6.03a Linear momentum: p = mv

7 Two small smooth spheres, \(A\) and \(B\), are the same size and have masses \(2 m\) and \(m\) respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere \(A\) receives an impulse of magnitude \(J\) and moves with speed \(2 u\) directly towards \(B\).

1. \(\quad\) Find \(J\) in terms of \(m\) and \(u\).
2. The sphere \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). Find, in terms of \(u\), the speeds of \(A\) and \(B\) immediately after the collision.
3. At the instant of collision, the centre of \(B\) is at a distance \(s\) from a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, \(B\) collides with the wall. The radius of each sphere is \(r\).
   Show that the distance of the centre of \(A\) from the wall at the instant that \(B\) hits the wall is \(\frac { 3 s + 12 r } { 5 }\).
4. The diagram below shows the positions of \(A\) and \(B\) when \(B\) hits the wall. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere \(B\) collides with \(A\) again after rebounding from the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\). Find the distance of the centre of \(\boldsymbol { B }\) from the wall at the instant when \(A\) and \(B\) collide again.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}

6 \includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 2 kg and \(m \mathrm {~kg}\) respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving towards \(B\) at an angle of \(\alpha\) to the line of centres, where \(\cos \alpha = 0.6\). \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(A\) along the line of centres (see diagram). As a result of the collision, \(A\) 's loss of kinetic energy is \(7.56 \mathrm {~J} , B\) 's direction of motion is reversed and \(B\) 's speed after the collision is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) after the collision,
2. the component of \(A\) 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,
3. the value of \(m\),
4. the coefficient of restitution between \(A\) and \(B\). \(7 S _ { A }\) and \(S _ { B }\) are light elastic strings. \(S _ { A }\) has natural length 2 m and modulus of elasticity \(120 \mathrm {~N} ; S _ { B }\) has natural length 3 m and modulus of elasticity 180 N . A particle \(P\) of mass 0.8 kg is attached to one end of each of the strings. The other ends of \(S _ { A }\) and \(S _ { B }\) are attached to fixed points \(A\) and \(B\) respectively, on a smooth horizontal table. The distance \(A B\) is \(6 \mathrm {~m} . P\) is released from rest at the point of the line segment \(A B\) which is 2.9 m from \(A\).
5. For the subsequent motion, show that the total elastic potential energy of the strings is the same when \(A P = 2.1 \mathrm {~m}\) and when \(A P = 2.9 \mathrm {~m}\). Deduce that neither string becomes slack.
6. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
7. State, giving a reason, the type of motion of \(P\) and find the time taken between successive occasions when \(P\) is instantaneously at rest. For the instant 0.6 seconds after \(P\) is released, find
8. the distance travelled by \(P\),
9. the speed of \(P\).

4 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.2 kg . Immediately before the collision \(A\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving away from \(A\) with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_422_844_431_612} The coefficient of restitution between the spheres is 0.8 . Find

1. the velocity of \(A\) immediately after the collision,
2. the angle turned through by the direction of motion of \(B\) as a result of the collision.

1 An empty railway truck of mass \(m _ { 0 }\) is moving along a straight horizontal track at speed \(v _ { 0 }\). The point P is at the front of the truck. The horizontal forces on the truck are negligible. As P passes a fixed point O , sand starts to fall vertically into the truck at a constant mass rate \(k\). At time \(t\) after P passes O the speed of the truck is \(v\) and \(\mathrm { OP } = x\).

1. Find an expression for \(v\) in terms of \(m _ { 0 } , v _ { 0 } , k\) and \(t\), and show that \(x = \frac { m _ { 0 } v _ { 0 } } { k } \ln \left( 1 + \frac { k t } { m _ { 0 } } \right)\).
2. Find the speed of the truck and the distance OP when the mass of sand in the truck is \(2 m _ { 0 }\).

2 On a building site, a pulley system is used for moving small amounts of material up to roof level. A light pulley, which can rotate freely, is attached with its axis horizontal to the top of some scaffolding. A light inextensible rope hangs over the pulley with a counterweight of mass \(m _ { 1 } \mathrm {~kg}\) attached to one end. Attached to the other end of the rope is a bag of negligible mass into which \(m _ { 2 } \mathrm {~kg}\) of roof tiles are placed, where \(m _ { 2 } < m _ { 1 }\). This situation is shown in Fig. 2. \begin{figure}[h] \end{figure} Initially the system is held at rest with the rope taut, the counterweight at the top of the scaffolding and the bag of tiles on the ground. When the counterweight is released, the bag ascends towards the top of the scaffolding. At time \(t \mathrm {~s}\) the velocity of the counterweight is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards. The counterweight is made from a bag of negligible mass filled with sand. At the moment the counterweight is released, this bag is accidentally ripped and after this time the sand drops out at a constant rate of \(\lambda \mathrm { kg } \mathrm { s } ^ { - 1 }\).

1. Find the equation of motion for the counterweight while it still contains sand, and hence show that $$v = g t + \frac { 2 g m _ { 2 } } { \lambda } \ln \left( 1 - \frac { \lambda t } { m _ { 1 } + m _ { 2 } } \right) .$$
2. Given that the sand would run out after 10 seconds and that \(m _ { 2 } = \frac { 4 } { 5 } m _ { 1 }\), find the maximum velocity attained by the counterweight towards the ground. You may assume that the scaffolding is sufficiently high that the counterweight does not hit the ground before this velocity is reached.

