6.03b Conservation of momentum: 1D two particles

8. Two ships \(A\) and \(B\), of masses \(m\) and km respectively, are moving towards each other in heavy fog along the same straight line, both with speed \(u\). The ships collide and immediately after the collision they drift away from each other, both their directions of motion having been reversed. The speed of \(A\) after the impact is \(\frac { 1 } { 5 } u\) and the speed of \(B\) after the impact is \(v\).

1. Show that \(v = u \left( \frac { 6 } { 5 k } - 1 \right)\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
2. Show that \(v = u \left( 2 e - \frac { 1 } { 5 } \right)\).
3. Use your answers to parts (a) and (b) to find the rational numbers \(p\) and \(q\) such that \(p \leq k < q\).
   (9 marks)

7 Two small spheres \(A\) and \(B\), with masses 0.3 kg and \(m \mathrm {~kg}\) respectively, lie at rest on a smooth horizontal surface. \(A\) is projected directly towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and hits \(B\). The direction of motion of \(A\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(1 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that \(m = 0.7\).
2. Find \(e\). B continues to move at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The coefficient of restitution between \(B\) and the wall is \(f\).
3. Find the range of values of \(f\) for which there will be a second collision between \(A\) and \(B\).
4. Find, in terms of \(f\), the magnitude of the impulse that the wall exerts on \(B\).
5. Given that \(f = \frac { 3 } { 4 }\), calculate the final speeds of \(A\) and \(B\), correct to 1 decimal place.

7 \includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_305_1301_1708_424} Two small spheres \(A\) and \(B\) of masses 2 kg and 3 kg respectively lie at rest on a smooth horizontal platform which is fixed at a height of 4 m above horizontal ground (see diagram). Sphere \(A\) is given an impulse of 6 N s towards \(B\), and \(A\) then strikes \(B\) directly. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. Show that the speed of \(B\) after it has been hit by \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\). Sphere \(B\) leaves the platform and follows the path of a projectile.
2. Calculate the speed and direction of motion of \(B\) at the instant when it hits the ground.

6 Two uniform spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.4 kg and the mass of \(B\) is 0.2 kg . The spheres \(A\) and \(B\) are travelling in the same direction in a straight line on a smooth horizontal surface, \(A\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(B\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < 5\). A collides directly with \(B\) and the impulse between them has magnitude 0.9 Ns . Immediately after the collision, the speed of \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate \(v\). \(B\) subsequently collides directly with a stationary sphere \(C\) of mass 0.1 kg and the same radius as \(A\) and \(B\). The coefficient of restitution between \(B\) and \(C\) is 0.6 .
2. Determine whether there will be a further collision between \(A\) and \(B\).

4 Two small spheres \(A\) and \(B\) are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide directly. The direction of motion of \(B\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. (a) Show that the direction of motion of \(A\) is unchanged by the collision.
   (b) Calculate the coefficient of restitution between \(A\) and \(B\). The mass of \(B\) is 0.2 kg .
2. Find the mass of \(A\). \(B\) continues to move at \(2.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The wall exerts an impulse of magnitude 0.68 N s on \(B\).
3. Calculate the coefficient of restitution between \(B\) and the wall.

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 \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_243_1179_1580_443} The masses of two particles \(A\) and \(B\) are 0.2 kg and \(m \mathrm {~kg}\) respectively. The particles are moving with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(u \mathrm {~ms} ^ { - 1 }\) in the same horizontal line and in the same direction (see diagram). The two particles collide and the coefficient of restitution between the particles is \(e\). After the collision, \(A\) and \(B\) continue in the same direction with speeds \(4 \left( 1 - e + e ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively.

