6.03b Conservation of momentum: 1D two particles

1

1. Two model railway trucks are moving freely on a straight horizontal track when they are in a direct collision. The trucks are P of mass 0.5 kg and Q of mass 0.75 kg . They are initially travelling in the same direction. Just before they collide P has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and Q has a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_263_640_484_715} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. Suppose that the speed of P is halved in the collision and that its direction of motion is not changed. Find the speed of Q immediately after the collision and find the coefficient of restitution.
   2. Show that it is not possible for both the speed of P to be halved in the collision and its direction of motion to be reversed. Both of the model trucks have flat horizontal tops. They are each travelling at the speeds they had immediately after the collision. Part of the mass of Q is a small object of mass 0.1 kg at rest at the edge of the top of Q nearest P . The object falls off, initially with negligible velocity relative to Q .
   3. Determine the speed of Q immediately after the object falls off it, making your reasoning clear. Part of the mass of P is an object of mass 0.05 kg that is fired horizontally from the top of P , parallel to and in the opposite direction to the motion of P . Immediately after the object is fired, it has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to P .
   4. Determine the speed of P immediately after the object has been fired from it.
2. The velocities of a small object immediately before and after an elastic collision with a horizontal plane are shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_172_741_1987_644} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} Show that the plane cannot be smooth.

2

1. A bullet of mass 0.04 kg is fired into a fixed uniform rectangular block along a line through the centres of opposite parallel faces, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_209_1287_342_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} The bullet enters the block at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to rest after travelling 0.2 m into the block.
   1. Calculate the resistive force on the bullet, assuming that this force is constant. Another bullet of the same mass is fired, as before, with the same speed into a similar block of mass 3.96 kg . The block is initially at rest and is free to slide on a smooth horizontal plane.
   2. By considering linear momentum, find the speed of the block with the bullet embedded in it and at rest relative to the block.
   3. By considering mechanical energy, find the distance the bullet penetrates the block, given the resistance of the block to the motion of the bullet is the same as in part (i).
2. Fig. 2.2 shows a block of mass 6 kg on a uniformly rough plane that is inclined at \(30 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_348_636_1382_712} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A string with a constant tension of 91.5 N parallel to the plane pulls the block up a line of greatest slope. The speed of the block increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a distance of 8 m .

7. A smooth sphere \(A\) of mass \(4 m\) is moving on a smooth horizontal plane with speed \(u\). It collides directly with a stationary smooth sphere \(B\) of mass \(5 m\) and with the same radius as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).

1. Show that after the collision the speed of \(B\) is 4 times greater than the speed of \(A\).
   (7 marks)
   Sphere \(B\) subsequently hits a smooth vertical wall at right angles. After rebounding from the wall, \(B\) collides with \(A\) again and as a result of this collision, \(B\) comes to rest. Given that the coefficient of restitution between \(B\) and the wall is \(e\),
2. find \(e\). END

6. Three uniform spheres \(A , B\) and \(C\) of equal radius have masses \(3 m , 2 m\) and \(2 m\) respectively. Initially, the spheres are at rest on a smooth horizontal table with their centres in a straight line and with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\),

1. show that the speeds of \(A\) and \(B\) after the collision are \(\frac { 1 } { 3 } u\) and \(u\) respectively.
   (6 marks)
   The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that \(A\) and \(B\) collide again,
2. show that \(e > \frac { 1 } { 3 }\).
   (8 marks)

2. Two smooth spheres \(P\) and \(Q\) of equal radius and of mass \(2 m\) and \(5 m\) respectively, are moving towards each other along a horizontal straight line when they collide. After the collision, \(P\) and \(Q\) travel in opposite directions with speeds of \(3 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively. Given that the coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\), find the speeds of \(P\) and \(Q\) before the collision.
(6 marks)

6. Particle \(S\) of mass \(2 M\) is moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it collides directly with a particle \(T\) of mass \(5 M\) which is lying at rest on the plane. The coefficient of restitution between \(S\) and \(T\) is \(\frac { 3 } { 4 }\). Given that the speed of \(T\) after the collision is \(4 \mathrm {~ms} ^ { - 1 }\),

1. find \(U\). As a result of the collision, \(T\) is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. \(T\) strikes the ground at the point \(X\) as shown in Figure 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Find the time taken for \(T\) to reach \(X\).
3. Show that the angle between the horizontal and the direction of motion of \(T\), just before it strikes the ground at \(X\), is \(78.5 ^ { \circ }\) correct to 3 significant figures.
   (4 marks)

7. Particle \(A\) of mass 7 kg is moving with speed \(u _ { 1 }\) on a smooth horizontal surface when it collides directly with particle \(B\) of mass 4 kg moving in the same direction as \(A\) with speed \(u _ { 2 }\). After the impact, \(A\) continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is \(e\),

1. show that \(8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )\). Given also that \(u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. find \(e\),
3. show that the percentage of the kinetic energy of the particles lost as a result of the impact is \(9.6 \%\), correct to 2 significant figures.

2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).

