Multiple sequential collisions

4 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).

1. Find the speed of \(B\) after the collision.
   A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).
2. Show that there is another collision between \(A\) and \(B\).
3. \(\quad A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions.

3 Three particles \(P , Q\) and \(R\), of masses \(0.1 \mathrm {~kg} , 0.2 \mathrm {~kg}\) and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle \(P\) is projected towards \(Q\) at a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(P\) and \(Q\) collide, \(P\) rebounds with speed \(1 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(Q\) immediately after the collision with \(P\).
   \(Q\) now collides with \(R\). Immediately after the collision with \(Q , R\) begins to move with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Given that there is no subsequent collision between \(P\) and \(Q\), find the greatest possible value of \(V\).

1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) after the collision.
   After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
2. Find \(m\).

1 Two particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(5 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the same direction as it was originally moving.

1. Find, in terms of \(m\), the speed of \(Q\) after the collision.
   After this collision, \(Q\) moves directly towards a third particle \(R\), of mass 0.6 kg , which is at rest on the plane. \(Q\) is brought to rest in the collision with \(R\), and \(R\) begins to move with a speed of \(1.5 \mathrm {~ms} ^ { - 1 }\).
2. Find the value of \(m\).

6 Three particles \(A , B\) and \(C\) of masses \(5 \mathrm {~kg} , 1 \mathrm {~kg}\) and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(X Y Z\). Initially \(A\) is at \(X , B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v \mathrm {~ms} ^ { - 1 }\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).

1. Show that after the second collision the speed of \(E\) is \(\frac { 15 - v } { 4 } \mathrm {~ms} ^ { - 1 }\).
2. The total loss of kinetic energy of the system due to the two collisions is 63 J . Use the result from (a) to show that \(v = 3\).
3. It is given that the distance \(X Y\) is 36 m and the distance \(Y Z\) is 98 m .
   1. Find the time between the two collisions.
   2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\).

7
\includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-10_289_1191_269_475} The diagram shows a smooth track which lies in a vertical plane. The section \(A B\) is a quarter circle of radius 1.8 m with centre \(O\). The section \(B C\) is a horizontal straight line of length 7.0 m and \(O B\) is perpendicular to \(B C\). The section \(C F E\) is a straight line inclined at an angle of \(\theta ^ { \circ }\) above the horizontal. A particle \(P\) of mass 0.5 kg is released from rest at \(A\). Particle \(P\) collides with a particle \(Q\) of mass 0.1 kg which is at rest at \(B\). Immediately after the collision, the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(B C\). You should assume that \(P\) is moving horizontally when it collides with \(Q\).

1. Show that the speed of \(Q\) immediately after the collision is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   When \(Q\) reaches \(C\), it collides with a particle \(R\) of mass 0.4 kg which is at rest at \(C\). The two particles coalesce. The combined particle comes instantaneously to rest at \(F\). You should assume that there is no instantaneous change in speed as the combined particle leaves \(C\), nor when it passes through \(C\) again as it returns down the slope.
2. Given that the distance \(C F\) is 0.4 m , find the value of \(\theta\).
3. Find the distance from \(B\) at which \(P\) collides with the combined particle.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Three particles \(A , B\) and \(C\) of masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively lie at rest in a straight line on a smooth horizontal plane. The distance between \(B\) and \(C\) is \(2.1 \mathrm {~m} . A\) is projected directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(A\) collides with \(B\) the speed of \(A\) is reduced to \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), still moving in the same direction.

1. Show that the speed of \(B\) after the collision is \(1.05 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   After the collision between \(A\) and \(B , B\) moves directly towards \(C\). Particle \(B\) now collides with \(C\). After this collision, the two particles coalesce and have a combined speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find \(m\).
3. Find the time that it takes, from the instant when \(B\) and \(C\) collide, until \(A\) collides with the combined particle.

3 Th ee small smo h se res \(A , B\) ad \(C 6\) eq l radi ad 6 masses \(4 \mathrm {~g} \quad 2 \mathrm {~g}\) ad 3 g resp ctie ly, lie in th todr in a strait lie o a smo hb izt al p ae. In tially, \(B\) ad \(C\) are at rest ad \(A\) is mi g ard \(B\) with sp ed \(6 \mathrm {~ms} ^ { - 1 }\). After th cb liso with \(B\), se re \(A\) co in s to mo in the same d rectim withs p ed \(\mathrm { ms } ^ { - 1 }\).

1. Fid b sp e \(\boldsymbol { \Phi } \quad B\) after th s cb liso Se re \(B\) cb lid s with \(C\).I it \(h\) s cb lisd \(\mathbf { b }\) se two se res co lesce tof \(\mathbf { o }\) m am \(\mathbf { b }\) ect \(D\).
2. Fid b sp e \(D\) after th s cb lisin
   \includegraphics[max width=\textwidth, alt={}, center]{0a1cec7f-f9d1-4628-b979-443514c73eb9-05_65_1652_1146_242}

2
\includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-2_221_1153_1340_497} Three small uniform spheres \(A , B\) and \(C\) have masses \(0.4 \mathrm {~kg} , 1.2 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres move in the same straight line on a smooth horizontal table, with \(B\) between \(A\) and \(C\). Sphere \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 } , B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(C\) is moving towards \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Spheres \(A\) and \(B\) collide. After this collision \(B\) moves with speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(C\).

