Three-particle sequential collisions

7. Three particles \(A , B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5 u\) and \(4 u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3 u\). In the subsequent motion, \(A , B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find (i) the speed of \(A\) immediately after the collision,
   (ii) the speed of \(B\) immediately after the collision. Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),
2. find the set of possible values of \(e\).

7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal surface with \(Q\) between \(P\) and \(R\). Particle \(P\) has mass \(m\), particle \(Q\) has mass \(2 m\) and particle \(R\) has mass \(3 m\). The coefficient of restitution between each pair of particles is \(e\). Particle \(P\) is projected towards \(Q\) with speed \(3 u\) and collides directly with \(Q\).

1. Find, in terms of \(u\) and \(e\),
   1. the speed of \(Q\) immediately after the collision,
   2. the speed of \(P\) immediately after the collision.
2. Find the range of values of \(e\) for which the direction of motion of \(P\) is reversed as a result of the collision with \(Q\). Immediately after the collision between \(P\) and \(Q\), particle \(R\) is projected towards \(Q\) with speed \(u\) so that \(R\) and \(Q\) collide directly. Given that \(e = \frac { 2 } { 3 }\)
3. show that there will be a second collision between \(P\) and \(Q\).

7. Three particles \(A , B\) and \(C\) lie at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The particles \(A\), \(B\) and \(C\) have mass \(6 m\), 4 \(m\) and \(m\) respectively. Particle \(A\) is projected towards \(B\) with speed \(3 u\) and \(A\) collides directly with \(B\). Immediately after this collision, the speed of \(B\) is \(w\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 6 }\).

1. Show that \(w = \frac { 21 } { 10 } u\).
2. Express the total kinetic energy of \(A\) and \(B\) lost in the collision as a fraction of the total kinetic energy of \(A\) and \(B\) immediately before the collision. After being struck by \(A\), the particle \(B\) collides directly with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). After the collision between \(B\) and \(C\), there are no further collisions between the particles.
3. Find the range of possible values of \(e\).
   

7. Three particles \(A\), \(B\) and \(C\) have masses \(2 m , 3 m\) and \(4 m\) respectively. The particles lie at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). The kinetic energy of \(A\) immediately after the collision is one ninth of the kinetic energy of \(A\) immediately before the collision. Given that the direction of motion of \(A\) is unchanged by the collision,

1. find the value of \(e\). After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\). The coefficient of restitution between \(B\) and \(C\) is \(f\), where \(f < \frac { 3 } { 4 }\). The speed of \(B\) immediately after the collision with \(C\) is \(V\).
2. 1. Express \(V\) in terms of \(f\) and \(u\).
   2. Hence show that there will be a second collision between \(A\) and \(B\).
      

1. Particles \(A , B\) and \(C\), of masses \(2 m , m\) and \(3 m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(2 u\) and collides directly with \(B\).

The coefficient of restitution between each pair of particles is \(e\).

1. 1. Show that the speed of \(B\) immediately after the collision with \(A\) is \(\frac { 4 } { 3 } u ( 1 + e )\)
   2. Find the speed of \(A\) immediately after the collision with \(B\). At the instant when \(A\) collides with \(B\), particle \(C\) is projected with speed \(u\) towards \(B\) so that \(B\) and \(C\) collide directly.
2. Show that there will be a second collision between \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-27_2644_1840_118_111}

8. Particles \(A , B\) and \(C\), of masses \(4 m , k m\) and \(2 m\) respectively, lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(3 u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\) Find

1. the speed of \(A\) immediately after the collision with \(B\), giving your answer in terms of \(u\) and \(k\),
2. the range of values of \(k\) for which \(A\) and \(B\) will both be moving in the same direction immediately after they collide. After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\). Given that \(k = 4\),
3. show that there will not be a second collision between \(A\) and \(B\).
   DO NOT WRITEIN THIS AREA

4. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal table. Another particle \(Q\) of mass \(k m\) is at rest on the table. The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. After the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(3 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 2 }\).

