6.03f Impulse-momentum: relation

4. A lift of mass 70 kg is supported by a cable which remains taut at all times. A man of mass 90 kg gets into the lift and it begins to descend vertically from rest with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate, giving your answers correct to 3 significant figures,

1. the magnitude of the force which the lift exerts on the man,
2. the tension in the cable. Prior to slowing down, the lift is moving at \(2 \mathrm {~ms} ^ { - 1 }\). It then uniformly decelerates until it is brought to rest.
3. Find the impulse exerted by the cable on the lift in bringing the lift to rest.
4. Given that it takes 2 seconds to come to rest, use your answer to part (c) to calculate the magnitude of the force exerted by the cable on the lift in bringing the lift to rest.
   (2 marks)

4. A bullet of mass 50 g is fired horizontally at a wooden block of mass 4.95 kg which is lying at rest on a rough horizontal surface. The bullet enters the block at \(400 \mathrm {~ms} ^ { - 1 }\) and becomes embedded in the block.

1. Find the speed with which the block begins to move. Given that the block decelerates uniformly to rest over a distance of 4 m ,
2. show that the coefficient of friction is \(\frac { 2 } { g }\).

5. A sledgehammer of mass 12 kg is being used to drive a wooden post of mass 4 kg into the ground. A labourer moves the sledgehammer from rest at a point 0.5 m vertically above the post with constant acceleration \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directed towards the post.

1. Find the velocity with which the sledgehammer hits the post. When the sledgehammer hits the post, they both move together with common speed, \(V\).
2. Show that \(V = 3 \mathrm {~ms} ^ { - 1 }\). As the sledgehammer hits the post, the labourer relaxes his grip and applies no further force. The sledgehammer and post are brought to rest by the action of a resistive force from the ground of magnitude 1500 N .
3. Find, in centimetres, the total distance that the sledgehammer and the post travel together before coming to rest.

3. A cannon of mass 600 kg lies on a rough horizontal surface and is used to fire a 3 kg shell horizontally at \(200 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse which the shell exerts on the cannon.
2. Find the speed with which the cannon recoils. Given that the coefficient of friction between the cannon and the surface is 0.75 ,
3. calculate, to the nearest centimetre, the distance that the cannon travels before coming to rest.

3. A small van of mass 1500 kg is used to tow a car of mass 750 kg by means of a rope of length 9 m joined to both vehicles. The van sets off with the rope slack and reaches a speed of \(2 \mathrm {~ms} ^ { - 1 }\) just before the rope becomes taut and jerks the car into motion. Immediately after the rope becomes taut, the van and car travel with common speed \(V \mathrm {~ms} ^ { - 1 }\).

1. Show that \(V = \frac { 4 } { 3 }\).
2. Calculate the magnitude of the impulse on the car when the rope tightens. The van and car eventually reach a steady speed of \(18 \mathrm {~ms} ^ { - 1 }\) with the rope taut when a child runs out into the road, 30 m in front of the van. The van driver brakes sharply and decelerates uniformly to rest in a distance of 27 m . It takes the driver of the car 1 second to react to the van starting to brake. He then brakes and the car decelerates uniformly at \(f \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest before colliding with the van.
3. Find the set of possible values of \(f\).
   (5 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.

1 A particle, of mass 0.8 kg , moves along a smooth horizontal surface. It hits a vertical wall, which is at right angles to the direction of motion of the particle, and rebounds. The speed of the particle as it hits the wall is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of restitution between the particle and the wall is 0.3 . Find

1. the impulse that the wall exerts on the particle,
2. the kinetic energy lost in the impact.

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 \(m\).

5 A small sphere of mass 0.2 kg is projected vertically downwards with a speed of \(5 \mathrm {~ms} ^ { - 1 }\) from a height of 1.6 m above horizontal ground. It hits the ground and rebounds vertically upwards coming to instantaneous rest at a height of \(h \mathrm {~m}\) above the ground. The coefficient of restitution between the sphere and the ground is 0.7 .

1. Find \(h\).
2. Find the magnitude and direction of the impulse exerted on the sphere by the ground.
3. Find the loss of energy of the sphere between the instant of projection and the instant it comes to instantaneous rest at height \(h \mathrm {~m}\).

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. 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
   Coordinates are referred to the axes shown in the figure.

1. A ball of mass 0.6 kg bounces against a wall and is given an impulse of \(( 12 \mathbf { i } - 9 \mathbf { j } ) \mathrm { Ns }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors. The velocity of the particle after the impact is \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the velocity of the particle before the impact.
(4 marks)

1. An ice hockey puck of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity \(( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)

3 A ball of mass 0.45 kg is travelling horizontally with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time \(t\) seconds after first coming into contact with the bat, the force exerted on the ball by the bat is \(1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)\) newtons, where \(0 \leqslant t \leqslant 0.1\). In this simple model, ignore the weight of the ball and model the ball as a particle.

1. Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
2. Find, to two significant figures, the speed of the ball immediately after the impact.
3. Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.

3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).

1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
2. Find the speed of \(P\) when \(t = 3\).
3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).

3 A particle of mass 0.2 kg lies at rest on a smooth horizontal table. A horizontal force of magnitude \(F\) newtons acts on the particle in a constant direction for 0.1 seconds. At time \(t\) seconds, $$F = 5 \times 10 ^ { 3 } t ^ { 2 } , \quad 0 \leqslant t \leqslant 0.1$$ Find the value of \(t\) when the speed of the particle is \(2 \mathrm {~ms} ^ { - 1 }\).
(4 marks)

4 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(m\) and \(2 m\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) when it collides with the sphere \(B\) which has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } )\). After the collision, the velocity of \(B\) is \(( \mathbf { i } + \mathbf { j } )\).

1. Find the velocity of \(A\) immediately after the collision.
2. Find the angle between the velocities of \(A\) and \(B\) immediately after the collision.
3. Find the impulse exerted by \(B\) on \(A\).
4. State, as a vector, the direction of the line of centres of \(A\) and \(B\) when they collide.
   (1 mark)

6 A small smooth ball of mass \(m\), moving on a smooth horizontal surface, hits a smooth vertical wall and rebounds. The coefficient of restitution between the wall and the ball is \(\frac { 3 } { 4 }\). Immediately before the collision, the ball has velocity \(u\) and the angle between the ball's direction of motion and the wall is \(\alpha\). The ball's direction of motion immediately after the collision is at right angles to its direction of motion before the collision, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-4_483_344_657_854}

1. Show that \(\tan \alpha = \frac { 2 } { \sqrt { 3 } }\).
2. Find, in terms of \(u\), the speed of the ball immediately after the collision.
3. The force exerted on the ball by the wall acts for 0.1 seconds. Given that \(m = 0.2 \mathrm {~kg}\) and \(u = 4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the average force exerted by the wall on the ball.

3 Three smooth spheres, \(A , B\) and \(C\), of equal radii have masses \(1 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(x \mathrm {~kg}\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(3 u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is \(\frac { 1 } { 3 }\).

1. Show that \(A\) is brought to rest by the impact and find the speed of \(B\) immediately after the collision in terms of \(u\).
2. Subsequently, \(B\) collides with \(C\). Show that the speed of \(C\) immediately after the collision is \(\frac { 4 u } { 3 + x }\).
   Find the speed of \(B\) immediately after the collision in terms of \(u\) and \(x\).
3. Show that \(B\) will collide with \(A\) again if \(x > 9\).
4. Given that \(x = 5\), find the magnitude of the impulse exerted on \(C\) by \(B\) in terms of \(u\).
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}