1. Find \(u\) and \(m\) in terms of \(e\).
2. Find the value of \(e\) for which the speed of \(A\) after the collision is least and find, in this case, the total loss in kinetic energy due to the collision.
3. Find the possible values of \(e\) for which the magnitude of the impulse that \(B\) exerts on \(A\) is 0.192 Ns . \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-5_744_887_264_589} The diagram shows a surface consisting of a horizontal part \(O A\) and a plane \(A B\) inclined at an angle of \(70 ^ { \circ }\) to the horizontal. A particle is projected from the point \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal \(O A\). The particle hits the plane \(A B\) at the point \(P\), with speed \(14 \mathrm {~ms} ^ { - 1 }\) and at right angles to the plane, 1.4 s after projection.

1. Show that the value of \(u\) is 15.9 , correct to 3 significant figures, and find the value of \(\theta\).
2. Find the height of \(P\) above the level of \(A\). The particle rebounds with speed \(v \mathrm {~ms} ^ { - 1 }\). The particle next lands at \(A\).
3. Find the value of \(v\).
4. Find the coefficient of restitution between the particle and the plane at \(P\).

6 Two small spheres \(A\) and \(B\), of masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) before they collide. The coefficient of restitution between \(A\) and \(B\) is 0.4 .

1. Find the speed of each sphere after the collision.
2. Find, in terms of \(m\), the loss of kinetic energy during the collision.
3. Given that the magnitude of the impulse exerted on \(A\) by \(B\) during the collision is 2.52 Ns , find \(m\).

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.24 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a horizontal table, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), 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 \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse exerted by \(A\) on \(B\).
2. Show that \(m \leqslant 0.08\). It is given that \(m = 0.06\).
3. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
4. Find the speeds of \(A\) and \(B\) immediately after the collision.

1

1. Roger of mass 70 kg and Sheuli of mass 50 kg are skating on a horizontal plane containing the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\). The resistances to the motion of the skaters are negligible. The two skaters are locked in a close embrace and accelerate from rest until they reach a velocity of \(2 \mathrm { ims } ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-002_191_181_543_740} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-002_177_359_589_1051} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. What impulse has acted on them? During a dance routine, the skaters separate on three occasions from their close embrace when travelling at a constant velocity of \(2 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
   2. Calculate the velocity of Sheuli after the separation in the following cases.
      (A) Roger has velocity \(\mathrm { ims } ^ { - 1 }\) after the separation.
      (B) Roger and Sheuli have equal speeds in opposite senses after the separation, with Roger moving in the \(\mathbf { i }\) direction.
      (C) Roger has velocity \(4 ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) after the separation.
2. Two discs with masses 2 kg and 3 kg collide directly in a horizontal plane. Their velocities just before the collision are shown in Fig. 1.2. The coefficient of restitution in the collision is 0.5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-002_278_970_1759_594} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
   1. Calculate the velocity of each disc after the collision. The disc of mass 3 kg moves freely after the collision and makes a perfectly elastic collision with a smooth wall inclined at \(60 ^ { \circ }\) to its direction of motion, as shown in Fig. 1.2.
   2. State with reasons the speed of the disc and the angle between its direction of motion and the wall after the collision.

1 When a stationary firework P of mass 0.4 kg is set off, the explosion gives it an instantaneous impulse of 16 N s vertically upwards.

1. Calculate the speed of projection of P . While travelling vertically upwards at \(32 \mathrm {~ms} ^ { - 1 } , P\) collides directly with another firework \(Q\), of mass 0.6 kg , that is moving directly downwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1. The coefficient of restitution in the collision is 0.1 and Q has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards immediately after the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-2_520_422_753_817} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
2. Show that \(u = 18\) and calculate the speed and direction of motion of P immediately after the collision. Another firework of mass 0.5 kg has a velocity of \(( - 3.6 \mathbf { i } + 5.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors, respectively. This firework explodes into two parts, C and D . Part C has mass 0.2 kg and velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) immediately after the explosion.
3. Calculate the velocity of D immediately after the explosion in the form \(a \mathbf { i } + b \mathbf { j }\). Show that C and D move off at \(90 ^ { \circ }\) to one another.
   [0pt] [8]