1. Find, in terms of \(v\) only, the speed of \(P\) before the collision.
2. Find the value of \(k\). After being struck by \(P\), the particle \(Q\) collides directly with a particle \(R\) of mass \(11 m\) which is at rest on the table. After this second collision, \(Q\) and \(R\) have the same speed and are moving in opposite directions. Show that
3. the coefficient of restitution between \(Q\) and \(R\) is \(\frac { 3 } { 4 }\),
4. there will be a further collision between \(P\) and \(Q\).

6. Three identical particles, \(A , B\) and \(C\), lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The mass of each particle is \(m\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\).

1. Find, in terms of \(u\),
   1. the speed of \(A\) after this collision,
   2. the speed of \(B\) after this collision.
2. Show that the kinetic energy lost in this collision is \(\frac { 5 } { 36 } m u ^ { 2 }\) After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\).
3. Find, in terms of \(u\), the speed of \(C\) immediately after this collision between \(B\) and \(C\).

7. A particle \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal floor when it collides directly with another particle \(B\), of mass \(3 m\), which is at rest on the floor. The coefficient of restitution between the particles is \(e\). The direction of motion of \(A\) is reversed by the collision.

1. Find, in terms of \(e\) and \(u\),
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision. After being struck by \(A\) the particle \(B\) collides directly with another particle \(C\), of mass \(4 m\), which is at rest on the floor. The coefficient of restitution between \(B\) and \(C\) is \(2 e\). Given that the direction of motion of \(B\) is reversed by this collision,
2. find the range of possible values of \(e\),
3. determine whether there will be a second collision between \(A\) and \(B\).

6. A particle \(A\) of mass \(2 m\) is moving with speed \(2 u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4 m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).

1. Show that the speed of \(B\) after the collision is \(\frac { 3 } { 2 } u\).
2. Find the speed of \(A\) after the collision. Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,
3. find the range of possible values for \(e\).
   (8)

7. A uniform sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another uniform sphere \(B\) of mass \(2 m\) which is at rest on the table. The spheres are of equal radius and the coefficient of restitution between them is \(e\). The direction of motion of \(A\) is unchanged by the collision.

1. Find the speeds of \(A\) and \(B\) immediately after the collision.
2. Find the range of possible values of \(e\). After being struck by \(A\), the sphere \(B\) collides directly with another sphere \(C\), of mass \(4 m\) and of the same size as \(B\). The sphere \(C\) is at rest on the table immediately before being struck by \(B\). The coefficient of restitution between \(B\) and \(C\) is also \(e\).
3. Show that, after \(B\) has struck \(C\), there will be a further collision between \(A\) and \(B\).
   (6) \section*{END}

5. Two small spheres \(A\) and \(B\) have mass \(3 m\) and \(2 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane, both with speed \(2 u\), when they collide directly. As a result of the collision, the direction of motion of \(B\) is reversed and its speed is unchanged.

1. Find the coefficient of restitution between the spheres. Subsequently, \(B\) collides directly with another small sphere \(C\) of mass \(5 m\) which is at rest. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 3 } { 5 }\).
2. Show that, after \(B\) collides with \(C\), there will be no further collisions between the spheres.

3 Three small smooth spheres \(A , B\) and \(C\) have equal radii and have masses \(m , 9 m\) and \(k m\) respectively. They are at rest on a smooth horizontal table and lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any pair of the spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that half of the total kinetic energy is lost as result of the collision between \(A\) and \(B\), find the value of \(e\). After \(B\) and \(C\) collide they move in the same direction and the speed of \(C\) is twice the speed of \(B\). Find the value of \(k\).

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.

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 \(m\).

6 Three particles \(A , B\) and \(C\) are in a straight line on a smooth horizontal surface. The particles have masses \(0.2 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and 0.6 kg respectively. \(B\) is at rest. \(A\) is projected towards \(B\) with a speed of \(1.8 \mathrm {~ms} ^ { - 1 }\) and collides with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).

