the spheres is \(\frac { 5 } { 9 }\).

1. Find the value of \(\alpha\).
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) during the collision.
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-23_2349_1707_221_153}

7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it collides with the sphere \(B\), which has velocity \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision, the velocity of the sphere \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Show that the impulse exerted on \(B\) in the collision is \(( 6 m \mathbf { j } )\) Ns.
3. Find the coefficient of restitution between the two spheres.
4. After the collision, each sphere moves in a straight line with constant speed. Given that the radius of each sphere is 0.05 m , find the time taken, from the collision, until the centres of the spheres are 1.10 m apart.

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 4 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to unit vector \(\mathbf { i }\). The direction of motion of \(B\) is changed through \(90 ^ { \circ }\) by the collision, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-14_332_1184_566_543}

1. Show that the velocity of \(B\) immediately after the collision is \(\left( \frac { 9 } { 2 } \mathbf { i } - 3 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the coefficient of restitution between the spheres.
3. Find the impulse exerted on \(B\) during the collision. State the units of your answer.

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 4 kg respectively. The spheres are moving on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres of the spheres, and \(B\) has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-16_291_844_607_468} Just after the collision, \(B\) moves in a direction perpendicular to the line of centres.

1. Find the speed of \(A\) immediately after the collision.
2. Find the acute angle, correct to the nearest degree, between the velocity of \(A\) and the line of centres immediately after the collision.
3. Find the coefficient of restitution between the spheres.
4. Find the magnitude of the impulse exerted on \(B\) during the collision.

5 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 1 kg respectively. The spheres move on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\alpha\) to the line of centres of the spheres, and \(B\) has velocity \(2.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\beta\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-14_458_1068_541_625} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 7 }\).
Given that \(\sin \alpha = \frac { 4 } { 5 }\) and \(\sin \beta = \frac { 12 } { 13 }\), find the speeds of \(A\) and \(B\) immediately after the collision.
[0pt] [11 marks]

4 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

4 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-3_446_821_1007_664} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg , and \(B\) has mass \(m \mathrm {~kg}\). Immediately before the collision, \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, and \(B\) is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(30 ^ { \circ }\) to the line of centres. Immediately after the collision \(A\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(45 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres (see diagram).

1. Find \(u\).
2. Given that \(\theta = 88.1 ^ { \circ }\) correct to 1 decimal place, calculate the approximate values of \(v\) and \(m\).
3. The coefficient of restitution is 0.75 . Show that the exact value of \(\theta\) is a root of the equation \(8 \sin \theta - 6 \cos \theta = 9 \cos 30 ^ { \circ }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of \(A\) has components \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(B\), and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres. \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\), the component of the velocity of \(A\) along the line of centres immediately after the collision.
2. Given that the speeds of \(A\) and \(B\) are the same immediately after the collision, and that \(3 e ^ { 2 } = 1\), find \(v\).

5 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-4_369_953_269_596} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 3 kg and 4 kg respectively. They are moving on a horizontal surface, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. The directions of motion of \(A\) and \(B\) make angles \(\alpha\) and \(\beta\) respectively with the line of centres of the spheres, where \(\sin \alpha = \cos \beta = 0.6\) (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the angle that the velocity of \(A\) makes, immediately after impact, with the line of centres of the spheres.
[0pt] [10]

2 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_421_759_936_694} 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, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and \(B\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres (see diagram). The coefficient of restitution is 0.6 . The direction of motion of \(B\) after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Find the value of \(v\).

4 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-3_497_1157_255_493} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg and \(B\) has mass 0.3 kg . Immediately before the collision \(A\) is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\), and \(B\) is moving with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.7. Find

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

2 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-2_544_816_781_603} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. They are moving in opposite directions on a horizontal surface and they collide. Immediately before the collision, each sphere has speed \(u \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.5 .

1. Show that the speed of \(B\) is unchanged as a result of the collision.
2. Find the direction of motion of each of the spheres after the collision.

2 Two uniform smooth spheres \(A\) and \(B\), of equal radius and equal mass, are moving towards each other on a horizontal surface. Immediately before they collide, \(A\) has speed \(0.3 \mathrm {~ms} ^ { - 1 }\) along the line of centres and \(B\) has speed \(0.6 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_302_1013_1247_502} After the collision, the direction of motion of \(B\) is at right angles to its original direction of motion. Find

1. the speed of \(B\) after the collision,
2. the speed and direction of motion of \(A\) after the collision,
3. the coefficient of restitution between \(A\) and \(B\).