1

1. An object P , with mass 6 kg and speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is sliding on a smooth horizontal table. Object P explodes into two small parts, Q and \(\mathrm { R } . \mathrm { Q }\) has mass 4 kg and R has mass 2 kg and speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of motion of P before the explosion. This information is shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_346_1267_429_479} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. Calculate the velocity of Q . Just as object R reaches the edge of the table, it collides directly with a small object S of mass 3 kg that is travelling horizontally towards R with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This information is shown in Fig. 1.2. The coefficient of restitution in this collision is 0.1 . \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-2_506_647_1215_790} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   2. Calculate the velocities of R and S immediately after the collision. The table is 0.4 m above a horizontal floor. After the collision, R and S have no contact with the table.
   3. Calculate the distance apart of R and S when they reach the floor.
2. A particle of mass \(m \mathrm {~kg}\) bounces off a smooth horizontal plane. The components of velocity of the particle just before the impact are \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the plane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the plane. The coefficient of restitution is \(e\). Show that the mechanical energy lost in the impact is \(\frac { 1 } { 2 } m v ^ { 2 } \left( 1 - e ^ { 2 } \right) \mathrm { J }\).

4

1. A large nail of mass 0.02 kg has been driven a short distance horizontally into a fixed block of wood, as shown in Fig. 4.1, and is to be driven horizontally further into the block. The wood produces a constant resistance of 2.43 N to the motion of the nail. The situation is modelled by assuming that linear momentum is conserved when the nail is struck, that all the impacts with the nail are direct and that the head of the nail never reaches the wood. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_279_711_482_676} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The nail is first struck by an object of mass 0.1 kg that is moving parallel to the nail with linear momentum of magnitude 0.108 Ns . The object becomes firmly attached to the nail.
   1. Calculate the speed of the nail and object immediately after the impact.
   2. Calculate the time for which the nail and object move, and the distance they travel in that time. On a second attempt to drive in the nail, it is struck by the same object of mass 0.1 kg moving parallel to the nail with the same linear momentum of magnitude 0.108 Ns . This time the object does not become attached to the nail and after the contact is still moving parallel to the nail. The coefficient of restitution in the impact is \(\frac { 1 } { 3 }\).
   3. Calculate the speed of the nail immediately after this impact.
2. A small ball slides on a smooth horizontal plane and bounces off a smooth straight vertical wall. The speed of the ball is \(u\) before the impact and, as shown in Fig. 4.2, the impact turns the path of the ball through \(90 ^ { \circ }\). The coefficient of restitution in the collision between the ball and the wall is \(e\). Before the collision, the path is inclined at \(\alpha\) to the wall. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_294_590_1804_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Write down, in terms of \(u , e\) and \(\alpha\), the components of the velocity of the ball parallel and perpendicular to the wall before and after the impact.
   2. Show that \(\tan \alpha = \frac { 1 } { \sqrt { e } }\).
   3. Hence show that \(\alpha \geqslant 45 ^ { \circ }\).

1

1. Fig. 1.1 shows the velocities of a tanker of mass 120000 tonnes before and after it changed speed and direction. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-2_237_917_360_577} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Calculate the magnitude of the impulse that acted on the tanker.
2. An object of negligible size is at rest on a horizontal surface. It explodes into two parts, P and Q , which then slide along the surface. Part P has mass 0.4 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Part Q has mass 0.5 kg .
   1. Calculate the speed of Q immediately after the explosion. State how the directions of motion of P and Q are related. The explosion takes place at a distance of 0.75 m from a raised vertical edge, as shown in Fig. 1.2. P travels along a line perpendicular to this edge. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-2_238_1205_1366_429} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure} After the explosion, P has a perfectly elastic direct collision with the raised edge and then collides again directly with Q . The collision between P and Q occurs \(\frac { 2 } { 3 } \mathrm {~s}\) after the explosion. Both collisions are instantaneous. The contact between P and the surface is smooth but there is a constant frictional force between Q and the surface.
   2. Show that Q has speed \(2.7 \mathrm {~ms} ^ { - 1 }\) just before P collides with it.
   3. Calculate the coefficient of friction between Q and the surface.
   4. Given that the coefficient of restitution between P and Q is \(\frac { 1 } { 8 }\), calculate the speed of Q immediately after its collision with P .