1. Show that the speed of \(B\) after the collision is \(0.8 \mathrm {~ms} ^ { - 1 }\) and find the speed of \(A\) after the collision. \(C\) is moving with speed \(0.2 \mathrm {~ms} ^ { - 1 }\) in the same direction as \(B\). Particle \(B\) subsequently collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\).
2. Find the set of values for \(e\) such that \(B\) does not collide again with \(A\).

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)

6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.
The particles have masses \(5 \mathrm {~kg} , 3 \mathrm {~kg}\) and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of \(u \mathrm {~ms} ^ { - 1 }\) and collides with B . The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Particle B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 3 }\).

1. Determine whether any further collisions occur.
2. Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of \(u\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure} Fig. 7 shows a uniform rod AB of length \(4 a\) and mass \(m\).
   The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where \(\mathrm { AC } = 4 a\). The plane ABC is perpendicular to the wall and the angle ABC is \(30 ^ { \circ }\). The system is in limiting equilibrium. Find the coefficient of friction between the wall and the rod. \section*{END OF QUESTION PAPER}

4 The diagram shows three beads, A, B and C, of masses \(0.3 \mathrm {~kg} , 0.5 \mathrm {~kg}\) and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section. \includegraphics[max width=\textwidth, alt={}, center]{a87d62b8-406d-44cd-9ffa-384005329566-5_361_961_450_248} Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B with a speed of \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the subsequent collision between A and \(\mathrm { B } , \mathrm { A }\) is brought to rest.

1. Show that the coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Bead B next collides with C.
2. Show that the speed of B before this collision is \(4.37 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures. In this collision between B and C , B is brought to rest.
3. Determine whether C next collides with A or with B .
4. Explain why, if B has a greater mass than C , B could not be brought to rest in their collision.

4 Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of \(\mathrm { A } , \mathrm { B }\) and C are \(2 \mathrm {~kg} , 4 \mathrm {~kg}\) and 1 kg respectively. Initially the three spheres are at rest.
Spheres \(A\) and \(C\) are each given impulses so that \(A\) moves towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-5_325_1591_603_239} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
It is given that the first collision occurs between A and B .

1. State how you can tell from the information given above that kinetic energy is lost when A collides with B .
2. Show that the combined kinetic energy of A and B decreases by \(24 \%\) during their collision. Sphere B next collides with C. The coefficient of restitution between B and C is \(\frac { 2 } { 3 }\).
3. Given that a third collision occurs, determine the range of possible values for \(u\).
4. State one limitation of the model used in this question.

5 Throughout this question it may be assumed that there are no resistances to motion.
Model trucks A and B, with masses 5 kg and 3 kg respectively, rest on a set of straight, horizontal rails. Truck A is given an impulse of 3.8 Ns towards B .

1. Calculate the initial speed of A. Truck A collides directly with B. After the collision, B moves with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\).
2. Determine
   1. the velocity of A after the collision,
   2. the kinetic energy lost due to the collision.
3. B continues to move with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\) and collides with a model truck C, of mass 4 kg , which is travelling at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\) towards B on the same set of rails. After the collision between B and C , the speeds of B and C are in the ratio 1 to 2 . Determine the two possible values of the coefficient of restitution between B and C .

3 Three small uniform spheres A, B and C have masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively. The spheres move in the same straight line on a smooth horizontal table, with B between A and C . Sphere A moves towards B with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 } , \mathrm {~B}\) is at rest and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-3_181_1291_461_251} Spheres A and B collide. Collisions between A and B can be modelled as perfectly elastic.

1. Determine the magnitude of the impulse of A on B in this collision.
2. Use this collision to verify that in a perfectly elastic collision no kinetic energy is lost. After the collision between A and B, sphere B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 4 }\).
3. Show that, after the collision between B and C , B has a speed of \(( 1.225 - 0.78125 \mathrm { u } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) towards C.
4. Determine the range of values for \(u\) for there to be a second collision between A and B .