6.03l Newton's law: oblique impacts

4

1. A pulley consists of a central cylinder of wood and an outer ring of steel. The density of the wood is \(700 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the density of the steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). The pulley has a radius of 20 cm and is 10 cm thick (see Fig. 4.1). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-4_359_661_404_742} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Find the radius that the central cylinder must have in order that the moment of inertia of the pulley about the axis of symmetry shown in Fig. 4.1 is \(1.5 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
2. Two blocks P and Q of masses 10 kg and 20 kg are connected by a light inextensible string. The string passes over a heavy rough pulley of radius 25 cm . The pulley can rotate freely and the string does not slip. Block P is held at rest in smooth contact with a plane inclined at \(30 ^ { \circ }\) to the horizontal, and block Q is at rest below the pulley (see Fig. 4.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c3ac9277-d34d-4d0e-9f9b-d0bce8c741af-5_341_917_438_541} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} At \(t \mathrm {~s}\) after the system is released from rest, the pulley has angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and block P has constant acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up the slope.
   1. Show that the net loss of energy of the two blocks in the first \(t\) seconds of motion is \(87 t ^ { 2 } \mathrm {~J}\) and use the principle of conservation of energy to show that the moment of inertia of the pulley about its axis of rotation is \(\frac { 87 } { 32 } \mathrm {~kg} \mathrm {~m} ^ { 2 }\). When \(t = 3\) a resistive couple is applied to the pulley. This resistive couple has magnitude \(( 2 \omega + k ) \mathrm { Nm }\), where \(k\) is a constant. The couple on the pulley due to tensions in the sections of string is \(\left( \frac { 147 } { 4 } - \frac { 15 } { 8 } \frac { \mathrm {~d} \omega } { \mathrm {~d} t } \right) \mathrm { Nm }\) in the direction of positive \(\omega\).
   2. Write down a first order differential equation for \(\omega\) when \(t \geqslant 3\) and show by integration that $$\omega = \frac { 1 } { 8 } \left( ( 45 + 4 k ) \mathrm { e } ^ { \frac { 64 } { 147 } ( 3 - t ) } + 147 - 4 k \right) .$$
   3. By considering the equation given in part (ii), find the value or set of values of \(k\) for which the pulley
      (A) continues to rotate with constant angular velocity,
      (B) rotates with decreasing angular velocity without coming to rest,
      (C) rotates with decreasing angular velocity and comes to rest if there is sufficient distance between P and the pulley. \section*{END OF QUESTION PAPER}

5 A small ball is held at a height of 160 cm above a horizontal surface. The ball is released from rest and rebounds from the surface. After its first bounce on the surface the ball reaches a height of 122.5 cm .

1. Find the height reached by the ball after its second bounce on the surface. After \(n\) bounces the height reached by the ball is less than 10 cm .
2. Find the minimum possible value of \(n\).
3. State what would happen if the same ball is released from rest from a height of 160 cm above a different horizontal surface and
   (A) the coefficient of restitution between the ball and the new surface is 0 ,
   (B) the coefficient of restitution between the ball and the new surface is 1 .

6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.
The particles have masses \(5 \mathrm {~kg} , 3 \mathrm {~kg}\) and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of \(u \mathrm {~ms} ^ { - 1 }\) and collides with B . The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Particle B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 3 }\).

1. Determine whether any further collisions occur.
2. Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of \(u\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure} Fig. 7 shows a uniform rod AB of length \(4 a\) and mass \(m\).
   The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where \(\mathrm { AC } = 4 a\). The plane ABC is perpendicular to the wall and the angle ABC is \(30 ^ { \circ }\). The system is in limiting equilibrium. Find the coefficient of friction between the wall and the rod. \section*{END OF QUESTION PAPER}

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.

4 The diagram shows three beads, A, B and C, of masses \(0.3 \mathrm {~kg} , 0.5 \mathrm {~kg}\) and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section. \includegraphics[max width=\textwidth, alt={}, center]{a87d62b8-406d-44cd-9ffa-384005329566-5_361_961_450_248} Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B with a speed of \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the subsequent collision between A and \(\mathrm { B } , \mathrm { A }\) is brought to rest.

