6.03b Conservation of momentum: 1D two particles

5 A particle \(P\) of mass \(2 m\) is moving on a smooth horizontal surface with speed \(u\) when it collides directly with a particle \(Q\) of mass \(k m\) whose speed is \(3 u\) in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of \(P\) is halved.

1. Find, in terms of \(u\) and \(k\), the speed of \(Q\) after the collision. Hence write down the range of possible values of \(k\).
2. Calculate the magnitude of the impulse which \(Q\) exerts on \(P\).
3. Given that \(k = \frac { 1 } { 2 }\), calculate the coefficient of restitution between \(P\) and \(Q\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} One end of a light inextensible string is attached to a point \(P\). The other end is attached to a point \(Q , 1.96 \mathrm {~m}\) vertically below \(P\). A small smooth bead \(B\), of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre \(Q\) and radius \(1.96 \mathrm {~m} . B\) rotates about \(Q\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see Fig. 1).

5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.

4 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_168_956_246_593} Three smooth spheres \(A , B\) and \(C\), of equal radius and of masses \(m \mathrm {~kg} , 2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the coefficient of restitution between \(A\) and \(B\).
2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.
3. Show that there will be another collision.

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
3. Find the kinetic energy lost due to the collision.

4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).

14 J
18J
42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).
5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) Find the maximum power output of the engine.
Fully justify your answer.
6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is a constant.
6

1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
   6
2. The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where \(a , b\) and \(c\) are constants.
   Determine the values of \(a , b\) and \(c\).
   7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) As part of a competition, Jo-Jo makes a small pop-up rocket.
   It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
   The mass of the rocket is 18 grams and the stiffness constant of the spring is \(60 \mathrm { Nm } ^ { - 1 }\) Initially the spring is compressed by 3 cm
   7
   1. Find the speed of the rocket when the spring first reaches its natural length.
      7
   2. By considering energy find the distance that the rocket rises. 7
   3. In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
      8 Two smooth spheres \(A\) and \(B\) have the same radius and are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(2 m\) and \(m\) respectively.
      Both \(A\) and \(B\) are initially at rest.
      The sphere \(A\) is set in motion directly towards \(B\) with speed \(3 u\) and at the same time \(B\) is set in motion directly towards \(A\) with speed \(2 u\). Subsequently \(A\) and \(B\) collide directly. \(A\) The coefficient of restitution between the spheres is \(e\).
      8
   4. Show that the speed of \(B\) after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
   5. Given that the direction of the velocity of \(A\) is reversed during the collision, find the range of possible values of \(e\). Fully justify your answer.
      [0pt] [4 marks]
      8
   6. Given that the magnitude of the impulse that \(A\) exerts on \(B\) is \(\frac { 19 m u } { 3 }\), find the value of \(e\).
      

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5 Two small smooth discs, \(C\) and \(D\), have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc \(C\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and disc \(D\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they collide. The coefficient of restitution between \(C\) and \(D\) is 0.6 The diagram shows the discs, viewed from above, before the collision. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

1. Show that the speed of \(D\) immediately after the collision is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 2 significant figures.
   5
2. Find the speed of \(C\) immediately after the collision.
   [0pt] [2 marks]
   5
3. In fact the horizontal surface on which the discs are sliding is not smooth.
   Explain how the introduction of friction will affect your answer to part (b).
   [0pt] [2 marks]
   Turn over for the next question

4 \includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-3_218_1335_251_367} Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively (see diagram). The coefficient of restitution in collisions between any two of them is \(\frac { 3 } { 4 }\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving with a velocity of \(4.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\).

1. Show that immediately after the collision between \(A\) and \(B\) the speed of \(B\) is \(2.8 \mathrm {~ms} ^ { - 1 }\).
2. Find the velocity of \(A\) immediately after this collision. \(B\) subsequently collides with \(C\).
3. Find, in terms of \(m\), the velocity of \(B\) after its collision with \(C\).
4. Given that the direction of motion of \(B\) is reversed by the collision with \(C\), find the range of possible values of \(m\).

