6.03d Conservation in 2D: vector momentum

6 \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.

1. Find the value of \(k\).
2. Find the loss in the total kinetic energy as a result of the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.

1. Find the value of \(k\).
2. Find the loss in the total kinetic energy as a result of the collision.

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

6. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of weight 40 N and length 7 m , rests with end \(A\) on rough horizontal ground. The beam rests on a smooth horizontal peg at \(C\), with \(A C = 5 \mathrm {~m}\), as shown in Figure 5.
The beam is inclined at an angle \(\theta\) to the ground, where \(\sin \theta = \frac { 3 } { 5 }\) The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg.
The normal reaction between the beam and the peg at \(C\) has magnitude \(P\) newtons.
Using the model,

1. show that \(P = 22.4\)
2. find the magnitude of the resultant force acting on the beam at \(A\).

6. Three particles \(P , Q\) and \(R\) have masses \(3 m , k m\) and 7.5m respectively. The three particles lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 9 }\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 10 u } { 3 ( 3 + k ) }\).
2. Find the range of values of \(k\) for which the direction of motion of \(P\) is reversed as a result of the collision. Following the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\). Given that \(k = 7\) and that \(Q\) is brought to rest by the collision with \(R\),
3. find the total kinetic energy lost in the collision between \(Q\) and \(R\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(A\) has mass 2 kg and a particle \(B\) has mass 3 kg . The particles are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocity of \(A\) is \(5 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(B\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the total kinetic energy of the two particles before the collision.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse received by \(A\) in the collision. Given that, in the collision, the impulse of \(A\) on \(B\) is equal and opposite to the impulse of \(B\) on \(A\),
3. find the velocity of \(B\) immediately after the collision.

7. A particle, \(P\), of mass \(k m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal surface. Particle \(P\) collides directly with another particle, \(Q\), of mass \(2 m\) which is moving with speed \(u\) in the same direction along the same straight line. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Given that immediately after the collision \(P\) and \(Q\) are moving in opposite directions and the speed of \(Q\) is \(\frac { 3 } { 2 } u\),

1. find the range of possible values of \(e\). It is now also given that \(e = \frac { 7 } { 8 }\).
2. Show that the kinetic energy lost by \(P\) in the collision with \(Q\) is \(\frac { 11 } { 8 } m u ^ { 2 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision, \(Q\) hits a fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\). Particle \(Q\) rebounds from the wall and moves so that \(P\) and \(Q\) collide directly at the point \(B\).
3. Find, in terms of \(d\) and \(u\), the time interval between the collision at \(A\) and the collision at \(B\).
   
   \includegraphics[max width=\textwidth, alt={}]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-28_2639_1833_121_118}

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 \(e\).

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 smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide

• the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
• the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 6).

\begin{figure}[h] \end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision, find

1. the coefficient of restitution between \(A\) and \(B\),
2. the total loss of kinetic energy as a result of the collision.

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.

6 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a smooth horizontal surface when they collide. \(A\) has mass 2.5 kg and \(B\) has mass 3 kg . Immediately before the collision \(A\) and \(B\) each has speed \(u \mathrm {~ms} ^ { - 1 }\) and each moves in a direction at an angle \(\theta\) to their line of centres, as indicated in Fig. 1. Immediately after the collision \(A\) has speed \(v _ { 1 } \mathrm {~ms} ^ { - 1 }\) and moves in a direction at an angle \(\alpha\) to the line of centres, and \(B\) has speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a direction at an angle \(\beta\) to the line of centres as indicated in Fig. 2. The coefficient of restitution between \(A\) and \(B\) is \(e\). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(\tan \beta = \frac { 11 \tan \theta } { 10 e - 1 }\).
2. Given that after impact sphere \(A\) moves at an angle of \(50 ^ { \circ }\) to the line of centres and \(B\) moves perpendicular to the line of centres, find \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_817_848_374_210} \caption{Fig. 3}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},width=\textwidth]{cf99660f-6103-47be-99d4-d7f9214e9e91-5_819_953_376_1062} \caption{Fig. 4}
   \end{figure} The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 32\) and the curve \(y = \mathrm { e } ^ { 0.8 x }\) for \(0 \leq x \leq \ln 32\), is occupied by a uniform lamina (see Fig. 3).
3. Show that the \(x\)-coordinate of the centre of mass of the lamina is given by \(\frac { 16 } { 3 } \ln 2 - \frac { 5 } { 4 }\).
4. Calculate the \(y\)-coordinate of the centre of mass of the lamina.
5. The region bounded by the \(x\)-axis, the line \(x = 16\) and the curve \(y = 1.25 \ln x\) for \(1 \leq x \leq 16\), is occupied by a second uniform lamina (see Fig. 4). By using your answer to part (i) find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of this second lamina. www.ocr.org.uk after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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1 Two particles, \(A\) of mass 7 kg and \(B\) of mass 3 kg , are moving on a smooth horizontal plane when they collide. Just before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision, the particles coalesce to form a single combined particle. Find the velocity of the single combined particle after the collision.

