6.03g Impulse in 2D: vector form

2 A particle \(P\) of mass 2 kg is moving on a large smooth horizontal plane when it collides with a fixed smooth vertical wall. Before the collision its velocity is \(( 5 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision its velocity is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. The impulse imparted on \(P\) by the wall is denoted by INs. Find the following.

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. 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 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
   • a thin uniform rectangular base PQRS, 30 cm by 40 cm with mass 15 kg ; the sides QR and PS are parallel to \(\mathrm { O } x\),
   • a thin uniform vertical rod of length 200 cm and mass 3 kg that is fixed to the base at O , the mid-point of PQ and the origin of coordinates,
   • a thin uniform top rod AB of length 50 cm and mass \(2 \mathrm {~kg} ; \mathrm { AB }\) is parallel to \(\mathrm { O } x\).
   Coordinates are referred to the axes shown in the figure.

1 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle \(P\) of mass 0.4 kg moving in a straight line has speed \(8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse applied to \(P\) deflects it through \(45 ^ { \circ }\) and reduces its speed to \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Calculate the magnitude and direction of the impulse exerted on \(P\). \(2 \quad O\) is a fixed point on a horizontal straight line. A particle \(P\) of mass 0.5 kg is released from rest at \(O\). At time \(t\) seconds after release the only force acting on \(P\) has magnitude \(\left( 1 + k t ^ { 2 } \right) \mathrm { N }\) and acts horizontally and away from \(O\) along the line, where \(k\) is a positive constant.

1. Find the speed of \(P\) in terms of \(k\) and \(t\).
2. Given that \(P\) is 2 m from \(O\) when \(t = 1\), find the value of \(k\) and the time taken by \(P\) to travel 20 m from \(O\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_231_971_539_587} When a tennis ball of mass 0.057 kg bounces it receives an impulse of magnitude \(I \mathrm {~N} \mathrm {~s}\) at an angle of \(\theta\) to the horizontal. Immediately before the ball bounces it has speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal. Immediately after the ball bounces it has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal (see diagram). Find \(I\) and \(\theta\).

1 A smooth horizontal surface lies in the \(x - y\) plane. A particle \(P\) of mass 0.5 kg is moving on the surface with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the \(x\)-direction when it is struck by a horizontal blow whose impulse has components - 3.5 N s and 2.4 N s in the \(x\)-direction and \(y\)-direction respectively.

1. Find the components in the \(x\)-direction and the \(y\)-direction of the velocity of \(P\) immediately after the blow. Hence show that the speed of \(P\) immediately after the blow is \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(P\) is struck by a second horizontal blow whose impulse is \(\mathbf { I }\).
2. Given that \(P\) 's direction of motion immediately after this blow is parallel to the \(x\)-axis, write down the component of \(\mathbf { I }\) in the \(y\)-direction.

1 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-2_385_741_269_701} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(6.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse of magnitude 2.6 N s applied to \(P\) deflects its direction of motion through an angle \(\theta\), and reduces its speed to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). By considering an impulse-momentum triangle, or otherwise,

1. show that \(\cos \theta = 0.6\),
2. find the angle that the impulse makes with the original direction of motion of \(P\).

1 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_323_639_255_753} A particle \(P\) of mass 0.4 kg is moving horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\), in a direction which makes an angle \(( 180 - \theta ) ^ { \circ }\) with the direction of motion of \(P\). Immediately after the impulse acts \(P\) moves horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is turned through an angle of \(60 ^ { \circ }\) by the impulse (see diagram). Find \(I\) and \(\theta\).

1 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_476_583_258_781} A ball of mass 0.5 kg is moving with speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line when it is struck by a bat. The impulse exerted by the bat has magnitude 15 N s and the ball is deflected through an angle of \(90 ^ { \circ }\) (see diagram). Find

1. the direction of the impulse,
2. the speed of the ball immediately after it is struck.

1 A particle \(P\) of mass 0.05 kg is moving on a smooth horizontal surface with speed \(2 \mathrm {~ms} ^ { - 1 }\), when it is struck by a horizontal blow in a direction perpendicular to its direction of motion. The magnitude of the impulse of the blow is \(I\) Ns. The speed of \(P\) after the blow is \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(I\). Immediately before the blow \(P\) is moving parallel to a smooth vertical wall. After the blow \(P\) hits the wall and rebounds from the wall with speed \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the coefficient of restitution between \(P\) and the wall.

1 \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_477_534_261_770} A ball of mass 0.6 kg is moving with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line. It is struck by an impulse \(I \mathrm { Ns }\) acting at an acute angle \(\theta\) to its direction of motion (see diagram). The impulse causes the direction of motion of the ball to change by an acute angle \(\alpha\), where \(\sin \alpha = \frac { 8 } { 17 }\). After the impulse acts the ball is moving with a speed of \(3.4 \mathrm {~ms} ^ { - 1 }\). Find \(I\) and \(\theta\).