1. Show that the coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Bead B next collides with C.
2. Show that the speed of B before this collision is \(4.37 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures. In this collision between B and C , B is brought to rest.
3. Determine whether C next collides with A or with B .
4. Explain why, if B has a greater mass than C , B could not be brought to rest in their collision.

4 Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of \(\mathrm { A } , \mathrm { B }\) and C are \(2 \mathrm {~kg} , 4 \mathrm {~kg}\) and 1 kg respectively. Initially the three spheres are at rest.
Spheres \(A\) and \(C\) are each given impulses so that \(A\) moves towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-5_325_1591_603_239} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
It is given that the first collision occurs between A and B .

1. State how you can tell from the information given above that kinetic energy is lost when A collides with B .
2. Show that the combined kinetic energy of A and B decreases by \(24 \%\) during their collision. Sphere B next collides with C. The coefficient of restitution between B and C is \(\frac { 2 } { 3 }\).
3. Given that a third collision occurs, determine the range of possible values for \(u\).
4. State one limitation of the model used in this question.

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 .

3 Three small uniform spheres A, B and C have masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively. The spheres move in the same straight line on a smooth horizontal table, with B between A and C . Sphere A moves towards B with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 } , \mathrm {~B}\) is at rest and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-3_181_1291_461_251} Spheres A and B collide. Collisions between A and B can be modelled as perfectly elastic.

1. Determine the magnitude of the impulse of A on B in this collision.
2. Use this collision to verify that in a perfectly elastic collision no kinetic energy is lost. After the collision between A and B, sphere B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 4 }\).
3. Show that, after the collision between B and C , B has a speed of \(( 1.225 - 0.78125 \mathrm { u } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) towards C.
4. Determine the range of values for \(u\) for there to be a second collision between A and B .

7 Rose and Steve collide while sitting firmly on trays that are sliding on smooth horizontal ice. There are no external driving forces. Fig. 7 shows the masses of Rose and of Steve with their trays, their velocities just before their collision and the line of their motion and of their impact. Immediately after the collision, Rose has a velocity of \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of her motion before the collision. \begin{figure}[h] \end{figure}

1. Find Steve's velocity after the collision.
2. Find the coefficient of restitution between Rose and Steve on their trays. Shortly after the collision, Steve catches Rose's hand, pulls her towards him with a horizontal impulse of 4.48 Ns and then lets go of her hand.
3. Calculate Rose's velocity after the pull. When they collide again they hold one another and move together with a common speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Calculate \(V\).
5. Why did you need to know that there are no driving forces and that the ice is smooth? {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.
   OCR is part of the }

4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.
Immediately before the collision, A is moving with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) with the line of centres, where \(\sin \alpha = 0.6\), and B is moving with speed \(0.3 \mathrm {~ms} ^ { - 1 }\) at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 . \begin{figure}[h] \end{figure}

1. Find the speeds of A and B immediately after the collision.
2. Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
3. Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
4. How does your answer to part (b) change if the contact between B and the wall is not smooth?

5 Two small uniform discs, A of mass \(2 m \mathrm {~kg}\) and B of mass \(3 m \mathrm {~kg}\), slide on a smooth horizontal surface and collide obliquely with smooth contact. Immediately before the collision, A is moving towards B along the line of centres with speed \(2 \mathrm {~ms} ^ { - 1 }\) and B is moving towards A with speed \(\sqrt { 3 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-5_366_976_539_244}

1. Explain how you know that the motion of A will be along the line of centres after the collision.
2. - Determine the maximum possible speed of A after the collision.
   When the speed of B after the collision is a minimum, the loss of kinetic energy in the collision is 1.4625 J .
3. Determine the value of \(m\).

4 Two small smooth discs, A of mass 0.5 kg and B of mass 0.4 kg , collide while sliding on a smooth horizontal plane. Immediately before the collision A and B are moving towards each other, A with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Before the collision the direction of motion of A makes an angle \(\alpha\) with the line of centres, where \(\tan \alpha = 0.75\), and the direction of motion of B makes an angle of \(60 ^ { \circ }\) with the line of centres, as shown in Fig. 4. \begin{figure}[h] \end{figure} After the collision, one of the discs moves in a direction perpendicular to the line of centres, and the other disc moves in a direction making an angle \(\beta\) with the line of centres.