1

1. Roger of mass 70 kg and Sheuli of mass 50 kg are skating on a horizontal plane containing the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\). The resistances to the motion of the skaters are negligible. The two skaters are locked in a close embrace and accelerate from rest until they reach a velocity of \(2 \mathrm { ims } ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-2_191_181_543_740} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-2_177_359_589_1051} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. What impulse has acted on them? During a dance routine, the skaters separate on three occasions from their close embrace when travelling at a constant velocity of \(2 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
   2. Calculate the velocity of Sheuli after the separation in the following cases.
      (A) Roger has velocity \(\mathrm { ims } ^ { - 1 }\) after the separation.
      (B) Roger and Sheuli have equal speeds in opposite senses after the separation, with Roger moving in the \(\mathbf { i }\) direction.
      (C) Roger has velocity \(4 ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) after the separation.
2. Two discs with masses 2 kg and 3 kg collide directly in a horizontal plane. Their velocities just before the collision are shown in Fig. 1.2. The coefficient of restitution in the collision is 0.5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-2_278_970_1759_594} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
   1. Calculate the velocity of each disc after the collision. The disc of mass 3 kg moves freely after the collision and makes a perfectly elastic collision with a smooth wall inclined at \(60 ^ { \circ }\) to its direction of motion, as shown in Fig. 1.2.
   2. State with reasons the speed of the disc and the angle between its direction of motion and the wall after the collision.

1

1. Two small spheres, \(P\) of mass 2 kg and \(Q\) of mass 6 kg , are moving in the same straight line along a smooth, horizontal plane with the velocities shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-2_252_647_404_708} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Consider the direct collision of P and Q in the following two cases.
   1. The spheres coalesce on collision.
      (A) Calculate the common velocity of the spheres after the collision.
      (B) Calculate the energy lost in the collision.
   2. The spheres rebound with a coefficient of restitution of \(\frac { 2 } { 3 }\) in the collision.
      (A) Calculate the velocities of P and Q after the collision.
      (B) Calculate the impulse on P in the collision.
2. A small ball bounces off a smooth, horizontal plane. The ball hits the plane with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\arcsin \frac { 12 } { 13 }\) to it. The ball rebounds at an angle of \(\arcsin \frac { 3 } { 5 }\) to the plane, as shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-2_238_545_1695_767} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} Calculate the speed with which the ball rebounds from the plane.
   Calculate also the coefficient of restitution in the impact.

1

1. Disc A of mass 6 kg and disc B of mass 0.5 kg are moving in the same straight line. The relative positions of the discs and the \(\mathbf { i }\) direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_282_1325_402_450} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} The discs collide directly. The impulse on A in the collision is \(- 12 \mathbf { i }\) Ns and after the collision A has velocity \(3 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B has velocity \(11 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Show that the velocity of A just before the collision is \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and find the velocity of B at this time.
   2. Calculate the coefficient of restitution in the collision.
   3. After the collision, a force of \(- 2 \mathbf { i } \mathrm {~N}\) acts on B for 7 seconds. Find the velocity of B after this time.
2. A ball bounces off a smooth plane. The angles its path makes with the plane before and after the impact are \(\alpha\) and \(\beta\), as shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_317_1082_1468_575} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The velocity of the ball before the impact is \(u \mathbf { i } - v \mathbf { j }\) and the coefficient of restitution in the impact is \(e\). Write down an expression in terms of \(u , v , e , \mathbf { i }\) and \(\mathbf { j }\) for the velocity of the ball immediately after the impact. Hence show that \(\tan \beta = e \tan \alpha\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_581_486_274_383} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_593_392_264_1370} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A uniform wire is bent to form a bracket OABCD . The sections \(\mathrm { OA } , \mathrm { AB }\) and BC lie on three sides of a square and CD is parallel to AB . This is shown in Fig. 2.1 where the dimensions, in centimetres, are also given.
   1. Show that, referred to the axes shown in Fig. 2.1, the \(x\)-coordinate of the centre of mass of the bracket is 3.6 . Find also the \(y\)-coordinate of its centre of mass.
   2. The bracket is now freely suspended from D and hangs in equilibrium. Draw a diagram showing the position of the centre of mass and calculate the angle of CD to the vertical.
   3. The bracket is now hung by means of vertical, light strings BP and DQ attached to B and to D , as shown in Fig. 2.2. The bracket has weight 5 N and is in equilibrium with OA horizontal. Calculate the tensions in the strings BP and DQ . The original bracket shown in Fig. 2.1 is now changed by adding the section OE, where AOE is a straight line. This section is made of the same type of wire and has length \(L \mathrm {~cm}\), as shown in Fig. 2.3. \(\begin{array} { l l l l } \begin{array} { l } \text { not to } \\ \text { scale } \end{array} & 2 & 6 & \\ \mathrm {~L} \longrightarrow & \mathrm {~L} & & \\ \mathrm {~L} & \mathrm { O } & 6 & \mathrm {~A} \end{array}\) Fig. 2.3 The value of \(L\) is chosen so that the centre of mass is now on the \(y\)-axis.
   4. Calculate \(L\).