1. Find the speed with which \(A\) moves after the collision and state the direction of motion of \(A\).
2. Spheres \(B\) and \(C\) now collide and move away from each other with speeds \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).

4
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-3_216_1146_269_502} Three uniform spheres \(L , M\) and \(N\) have masses \(0.8 \mathrm {~kg} , 0.6 \mathrm {~kg}\) and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(L\) after the collision.
2. \(M\) then collides with \(N\).
   (a) Find the total momentum of \(M\) and \(N\) in the direction of \(M\) 's motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision.
   (b) Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision.

6
\includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_168_1032_292_552} Three uniform spheres \(A , B\) and \(C\) have masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) (see diagram). Air resistance may be ignored.

1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed with which \(B\) starts to move.
2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).
3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\).
4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers.

3
\includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-2_153_1009_978_570} Three particles \(P , Q\) and \(R\), are travelling in the same direction in the same straight line on a smooth horizontal surface. \(P\) has mass \(m \mathrm {~kg}\) and speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 } , Q\) has mass 0.8 kg and speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(R\) has mass 0.4 kg and speed \(2.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).

1. A collision occurs between \(P\) and \(Q\), after which \(P\) and \(Q\) move in opposite directions, each with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
   (a) the value of \(m\),
   (b) the change in the momentum of \(P\).
2. When \(Q\) collides with \(R\) the two particles coalesce. Find their subsequent common speed.

1
\includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_135_917_274_575} Three particles \(P , Q\) and \(R\) have masses \(0.1 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and 0.6 kg respectively. The particles travel along the same straight line on a smooth horizontal table and have velocities \(1.5 \mathrm {~ms} ^ { - 1 } , 1.1 \mathrm {~ms} ^ { - 1 }\) and \(0.8 \mathrm {~ms} ^ { - 1 }\) respectively (see diagram). \(P\) collides with \(Q\) and then \(Q\) collides with \(R\). In the second collision \(Q\) and \(R\) coalesce and subsequently move with a velocity of \(1 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(Q\) immediately before the second collision.
2. Calculate the change in momentum of \(P\) in the first collision.

2
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_138_1118_680_463} Three particles \(P , Q\) and \(R\) with masses \(0.4 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are moving along the same straight line on a smooth horizontal surface. \(P\) and \(Q\) are moving towards each other with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(R\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the same direction as \(Q\) (see diagram).

1. Immediately after the collision between \(P\) and \(Q\) their directions of motion have been reversed, but their speeds are unchanged. Calculate \(u\). The next collision is between \(Q\) and \(R\). After the collision between \(Q\) and \(R\), particle \(Q\) is at rest and \(R\) has speed \(9 \mathrm {~ms} ^ { - 1 }\).
2. Calculate \(m\).
   \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_547_1506_1521_251} Two travellers \(A\) and \(B\) make the same journey on a long straight road. Each traveller walks for part of the journey and rides a bicycle for part of the journey. They start their journeys at the same instant, and they end their journeys simultaneously after travelling for \(T\) hours. \(A\) starts the journey cycling at a steady \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) for 1 hour. \(A\) then leaves the bicycle at the side of the road, and completes the journey walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \(B\) begins the journey walking at a steady \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). When \(B\) finds the bicycle where \(A\) left it, \(B\) cycles at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) to complete the journey (see diagram).

4
\includegraphics[max width=\textwidth, alt={}, center]{c6bac5bf-960e-4c3d-b9fa-c52de66ba719-2_144_1317_1655_372} Four particles \(A , B , C\) and \(D\) are on the same straight line on a smooth horizontal table. \(A\) has speed \(6 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(B\). The speed of \(B\) is \(2 \mathrm {~ms} ^ { - 1 }\) and \(B\) is moving towards \(A\). The particle \(C\) is moving with speed \(5 \mathrm {~ms} ^ { - 1 }\) away from \(B\) and towards \(D\), which is stationary (see diagram). The first collision is between \(A\) and \(B\) which have masses 0.8 kg and 0.2 kg respectively.

1. After the particles collide \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) in its original direction of motion. Calculate the speed of \(B\) after the collision. The second collision is between \(C\) and \(D\) which have masses 0.3 kg and 0.1 kg respectively.
2. The particles coalesce when they collide. Find the speed of the combined particle after this collision. The third collision is between \(B\) and the combined particle, after which no further collisions occur.
3. Calculate the greatest possible speed of the combined particle after the third collision.

6. \(A , B\) and \(C\) are three small spheres of equal radii and masses \(2 m , m\) and \(5 m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude 8 m Ns on \(B\).

1. Find the speed with which \(B\) starts to move.
2. Show that the speed of \(A\) after it collides with \(B\) is \(2 \mathrm {~ms} ^ { - 1 }\). After travelling \(3 \mathrm {~m} , B\) hits \(C\), which is then travelling towards \(B\) at \(2 \cdot 2 \mathrm {~ms} ^ { - 1 } . C\) is brought to rest by this impact.
3. Show that the direction of \(B\) 's motion is reversed and find its new speed.
4. Find how far \(B\) now travels before it collides with \(A\) again.
5. State a modelling assumption that you have made about the spheres.