1. Explain why the disc which moves perpendicular to the line of centres must be A .
2. Determine the value of \(\beta\).
3. Determine the kinetic energy lost in the collision.
4. Determine the value of the coefficient of restitution between A and B .

4 Two uniform circular discs with the same radius, A of mass 1 kg and B of mass 5.25 kg , slide on a smooth horizontal surface and collide obliquely with smooth contact. Fig. 4 gives information about the velocities of the discs just before and just after the collision.

• The line XY passes through the centres of the discs at the moment of collision
• The components parallel and perpendicular to XY of the velocities of A are shown
• Before the collision, B is at rest and after it is moving at \(2 \mathrm {~ms} ^ { - 1 }\) in the direction XY

\begin{figure}[h] \end{figure} The coefficient of restitution between the two discs is \(\frac { 2 } { 3 }\).

1. Find the values of \(U\) and \(u\).
2. What information in the question tells you that \(v = V\) ? The speed of disc A before the collision is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the speed of disc A after the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_398_396_397_475} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_399_332_399_945} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_305_326_493_1354} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} Fig. 5.1 shows a vertical light elastic spring. It is fixed to a horizontal table at one end. Fig 5.2 shows the spring with a particle of mass \(m \mathrm {~kg}\) attached to it at the other end. The system is in equilibrium when the spring is compressed by a distance \(h \mathrm {~m}\).

6 Stones A and B have masses \(m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively. They lie at rest on a large area of smooth horizontal ice and may move freely over the ice. Stone A is given a horizontal impulse of magnitude \(m u \mathrm {~N} s\) towards B so that the stones collide directly. After the collision the direction of motion of A is reversed. The coefficient of restitution between A and B is denoted by \(e\).

1. Find the range of possible values of \(e\). After the collision, B subsequently collides with a vertical smooth wall perpendicular to its path and rebounds. The coefficient of restitution between \(B\) and the wall is the same as the coefficient of restitution between A and B .
2. Show that A and B will collide again unless the collision between B and the wall is perfectly elastic.
3. Explain why modelling the collision between B and the wall as perfectly elastic is possibly unrealistic.
4. Given that the kinetic energy lost in the first collision between A and B is \(\frac { 5 } { 24 } m u ^ { 2 }\), determine the value of \(e\).
5. Given that B was 2 metres from the wall when the stones first collided, determine the distance of the stones from the wall when they next collide.

4 A child throws a ball of mass \(m \mathrm {~kg}\) vertically upwards with a speed of \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball leaves the child's hand at a height of 1.6 m above horizontal ground.

1. Ignoring any possible air resistance, use an energy method to determine the maximum height reached by the ball above the ground. In fact, the ball only reaches a height of 4.1 m above the ground. For the rest of this question you should assume that the air resistance may be modelled as a constant force acting in the opposite direction to the ball's motion.
2. Show that the ball does 0.568 mJ of work against air resistance per metre travelled.
3. Calculate the speed of the ball just before it hits the ground. The ball bounces off the ground and first comes instantaneously to rest 2.8 m above the ground.
4. Determine the coefficient of restitution between the ball and the ground. In the first impact between the ball and the ground, the magnitude of the impulse exerted on the ball by the ground is 12 Ns .
5. Determine the value of \(m\).

6 A block rests on a horizontal surface. The coefficient of friction between the block and the surface is \(\mu\).

1. Show that if the block is given an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it will move a distance of \(\frac { \mathrm { v } ^ { 2 } } { 2 \mu \mathrm {~g} }\) before coming to rest. Block B rests on the same horizontal surface as a sphere S . On the other side of S is a vertical wall, as shown below. The mass of \(B\) is 8 times the mass of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-8_211_1013_662_244} S is projected directly towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and hits B . It is given that
   Furthermore, you should model the contact between B and the surface as rough and model the contact between S and the surface as smooth.
2. Determine, in terms of \(u\), expressions for
   It is given that B has sufficient time to come to rest before each subsequent collision with S .
   Let \(\mathrm { X } _ { \mathrm { n } }\) be the distance B moves after the \(n\)th impact between S and B .
3. Explain why \(\mathrm { x } _ { \mathrm { n } + 1 } = \frac { 9 } { 25 } \mathrm { x } _ { \mathrm { n } }\).
4. Given that \(u = 11.2\) and the coefficient of friction between B and the surface is \(\frac { 1 } { 7 }\), show that B will travel a total distance that cannot exceed 2.8 m . \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .

7. Three spheres \(A , B , C\), of equal radii and each of mass \(m \mathrm {~kg}\), lie at rest on a smooth horizontal surface such that their centres are in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u \mathrm {~ms} ^ { - 1 }\) so that it collides with \(B\).

1. Find expressions, in terms of \(e\) and \(u\), for the speed of \(A\) and the speed of \(B\) after they collide. You are now given that \(e = \frac { 1 } { 2 }\).
2. Find, in terms of \(m\) and \(u\), the loss in kinetic energy due to the collision between \(A\) and \(B\).
3. After the collision between \(A\) and \(B\), sphere \(B\) then collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e _ { 1 }\). Show that there will be no further collisions if \(e _ { 1 } \leqslant \frac { 1 } { 3 }\).

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.

1. \begin{figure}[h] \end{figure} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W _ { 1 }\) and \(W _ { 2 }\) are two fixed parallel vertical walls. The walls are 3 metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leqslant 3\), from \(W _ { 1 }\) At time \(t = 0\), the particle is projected from \(O\) towards \(W _ { 1 }\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\) The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.

1. Show that \(T = \frac { 45 - 5 d } { 4 u }\) The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.
2. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer.

2. \begin{figure}[h] \end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are fixed vertical walls with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards \(A B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) on a path that makes an angle \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\). The ball hits \(A B\) and then hits \(B C\).
Immediately after hitting \(A B\), the ball is moving at an angle \(\beta\) to \(A B\), where \(\tan \beta = \frac { 1 } { 3 }\) The coefficient of restitution between the ball and \(A B\) is \(e\).
The coefficient of restitution between the ball and \(B C\) is \(\frac { 1 } { 2 }\) By modelling the ball as a particle and the floor and walls as being smooth,

1. show that the value of \(e = \frac { 1 } { 4 }\)
2. find the speed of the ball immediately after it hits \(B C\).
3. Suggest two ways in which the model could be refined to make it more realistic.

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

\begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a smooth horizontal floor, where \(A B\) represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.
Immediately before the impact the velocity of the ball is \(( 7 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Immediately after the impact the velocity of the ball is \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
The coefficient of restitution between the ball and the wall is \(e\).

1. Show that \(A B\) is parallel to \(( 2 \mathbf { i } + 3 \mathbf { j } )\).
2. Find the value of \(e\).

7. \begin{figure}[h] \end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(C D\) represent fixed vertical walls, with \(A B\) parallel to \(C D\). A small ball is projected along the floor towards wall \(A B\). Immediately before hitting wall \(A B\), the ball is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to \(A B\), where \(0 < \alpha < \frac { \pi } { 2 }\) The ball hits wall \(A B\) and then hits wall \(C D\).
After the impact with wall \(C D\), the ball is moving at angle \(\frac { 1 } { 2 } \alpha\) to \(C D\).
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 2 } { 3 }\) The coefficient of restitution between the ball and wall \(C D\) is also \(\frac { 2 } { 3 }\) The floor and the walls are modelled as being smooth. The ball is modelled as a particle.

1. Show that \(\tan \left( \frac { 1 } { 2 } \alpha \right) = \frac { 1 } { 3 }\)
2. Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

5. \begin{figure}[h] \end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) represent fixed vertical walls, with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards the wall \(A B\). Immediately before hitting the wall \(A B\) the ball is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to \(A B\). The ball hits the wall \(A B\) and then hits the wall \(B C\).
The coefficient of restitution between the ball and the wall \(A B\) is \(\frac { 1 } { 3 }\) The coefficient of restitution between the ball and the wall \(B C\) is \(e\).
The floor and the walls are modelled as being smooth.
The ball is modelled as a particle.
The ball loses half of its kinetic energy in the impact with the wall \(A B\).

1. Find the exact value of \(\cos \theta\). The ball loses half of its remaining kinetic energy in the impact with the wall \(B C\).
2. Find the exact value of \(e\).