1

1. Two small objects, P of mass \(m \mathrm {~kg}\) and Q of mass \(k m \mathrm {~kg}\), slide on a smooth horizontal plane. Initially, P and Q are moving in the same straight line towards one another, each with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After a direct collision with P , the direction of motion of Q is reversed and it now has a speed of \(\frac { 1 } { 3 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of P is now \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where the positive direction is the original direction of motion of P .
   1. Draw a diagram showing the velocities of P and Q before and after the impact.
   2. By considering the linear momentum of the objects before and after the collision, show that \(v = \left( 1 - \frac { 4 } { 3 } k \right) u\).
   3. Hence find the condition on \(k\) for the direction of motion of P to be reversed. The coefficient of restitution in the collision is 0.5 .
   4. Show that \(v = - \frac { 2 } { 3 } u\) and calculate the value of \(k\).
2. Particle \(A\) has a mass of 5 kg and velocity \(\binom { 3 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(B\) has mass 3 kg and is initially at rest. A force \(\binom { 1 } { - 2 } \mathrm {~N}\) acts for 9 seconds on B and subsequently (in the absence of the force), \(A\) and \(B\) collide and stick together to form an object \(C\) that moves off with a velocity \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. Show that \(\mathbf { V } = \binom { 3 } { - 1 }\). The object C now collides with a smooth barrier which lies in the direction \(\binom { 0 } { 1 }\). The coefficient of restitution in the collision is 0.5 .
   2. Calculate the velocity of C after the impact.

1 Two sledges P and Q, with their loads, have masses of 200 kg and 250 kg respectively and are sliding on a horizontal surface against negligible resistance. There is an inextensible light rope connecting the sledges that is horizontal and parallel to the direction of motion. Fig. 1 shows the initial situation with both sledges travelling with a velocity of \(5 \mathbf { i m ~ } \mathbf { m } ^ { - 1 }\). \begin{figure}[h] \end{figure} A mechanism on Q jerks the rope so that there is an impulse of \(250 \mathbf { i N s }\) on P .

1. Show that the new velocity of \(P\) is \(6.25 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\) and find the new velocity of \(Q\). There is now a direct collision between the sledges and after the impact P has velocity \(4.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Show that the velocity of Q becomes \(5.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the coefficient of restitution in the collision. Before the rope becomes taut again, the velocity of P is increased so that it catches up with Q . This is done by throwing part of the load from sledge P in the \(- \mathbf { i }\) direction so that P 's velocity increases to \(5.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). The part of the load thrown out is an object of mass 20 kg .
3. Calculate the speed of separation of the object from P . When the sledges directly collide again they are held together and move as a single object.
4. Calculate the common velocity of the pair of sledges, giving your answer correct to 3 significant figures. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1a605f0b-f595-4bb9-9624-f816c789ad86-3_987_524_258_769} \captionsetup{labelformat=empty} \caption{not to scale he lengths are}
   \end{figure} Fig. 2 Fig. 2 shows a stand on a horizontal floor and horizontal and vertical coordinate axes \(\mathrm { O } x\) and \(\mathrm { O } y\). The stand is modelled as
   Coordinates are referred to the axes shown in the figure.