1 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. During the collision, the two particles coalesce to form a single combined particle. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { r } 6 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } - 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of the combined particle after the collision.
2. Find the speed of the combined particle after the collision.

3 Two particles, \(A\) and \(B\), are moving on a smooth horizontal plane when they collide. The mass of \(A\) is 6 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } 7 \\ b \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find \(m\).
2. \(\quad\) Find \(b\).
   (2 marks)
   .......... \(\_\_\_\_\) \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_40_118_529_159} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_39_117_623_159}

4 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. The mass of \(A\) is 6 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Before the collision, the velocity of \(A\) is \(( 5 \mathbf { i } + 18 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and the velocity of \(B\) is \(V \mathbf { j } \mathrm {~ms} ^ { - 1 }\).

1. Find \(m\).
2. \(\quad\) Find \(V\).

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

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 \(\frac { 1 } { 2 } m v ^ { 2 } \left( 1 - e ^ { 2 } \right) \mathrm { J }\).

5 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 1 kg respectively. The spheres move on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\alpha\) to the line of centres of the spheres, and \(B\) has velocity \(2.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\beta\) to the line of centres, as shown in the diagram.
The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 7 }\).
Given that \(\sin \alpha = \frac { 4 } { 5 }\) and \(\sin \beta = \frac { 12 } { 13 }\), find the speeds of \(A\) and \(B\) immediately after the collision.
[0pt] [11 marks]

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 2 kg and sphere \(B\) has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to \(\mathbf { i }\). Before the collision \(A\) has velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision it has velocity \(( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Before the collision the velocity of \(B\) makes an angle \(\alpha\) with the line of centres, as shown in Fig. 1, where \(\tan \alpha = 2\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) before the collision.
(9)

2. Two small smooth spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(2 m \mathrm {~kg}\) and the mass of \(B\) is \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane and they collide. Immediately before the collision the velocity of \(A\) is \(( 2 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the speed of \(B\) immediately after the collision.
(5)

5. Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
3. Find the coefficient of restitution between \(A\) and \(B\).
   \section*{June 2009}

2. Two smooth uniform spheres \(S\) and \(T\) have equal radii. The mass of \(S\) is 0.3 kg and the mass of \(T\) is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of \(S\) is \(\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(T\) is \(\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of \(S\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(T\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when the spheres collide the line joining their centres is parallel to \(\mathbf { i }\),

1. find
   1. \(\mathbf { u } _ { 1 }\),
   2. \(\mathbf { u } _ { 2 }\). After the collision, \(T\) goes on to collide with a smooth vertical wall which is parallel to \(\mathbf { j }\). Given that the coefficient of restitution between \(T\) and the wall is also 0.5 , find
2. the angle through which the direction of motion of \(T\) is deflected as a result of the collision with the wall,
3. the loss in kinetic energy of \(T\) caused by the collision with the wall.

7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(- \mathbf { i } + \mathbf { j }\).
2. Find the velocity of \(B\) immediately after the collision.
3. Find the coefficient of restitution between \(A\) and \(B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-2_823_650_1318_751} A boat \(A\) is travelling with constant speed \(6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(075 ^ { \circ }\). Boat \(B\) is travelling with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(025 ^ { \circ }\). At one instant, \(A\) is 2500 m due north of \(B\) (see diagram).

1. Find the magnitude and bearing of the velocity of \(A\) relative to \(B\).
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.