1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).

4 Two small smooth spheres, \(A\) and \(B\), of equal radii have masses 0.3 kg and 0.2 kg respectively. They are moving on a smooth horizontal surface directly towards each other with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively when they collide. The coefficient of restitution between \(A\) and \(B\) is 0.8 .

1. Find the speeds of \(A\) and \(B\) immediately after the collision.
2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the path of the sphere. The coefficient of restitution between \(B\) and the wall is 0.7 . Show that \(B\) will collide again with \(A\).

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)

3 Three smooth spheres, \(A , B\) and \(C\), of equal radii have masses \(1 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(x \mathrm {~kg}\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(3 u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is \(\frac { 1 } { 3 }\).

1. Show that \(A\) is brought to rest by the impact and find the speed of \(B\) immediately after the collision in terms of \(u\).
2. Subsequently, \(B\) collides with \(C\). Show that the speed of \(C\) immediately after the collision is \(\frac { 4 u } { 3 + x }\).
   Find the speed of \(B\) immediately after the collision in terms of \(u\) and \(x\).
3. Show that \(B\) will collide with \(A\) again if \(x > 9\).
4. Given that \(x = 5\), find the magnitude of the impulse exerted on \(C\) by \(B\) in terms of \(u\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}

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}
   1. Find the components of the velocity of the ball, parallel and perpendicular to the plane, as it strikes the inclined plane at \(A\).
   2. 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.

4 A smooth sphere \(A\), of mass \(m\), is moving with speed \(4 u\) in a straight line on a smooth horizontal table. A smooth sphere \(B\), of mass \(3 m\), has the same radius as \(A\) and is moving on the table with speed \(2 u\) in the same direction as \(A\). \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-10_289_780_493_625} The sphere \(A\) collides directly with sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speeds of \(A\) and \(B\) immediately after the collision.
2. Show that the speed of \(B\) after the collision cannot be greater than \(3 u\).
3. Given that \(e = \frac { 2 } { 3 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) in the collision.

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.

7 Two small smooth spheres, \(A\) and \(B\), are the same size and have masses \(2 m\) and \(m\) respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere \(A\) receives an impulse of magnitude \(J\) and moves with speed \(2 u\) directly towards \(B\).

1. \(\quad\) Find \(J\) in terms of \(m\) and \(u\).
2. The sphere \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). Find, in terms of \(u\), the speeds of \(A\) and \(B\) immediately after the collision.
3. At the instant of collision, the centre of \(B\) is at a distance \(s\) from a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, \(B\) collides with the wall. The radius of each sphere is \(r\).
   Show that the distance of the centre of \(A\) from the wall at the instant that \(B\) hits the wall is \(\frac { 3 s + 12 r } { 5 }\).
4. The diagram below shows the positions of \(A\) and \(B\) when \(B\) hits the wall. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere \(B\) collides with \(A\) again after rebounding from the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\). Find the distance of the centre of \(\boldsymbol { B }\) from the wall at the instant when \(A\) and \(B\) collide again.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}

4 Three uniform smooth spheres, \(A , B\) and \(C\), have equal radii and masses \(m , 2 m\) and \(6 m\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 } { 9 } u\).
   2. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
2. Subsequently, \(B\) collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). Show that \(B\) will collide with \(A\) again if \(e > k\), where \(k\) is a constant to be determined.
3. Explain why it is not necessary to model the spheres as particles in this question.
   [0pt] [2 marks]

6 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-4_638_473_260_836} A particle \(P\), of mass 3.5 kg , is in equilibrium suspended from the top \(A\) of a smooth slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 40 } { 49 }\), by an elastic rope of natural length 4 m and modulus of elasticity 112 N (see diagram). Another particle \(Q\), of mass 0.5 kg , is released from rest at \(A\) and slides freely downwards until it reaches \(P\) and becomes attached to it.

1. Find the value of \(V ^ { 2 }\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) immediately before it becomes attached to \(P\), and show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(\frac { 1 } { 2 } \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The combined particles slide downwards for a distance of \(X \mathrm {~m}\), before coming instantaneously to rest at \(B\).
2. Show that \(28 X ^ { 2 } - 8 X - 5 = 0\).

5 A particle \(P\), of mass 2.5 kg , is in equilibrium suspended from a fixed point \(A\) by a light elastic string of natural length 3 m and modulus of elasticity 36.75 N . Another particle \(Q\), of mass 1 kg , is released from rest at \(A\) and falls freely until it reaches \(P\) and becomes attached to it.

1. Show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(2 \sqrt { 2 } \mathrm {~ms} ^ { - 1 }\). The combined particles fall a further distance \(X \mathrm {~m}\) before coming to instantaneous rest.
2. Find a quadratic equation satisfied by \(X\), and show that it simplifies to \(35 X ^ { 2 } - 56 X - 80 = 0\).

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 \(A\).

1. 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.
2. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
3. 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
4. the distance travelled by \(P\),
5. 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*{\(\mathrm { OCR } ^ { \text {勾 } }\)}