1

1. Sphere P , of mass 10 kg , and sphere Q , of mass 15 kg , move with their centres on a horizontal straight line and have no resistances to their motion. \(\mathrm { P } , \mathrm { Q }\) and the positive direction are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_332_803_434_712} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Initially, P has a velocity of \(- 1.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is acted on by a force of magnitude 13 N acting in the direction PQ . After \(T\) seconds, P has a velocity of \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has not reached Q .
   1. Calculate \(T\). The force of magnitude 13 N is removed. P is still travelling at \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with Q , which has a velocity of \(- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Suppose that P and Q coalesce in the collision to form a single object.
   2. Calculate their common velocity after the collision. Suppose instead that P and Q separate after the collision and that P has a velocity of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) afterwards.
   3. Calculate the velocity of Q after the collision and also the coefficient of restitution in the collision.
2. Fig. 1.2 shows a small ball projected at a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal over smooth horizontal ground. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_424_832_1918_699} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} The ball is initially 3.125 m above the ground. The coefficient of restitution between the ball and the ground is 0.6 . Calculate the angle with the horizontal of the ball's trajectory immediately after the second bounce on the ground.

1

1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

1

1. A particle, P , of mass 5 kg moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides with another particle, Q , of mass 30 kg travelling with a speed of \(\frac { u } { 3 } \mathrm {~ms} ^ { - 1 }\) towards P . The particles P and Q are moving in the same horizontal straight line with negligible resistance to their motion. As a result of the collision, the speed of P is halved and its direction of travel reversed; the speed of Q is now \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Draw a diagram showing this information. Find the velocity of Q immediately after the collision in terms of \(u\). Find also the coefficient of restitution between P and Q .
   2. Find, in terms of \(u\), the impulse of P on Q in the collision.
2. Fig. 1 shows a small object R of mass 5 kg travelling on a smooth horizontal plane at \(6 \mathrm {~ms} ^ { - 1 }\). It explodes into two parts of masses 2 kg and 3 kg . The velocities of these parts are in the plane in which R was travelling with the speeds and directions indicated. The angles \(\alpha\) and \(\beta\) are given by \(\cos \alpha = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-2_460_1450_1050_312} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
   1. Calculate \(u\) and \(v\).
   2. Calculate the increase in kinetic energy resulting from the explosion.

4

1. Two discs, P of mass 4 kg and Q of mass 5 kg , are sliding along the same line on a smooth horizontal plane when they collide. The velocity of P before the collision and the velocity of Q after the collision are shown in Fig. 4. P loses \(\frac { 5 } { 9 }\) of its kinetic energy in the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{71d839d8-12ca-4806-8f74-c572e7e21891-5_294_899_390_584} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Show that after the collision P has a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the opposite direction to its original motion. While colliding, the discs are in contact for \(\frac { 1 } { 5 } \mathrm {~s}\).
   2. Find the impulse on P in the collision and the average force acting on the discs.
   3. Find the velocity of Q before the collision and the coefficient of restitution between the two discs.
2. A particle is projected from a point 2.5 m above a smooth horizontal plane. Its initial velocity is \(5.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) below the horizontal, where \(\sin \theta = \frac { 15 } { 17 }\). The coefficient of restitution between the particle and the plane is \(\frac { 4 } { 5 }\).
   1. Show that, after bouncing off the plane, the greatest height reached by the particle is 2.5 m .
   2. Calculate the horizontal distance between the two points at which the particle is 2.5 m above the plane.