6.03j Perfectly elastic/inelastic: collisions

2. Two small smooth spheres \(P\) and \(Q\) are moving along a straight line in opposite directions, with equal speeds, and collide directly. Immediately after the impact, the direction of \(P\) 's motion has been reversed and its speed has been halved. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Express the speed of \(Q\) after the impact in the form \(a u ( b e + c )\), where \(a , b\) and \(c\) are constants to be found.
2. Deduce the range of values of \(e\) for which the direction of motion of \(Q\) remains unaltered.

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 \(h \mathrm {~m}\).

8 \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-4_342_981_255_525} Two small spheres, \(A\) and \(B\), are free to move on the inside of a smooth hollow cylinder, in such a way that they remain in contact with both the curved surface of the cylinder and its horizontal base. The mass of \(A\) is 0.4 kg , the mass of \(B\) is 0.5 kg and the radius of the cylinder is 0.6 m (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.35 . Initially, \(A\) and \(B\) are at opposite ends of a diameter of the base of the cylinder with \(A\) travelling at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) stationary. The magnitude of the force exerted on \(A\) by the curved surface of the cylinder is 6 N .

1. Show that \(v = 3\).
2. Calculate the speeds of the particles after \(A\) 's first impact with \(B\). Sphere \(B\) is removed from the cylinder and sphere \(A\) is now set in motion with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The magnitude of the total force exerted on \(A\) by the cylinder is 4.9 N .
3. Find \(\omega\). \section*{END OF QUESTION PAPER}

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.24 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a horizontal table, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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

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

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

1

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

1

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

1

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

1

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

4

1. A small heavy object of mass 10 kg travels the path ABCD which is shown in Fig. 4. ABCD is in a vertical plane; CD and AEF are horizontal. The sections of the path AB and CD are smooth but section BC is rough. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-5_368_1323_402_338} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} You should assume that
   • the object does not leave the path when travelling along ABCD and does not lose energy when changing direction
   • there is no air resistance.
   Initially, the object is projected from A at a speed of \(16.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope.
   1. Show that the object gets beyond B . The section of the path BC produces a constant resistance of 14 N to the motion of the object.
   2. Using an energy method, find the velocity of the object at D . At D , the object leaves the path and bounces on the smooth horizontal ground between E and F , shown in Fig. 4. The coefficient of restitution in the collision of the object with the ground is \(\frac { 1 } { 2 }\).
   3. Calculate the greatest height above the ground reached by the object after its first bounce.
2. A car of mass 1500 kg travelling along a straight, horizontal road has a steady speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its driving force has power \(P \mathrm {~W}\). When at this speed, the power is suddenly reduced by \(20 \%\). The resistance to the car's motion, \(F \mathrm {~N}\), does not change and the car begins to decelerate at \(0.08 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the values of \(P\) and \(F\).

4

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

1

1. Two model railway trucks are moving freely on a straight horizontal track when they are in a direct collision. The trucks are P of mass 0.5 kg and Q of mass 0.75 kg . They are initially travelling in the same direction. Just before they collide P has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and Q has a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_263_640_484_715} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure}
   1. Suppose that the speed of P is halved in the collision and that its direction of motion is not changed. Find the speed of Q immediately after the collision and find the coefficient of restitution.
   2. Show that it is not possible for both the speed of P to be halved in the collision and its direction of motion to be reversed. Both of the model trucks have flat horizontal tops. They are each travelling at the speeds they had immediately after the collision. Part of the mass of Q is a small object of mass 0.1 kg at rest at the edge of the top of Q nearest P . The object falls off, initially with negligible velocity relative to Q .
   3. Determine the speed of Q immediately after the object falls off it, making your reasoning clear. Part of the mass of P is an object of mass 0.05 kg that is fired horizontally from the top of P , parallel to and in the opposite direction to the motion of P . Immediately after the object is fired, it has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to P .
   4. Determine the speed of P immediately after the object has been fired from it.
2. The velocities of a small object immediately before and after an elastic collision with a horizontal plane are shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-2_172_741_1987_644} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure} Show that the plane cannot be smooth.

1. Two identical particles are approaching each other along a straight horizontal track. Just before they collide, they are moving with speeds \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between the particles is \(\frac { 1 } { 2 }\).

Find the speeds of the particles immediately after the impact.

7. A smooth sphere \(A\) of mass \(4 m\) is moving on a smooth horizontal plane with speed \(u\). It collides directly with a stationary smooth sphere \(B\) of mass \(5 m\) and with the same radius as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).