1 A particle \(P\) of mass 0.3 kg is moving on a smooth horizontal surface with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a horizontal impulse. The magnitude of the impulse is 0.6 Ns .

1. (a) Find the greatest possible speed of \(P\) after the impulse acts.
   (b) Find the least possible speed of \(P\) after the impulse acts.
2. In fact the speed of \(P\) after the impulse acts is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle the impulse makes with the original direction of travel of \(P\) and draw a sketch to make this direction clear.

7. \begin{figure}[h] \end{figure} A body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), with a uniform rod. The centres of the discs are fixed to the ends \(A\) and \(B\) of the rod, which has mass \(3 m\) and length 8a. The discs and the rod are coplanar, as shown in Fig. 2. The body is free to rotate in a vertical plane about a smooth fixed horizontal axis. The axis is perpendicular to the plane of the discs and passes through the point \(O\) of the rod, where \(A O = 3 a\).

1. Show that the moment of inertia of the body about the axis is \(54 m a ^ { 2 }\). The body is held at rest with \(A B\) horizontal and is then released. When the body has turned through an angle of \(30 ^ { \circ }\), the rod \(A B\) strikes a small fixed smooth peg \(P\) where \(O P = 3 a\). Given that the body rebounds from the peg with its angular speed halved by the impact,
2. show that the magnitude of the impulse exerted on the body by the peg at the impact is $$9 m \sqrt { \left( \frac { 5 g a } { 6 } \right) } .$$ END

3. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate about a fixed smooth axis which passes through \(A\) and is perpendicular to the rod. The rod has angular speed \(\omega\) when it strikes a particle \(P\) of mass \(m\) and adheres to it. Immediately before the rod strikes \(P , P\) is at rest and at a distance \(x\) from \(A\). Immediately after the rod strikes \(P\), the angular speed of the rod is \(\frac { 3 } { 4 } \omega\). Find \(x\) in terms of \(a\).
(5)

5. At time \(t = 0\) a rocket is launched. The rocket has initial mass \(M\), of which mass \(\lambda M\), \(0 < \lambda < 1\), is fuel. The rocket is launched vertically upwards, from rest, from the surface of the Earth. The rocket burns fuel and the burnt fuel is ejected vertically downwards with constant speed \(U\) relative to the rocket. At time \(t\), the rocket has mass \(m\) and velocity \(v\). Ignoring air resistance and any variation in \(g\),

1. show, from first principles, that until all the fuel is used, $$m \frac { \mathrm {~d} v } { \mathrm {~d} t } + U \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g$$ The rocket accelerates vertically upwards with constant acceleration \(g\).
2. Show that \(m = M \mathrm { e } ^ { \frac { - 2 g t } { U } }\)
3. Find, in terms of \(M , U\) and \(\lambda\), an expression for the kinetic energy of the rocket at the instant when all of the fuel has been used.

7. A pendulum consists of a uniform circular disc, of radius \(a\) and mass \(4 m\), whose centre is fixed to the end \(B\) of a uniform \(\operatorname { rod } A B\). The rod has mass \(3 m\) and length \(4 l\), where \(2 l > a\). The rod lies in the same plane as the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the disc. The moment of inertia of the pendulum about \(L\) is \(2 m \left( a ^ { 2 } + 40 l ^ { 2 } \right)\).

1. Find the approximate period of small oscillations of the pendulum about its position of stable equilibrium. The pendulum is held with \(B\) vertically above \(A\) and is then slightly displaced from rest. In the subsequent motion the midpoint of \(A B\) strikes a small peg, which is fixed at the same horizontal level as \(A\), and the pendulum rebounds upwards. Immediately before it strikes the peg, the angular speed of the pendulum is \(\omega\).
2. Show that \(\omega ^ { 2 } = \frac { 22 g l } { \left( a ^ { 2 } + 40 l ^ { 2 } \right) }\) Immediately after it strikes the peg, the angular speed of the pendulum is \(\frac { 1 } { 2 } \omega\).
3. Find, in terms of \(m , g , a\) and \(l\), the magnitude of the impulse exerted on the peg by the pendulum.
4. Show that the size of the angle turned through by the pendulum, between it hitting the peg and it next coming to rest, is \(\arcsin \frac { 1 } { 4 }\).
   \includegraphics[max width=\textwidth, alt={}]{1242d28a-a4bd-4754-ac49-9b48de95b880-24_2632_1830_121_121}

5 Fig. 5.1 shows a small smooth sphere A at rest on a smooth horizontal surface. At both ends of the surface is a smooth vertical wall. \begin{figure}[h] \end{figure} Sphere A is projected directly towards the left-hand wall at a speed of \(5 \mathrm {~ms} ^ { - 1 }\). Sphere A collides directly with the left-hand wall, rebounds, then collides directly with the right-hand wall. After this second collision A has a speed of \(3.2 \mathrm {~ms} ^ { - 1 }\).