2 A particle \(A\) of mass 3.6 kg is attached by a light inextensible string to a particle \(B\) of mass 2.4 kg . \(A\) and \(B\) are initially at rest, with the string slack, on a smooth horizontal surface. \(A\) is projected directly away from \(B\) with a speed of \(7.2 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the speed of \(A\) after the string becomes taut.
2. Find the impulse exerted on \(A\) at the instant that the string becomes taut.
3. Find the loss in kinetic energy as a result of the string becoming taut.

6 Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-5_224_828_354_244} A collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\). \(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.

1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
2. Find the coefficient of restitution for the collision between \(B\) and the wall.

1 Two stones, A and B , are sliding along the same straight line on a horizontal sheet of ice. Stone A, of mass 50 kg , is moving with a constant velocity of \(2.1 \mathrm {~ms} ^ { - 1 }\) towards stone B. Stone B, of mass 70 kg , is moving with a constant velocity of \(0.8 \mathrm {~ms} ^ { - 1 }\) towards stone A. A and B collide directly. Immediately after their collision stone A's velocity is \(0.35 \mathrm {~ms} ^ { - 1 }\) in the same direction as its velocity before the collision.

1. Find the speed of stone B immediately after the collision.
2. Find the coefficient of restitution for the collision.
3. Find the total loss of kinetic energy caused by the collision.
4. Explain whether the collision was perfectly elastic.

7 Two particles, \(P\) and \(Q\), are on a smooth horizontal floor. \(P\), of mass 1 kg , is moving with speed \(1.79 \mathrm {~ms} ^ { - 1 }\) directly towards a vertical wall. \(Q\), of mass 2.74 kg , is between \(P\) and the wall, moving directly towards \(P\) with speed \(0.08 \mathrm {~ms} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-4_232_830_1370_246} \(P\) and \(Q\) collide directly and the coefficient of restitution for this collision is denoted by \(e\).

1. Show that after this collision the speed of \(Q\) is given by \(0.42 + 0.5 e \mathrm {~ms} ^ { - 1 }\). After this collision, \(Q\) then goes on to collide directly with the wall. The coefficient of restitution for the collision between \(Q\) and the wall is also \(e\). There is then no subsequent collision between \(P\) and \(Q\).
2. Determine the range of possible values of \(e\).

1 Two particles \(A\), of mass \(m \mathrm {~kg}\), and \(B\), of mass \(3 m \mathrm {~kg}\), are connected by a light inextensible string and placed together at rest on a smooth horizontal surface with the string slack. \(A\) is projected along the surface, directly away from \(B\), with a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(B\) immediately after the string becomes taut.
2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) as a result of the string becoming taut.

5 Two identical spheres, \(A\) and \(B\), each of mass 4 kg , are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Before they collide, the speeds of \(A\) and \(B\) are \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. Immediately after they collide, the speed of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) and its direction of motion has been reversed.

1. 1. Determine the velocity of \(B\) immediately after \(A\) and \(B\) collide.
   2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).
   3. Calculate the total loss of kinetic energy due to this collision. Sphere \(B\) goes on to strike a fixed wall directly. As a result of this collision \(B\) moves along the same straight line with a speed of \(4 \mathrm {~ms} ^ { - 1 }\).
2. Find the coefficient of restitution between \(B\) and the wall, stating whether the collision between \(B\) and the wall is perfectly elastic.
3. Determine the magnitude of the impulse that \(B\) exerts on \(A\) the next time that they collide.

5 Two particles, \(A\) of mass \(m _ { A } \mathrm {~kg}\) and \(B\) of mass 5 kg , are moving directly towards each other on a smooth horizontal floor. Before they collide they have speeds \(\mathrm { u } _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after they collide the direction of motion of each particle has been reversed and \(A\) and \(B\) have speeds \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.75 . Before: \includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_711_552_283} After: \includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_707_552_1078}

1. Determine the value of \(m _ { A }\) and the value of \(u _ { A }\).
   [0pt] [5]
2. Show that approximately \(41 \%\) of the kinetic energy of the system is lost in this collision. After the collision between \(A\) and \(B\), \(B\) goes on to collide directly with a third particle \(C\) of mass 3 kg which is travelling towards \(B\) with a speed of \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(B\) and \(C\) is denoted by \(e\).
3. Given that, after \(B\) and \(C\) collide, there are no further collisions between \(A , B\) and \(C\) determine the range of possible values of \(e\).