1. Show that after the collision the speed of \(B\) is 4 times greater than the speed of \(A\).
   (7 marks)
   Sphere \(B\) subsequently hits a smooth vertical wall at right angles. After rebounding from the wall, \(B\) collides with \(A\) again and as a result of this collision, \(B\) comes to rest. Given that the coefficient of restitution between \(B\) and the wall is \(e\),
2. find \(e\). END

6. Three uniform spheres \(A , B\) and \(C\) of equal radius have masses \(3 m , 2 m\) and \(2 m\) respectively. Initially, the spheres are at rest on a smooth horizontal table with their centres in a straight line and with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\),

1. show that the speeds of \(A\) and \(B\) after the collision are \(\frac { 1 } { 3 } u\) and \(u\) respectively.
   (6 marks)
   The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that \(A\) and \(B\) collide again,
2. show that \(e > \frac { 1 } { 3 }\).
   (8 marks)

2. Two smooth spheres \(P\) and \(Q\) of equal radius and of mass \(2 m\) and \(5 m\) respectively, are moving towards each other along a horizontal straight line when they collide. After the collision, \(P\) and \(Q\) travel in opposite directions with speeds of \(3 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively. Given that the coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\), find the speeds of \(P\) and \(Q\) before the collision.
(6 marks)

7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(4 m\) and \(3 m\) respectively. The sphere \(A\) is moving on a smooth horizontal surface and collides with the sphere \(B\), which is stationary on the same surface. Just before the collision, \(A\) is moving with speed \(u\) at an angle of \(30 ^ { \circ }\) to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_449_927_1244_456} The coefficient of restitution between the spheres is \(\frac { 5 } { 9 }\).

1. Find the value of \(\alpha\).
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) during the collision.
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-23_2349_1707_221_153}

4 The diagram shows part of a horizontal snooker table of width 1.69 m . A player strikes the ball \(B\) directly, and it moves in a straight line. The ball hits the cushion of the table at \(C\) before rebounding and moving to the pocket at \(P\) at the corner of the table, as shown in the diagram. The point \(C\) is 1.20 m from the corner \(A\) of the table. The ball has mass 0.15 kg and, immediately before the collision with the cushion, it has velocity \(u\) in a direction inclined at \(60 ^ { \circ }\) to the cushion. The table and the cushion are modelled as smooth. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-08_517_963_719_511}

1. Find the coefficient of restitution between the ball and the cushion.
2. Show that the magnitude of the impulse on the cushion at \(C\) is approximately \(0.236 u\).
3. Find, in terms of \(u\), the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\).
4. Explain how you have used the assumption that the cushion is smooth in your answers.

5 A particle is projected from a point \(O\) on a smooth plane, which is inclined at \(25 ^ { \circ }\) to the horizontal. The particle is projected up the plane with velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(30 ^ { \circ }\) above the plane. The particle strikes the plane for the first time at a point \(A\). The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-12_518_839_552_630}

1. Find the time taken by the particle to travel from \(O\) to \(A\).
2. The coefficient of restitution between the particle and the inclined plane is \(\frac { 2 } { 3 }\). Find the speed of the particle as it rebounds from the inclined plane at \(A\). (8 marks)

7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it collides with the sphere \(B\), which has velocity \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision, the velocity of the sphere \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Show that the impulse exerted on \(B\) in the collision is \(( 6 m \mathbf { j } )\) Ns.
3. Find the coefficient of restitution between the two spheres.
4. After the collision, each sphere moves in a straight line with constant speed. Given that the radius of each sphere is 0.05 m , find the time taken, from the collision, until the centres of the spheres are 1.10 m apart.

4 A smooth sphere \(A\), of mass \(m\), is moving with speed \(4 u\) in a straight line on a smooth horizontal table. A smooth sphere \(B\), of mass \(3 m\), has the same radius as \(A\) and is moving on the table with speed \(2 u\) in the same direction as \(A\). \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-10_289_780_493_625} The sphere \(A\) collides directly with sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speeds of \(A\) and \(B\) immediately after the collision.
2. Show that the speed of \(B\) after the collision cannot be greater than \(3 u\).
3. Given that \(e = \frac { 2 } { 3 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) in the collision.

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 4 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to unit vector \(\mathbf { i }\). The direction of motion of \(B\) is changed through \(90 ^ { \circ }\) by the collision, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-14_332_1184_566_543}

1. Show that the velocity of \(B\) immediately after the collision is \(\left( \frac { 9 } { 2 } \mathbf { i } - 3 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the coefficient of restitution between the spheres.
3. Find the impulse exerted on \(B\) during the collision. State the units of your answer.

5 A small smooth ball is dropped from a height of \(h\) above a point \(A\) on a fixed smooth plane inclined at an angle \(\theta\) to the horizontal. The ball falls vertically and collides with the plane at the point \(A\). The ball rebounds and strikes the plane again at a point \(B\), as shown in the diagram. The points \(A\) and \(B\) lie on a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-12_318_636_548_712}

1. Explain whether or not the component of the velocity of the ball parallel to the plane is changed by the collision.
2. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(h , \theta , e\) and \(g\), the components of the velocity of the ball parallel to and perpendicular to the plane immediately after the collision.
3. Show that the distance \(A B\) is given by $$4 h e ( e + 1 ) \sin \theta$$

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 4 kg respectively. The spheres are moving on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres of the spheres, and \(B\) has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-16_291_844_607_468} Just after the collision, \(B\) moves in a direction perpendicular to the line of centres.

1. Find the speed of \(A\) immediately after the collision.
2. Find the acute angle, correct to the nearest degree, between the velocity of \(A\) and the line of centres immediately after the collision.
3. Find the coefficient of restitution between the spheres.
4. Find the magnitude of the impulse exerted on \(B\) during the collision.