1. Explain how it can be deduced that the collision between A and the left-hand wall was not inelastic. The coefficient of restitution between A and each wall is \(e\).
2. Calculate the value of \(e\). Sphere A is now brought to rest and a second identical sphere B is placed on the surface. The surface is 1 m long, and A and B are positioned so that they are both 0.5 m from each wall, as shown in Fig. 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-6_241_1307_1322_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Sphere A is projected directly towards the left-hand wall at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\). At the same time, B is projected directly towards the right-hand wall at a speed of \(0.3 \mathrm {~ms} ^ { - 1 }\). You may assume that the duration of impact of a sphere and a wall is negligible.
3. Calculate the distance of A and B from the left-hand wall when they meet again.

7 The diagram shows a cannon fixed to a trolley. The trolley runs on a smooth horizontal track. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-8_310_1086_296_520} A driver boards the trolley with two cannon balls. The combined mass of the trolley, driver, cannon and cannon balls is 320 kg . Each cannon ball has a mass of 5 kg . Initially the trolley is at rest. A force of 480 N acts on the trolley in the forward direction for 4 seconds.

1. 1. Calculate the magnitude of the impulse of the force on the trolley.
   2. Calculate the speed of the trolley after the force stops acting. The driver now fires a cannon ball horizontally in the backward direction. The cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
2. Show that, after the firing of the cannon ball, the trolley moves with a speed of \(7.41 \mathrm {~ms} ^ { - 1 }\), correct to \(\mathbf { 3 }\) significant figures. The driver now reverses the direction of the cannon and fires the second cannon ball horizontally in the forward direction. Again, the cannon ball and cannon separate at a rate of \(90 \mathrm {~ms} ^ { - 1 }\).
3. Calculate the overall percentage change in the kinetic energy of the trolley (alone) from before the first cannon ball is fired to after the second is fired, giving your answer correct to \(\mathbf { 2 }\) decimal places. You should make clear whether the change in kinetic energy is a gain or a loss.
4. Give a reason why one of the modelling assumptions that was required in answering parts (a), (b) and (c) may not have been appropriate. \section*{END OF QUESTION PAPER}

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 .

5. Two smooth spheres \(A\) and \(B\), of equal radii, are moving on a smooth horizontal plane when they collide. Immediately after the collision sphere \(A\) has velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, their line of centres is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the spheres is \(\frac { 2 } { 5 }\). Sphere \(A\) has mass 4 kg and sphere \(B\) has mass 2 kg .

1. Find the velocity of \(A\) and the velocity of \(B\) immediately before the collision. After the collision, sphere \(A\) continues to move with velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) until it collides with a smooth vertical wall. The impulse exerted by the wall on \(A\) is \(32 \mathbf { j }\) Ns.
2. State whether the wall is parallel to the vector \(\mathbf { i }\) or to the vector \(\mathbf { j }\). Give a reason for your answer.
3. Find the speed of \(A\) after the collision with the wall.
4. Calculate the loss of kinetic energy caused by the collision of sphere \(A\) with the wall.

5. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Two smooth spheres \(P\) and \(Q\), of equal radii, are moving on a smooth horizontal surface. The mass of \(P\) is 2 kg and the mass of \(Q\) is 6 kg . The velocity of \(P\) is \(( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). At a particular instant, \(Q\) is positioned 12 m east and 48 m south of \(P\).

1. Prove that \(P\) and \(Q\) will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\). Immediately after the collision, sphere \(Q\) has speed \(5 \mathrm {~ms} ^ { - 1 }\).
2. Determine the coefficient of restitution between the spheres and hence calculate the velocity of \(P\) immediately after the collision.
3. Find the magnitude of the impulse required to stop sphere \(P\) after the collision.

5. A particle \(A\), of mass \(m \mathrm {~kg}\), has position vector \(11 \mathbf { i } + 6 \mathbf { j }\) and a velocity \(2 \mathbf { i } + 7 \mathbf { j }\). At the same moment, second particle \(B\), of mass \(2 m \mathrm {~kg}\), has position vector \(7 \mathbf { i } + 10 \mathbf { j }\) and a velocity \(5 \mathbf { i } + 4 \mathbf { j }\).

1. If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
2. Determine the impulse exerted by \(A\) on \(B\).
3. Calculate the loss of kinetic energy caused by the collision.

1. A particle \(P\), of mass 0.5 kg , is moving with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse I of magnitude 2.5 Ns.

As a result of the impulse, the direction of motion of \(P\) is deflected through an angle of \(45 ^ { \circ }\) Given that \(\mathbf { I } = ( \lambda \mathbf { i } + \mu \mathbf { j } )\) Ns, find all the possible pairs of values of \(\lambda\) and \(\mu\).