4 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m .

1. Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~ms} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\).
2. Find the value of \(R\).
3. Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns .
4. 1. Find the speed of \(Q\) after the collision.
   2. Hence show that the collision is inelastic.

6 Three particles \(A , B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(3.3 \mathrm {~kg} , 2.2 \mathrm {~kg}\) and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \mathrm {~ms} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-4_221_1342_552_246}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and \(\mathrm { V } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively.
   After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).
2. Determine the range of possible values of \(e\).

3 A particle \(A\) of mass 0.5 kg is moving with a speed of \(3.15 \mathrm {~ms} ^ { - 1 }\) on a smooth horizontal surface when it collides directly with a particle \(B\) of mass 0.8 kg which is at rest on the surface. The velocities of \(A\) and \(B\) immediately after the collision are denoted by \(\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }\) and \(\mathrm { v } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. You are given that \(\mathrm { v } _ { \mathrm { B } } = 2 \mathrm { v } _ { \mathrm { A } }\).

1. Find the values of \(\mathrm { V } _ { \mathrm { A } }\) and \(\mathrm { V } _ { \mathrm { B } }\).
2. Find the coefficient of restitution between \(A\) and \(B\).
3. Explain why the coefficient of restitution is a dimensionless quantity.
4. Calculate the total loss of kinetic energy as a result of the collision.
5. State, giving a reason, whether or not the collision is perfectly elastic.
6. Calculate the impulse that \(B\) exerts on \(A\) in the collision.

6 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 \(\mathrm { I } = \frac { \sqrt { 3 } \mathrm { mv } } { 2 ( 1 + \mathrm { m } ) }\).
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.

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.

6 Two identical spheres, \(A\) and \(B\), each of mass \(m \mathrm {~kg}\), are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Just before they collide, the speeds of \(A\) and \(B\) are \(20 \mathrm {~ms} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. By finding, in terms of \(e\), an expression for the velocity of \(B\) after the collision, show that the direction of motion of \(B\) is reversed by the collision. After the collision between \(A\) and \(B\), which is not perfectly elastic, \(B\) goes on to collide directly with a fixed, vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 } e\). After the collision between \(B\) and the wall, there are no further collisions between \(A\) and \(B\).
2. Determine the range of possible values of \(e\). \(7 \quad\) A body \(B\) of mass 1.5 kg is moving along the \(x\)-axis. At the instant that it is at the origin, \(O\), its velocity is \(u \mathrm {~ms} ^ { - 1 }\) in the positive \(x\)-direction. At any instant, the resistance to the motion of \(B\) is modelled as being directly proportional to \(v ^ { 2 }\) where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of \(B\) at that instant. The resistance to motion is the only horizontal force acting on \(B\). At an instant when \(B\) 's velocity is \(2 \mathrm {~ms} ^ { - 1 }\), the resistance to its motion is 24 N .

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.

2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4 \\ 7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .

1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.

1 Two particles \(A\) and \(B\) have masses of 3 kg and 2 kg respectively. They are moving along a straight horizontal line towards each other. Each particle is moving with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when they collide. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-2_225_579_676_660}

1. If the particles coalesce during the collision to form a single particle, find the speed of the combined particle after the collision.
2. If, after the collision, \(A\) moves in the same direction as before the collision with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the speed of \(B\) after the collision.

4 Two particles, \(A\) and \(B\), are moving on a horizontal plane when they collide and coalesce to form a single particle. The mass of \(A\) is 5 kg and the mass of \(B\) is 15 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 2 U \\ U \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { c } V \\ - 1 \end{array} \right] \mathrm { ms } ^ { - 1 }\). After the collision, the velocity of the combined particle is \(\left[ \begin{array} { l } V \\ 0 \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find:
   1. \(U\);
   2. \(V\).
2. Find the speed of \(A\) before the collision.