1 A rocket is launched vertically upwards from rest. The initial mass of the rocket, including fuel and payload, is \(m _ { 0 }\) and the propulsion system ejects mass at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. The only other force acting on the rocket is its weight. The acceleration due to gravity is constant throughout the motion. At time \(t\) after launch the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket \(v = u \ln \left( \frac { m _ { 0 } } { m _ { 0 } - k t } \right) - g t\). The rocket initially has 2400 kg of fuel which is ejected at a constant rate of \(100 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\) with constant speed \(3000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket.
2. Given that the rocket must reach a speed of \(7910 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before releasing its payload, find the maximum possible value of \(m _ { 0 }\).

2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector \(( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N , acting at the origin, together with a couple \(\mathbf { G }\).

1. Find the values of \(p , q\) and \(r\).
2. Find \(\mathbf { G }\).

5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.

1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000 .$$
2. Find the speed of the rocket when burning stops.

5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).

6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that

1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
   (6)
   (Total 13 marks)

7. A motor boat of mass \(M\) is moving in a straight line, with its engine switched off, across a stretch of still water. The boat is moving with speed \(U\) when, at time \(t = 0\), it develops a leak. The water comes in at a constant rate so that at time \(t\), the mass of water in the boat is \(\lambda t\). At time \(t\) the speed of the boat is \(v\) and it experiences a total resistance to motion of magnitude \(2 \lambda v\).

1. Show that \(( M + \lambda t ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 \lambda v = 0\).
   (6)
2. Show that the time taken for the speed of the boat to reduce to \(\frac { 1 } { 2 } U\) is \(\frac { M } { \lambda } \left( 2 ^ { \frac { 1 } { 3 } } - 1 \right)\).
   (6) The boat sinks when the mass of water inside the boat is \(M\).
3. Show that the boat does not sink before the speed of the boat is \(\frac { 1 } { 2 } U\).

4. A particle \(P\), whose initial mass is \(m _ { 0 }\), is projected vertically upwards from the ground at time \(t = 0\) with speed \(\frac { g } { k }\), where \(k\) is a constant. As the particle moves upwards it gains mass by picking up small droplets of moisture from the atmosphere. The droplets are at rest before they are picked up. At time \(t\) the speed of \(P\) is \(v\) and its mass has increased to \(m _ { 0 } \mathrm { e } ^ { k t }\). Assuming that, during the motion, the acceleration due to gravity is constant,

1. show that, while \(P\) is moving upwards, $$k v + \frac { \mathrm { d } v } { \mathrm {~d} t } = - g$$
2. find, in terms of \(m _ { 0 }\), the mass of \(P\) when it reaches its greatest height above the ground.
   (6)

6. A firework rocket, excluding its fuel, has mass \(m _ { 0 } \mathrm {~kg}\). The rocket moves vertically upwards by ejecting burnt fuel vertically downwards with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } , u > 24.5\), relative to the rocket. The rocket starts from rest on the ground at time \(t = 0\). At time \(t\) seconds, \(t \leqslant 2\), the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the mass of the rocket including its fuel is \(m _ { 0 } ( 5 - 2 t ) \mathrm { kg }\). It is assumed that air resistance is negligible and the acceleration due to gravity is constant.

1. Show that, for \(t \leqslant 2\) $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 2 u } { 5 - 2 t } - 9.8$$
2. Find the speed of the rocket at the instant when all of its fuel has been burnt.

7. A uniform square lamina \(P Q R S\), of mass \(m\) and side \(2 a\), is free to rotate about a fixed smooth horizontal axis which passes through \(P\) and \(Q\). The lamina hangs at rest in a vertical plane with \(S R\) below \(P Q\) and is given a horizontal impulse of magnitude \(J\) at the midpoint of \(S R\). The impulse is perpendicular to \(S R\).

1. Find the initial angular speed of the lamina.
2. Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
3. Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}

6. A small object \(P\), of mass \(m _ { 0 }\), is projected vertically upwards from the ground with speed \(U\). As \(P\) moves upwards it picks up droplets of moisture from the atmosphere. The droplets are at rest immediately before they are picked up. In a model of the motion, \(P\) is modelled as a particle, air resistance is assumed to be negligible and the acceleration due to gravity is assumed to have the constant value of \(g\). When \(P\) is at a height \(x\) above the ground, the combined mass of \(P\) and the moisture is \(m _ { 0 } ( 1 + k x )\), where \(k\) is a constant, and the speed of \(P\) is \(v\).

1. Show that, while \(P\) is moving upwards $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( v ^ { 2 } \right) + \frac { 2 k v ^ { 2 } } { ( 1 + k x ) } = - 2 g$$ The general solution of this differential equation is given by \(v ^ { 2 } = \frac { A } { ( 1 + k x ) ^ { 2 } } - \frac { 2 g } { 3 k } ( 1 + k x )\),
   where \(A\) is an arbitrary constant. Given that \(U = \sqrt { 2 g h }\) and \(k = \frac { 7 } { 3 h }\)
2. find, in terms of \(h\), the height of \(P\) above the ground when \(P\) first comes to rest.

4 Two model railway trucks, A of mass 0.1 kg and B of mass 0.2 kg , are constrained to move on a smooth straight level track.
Initially B is stationary and A is moving towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide. The coefficient of restitution between A and B is \(e\).

1. Find the speed of A and the speed of B after the collision, giving your answers in terms of \(e\) and \(u\).
2. Show that the loss of kinetic energy in the collision is \(\frac { 1 } { 30 } u ^ { 2 } \left( 1 - e ^ { 2 } \right)\).
3. For the case in which the loss of kinetic energy is least

5 Point A lies 20 m vertically below a point B . A particle P of mass 4 m kg is projected upwards from A , at a speed of \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same time, a particle Q of mass \(m \mathrm {~kg}\) is released from rest at point B . The particles collide directly, and it is given that the coefficient of restitution in the collision between P and Q is 0.6 .

1. Show that, immediately after the collision, P continues to travel upwards at \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and determine, at this time, the corresponding velocity of Q . In another situation, a particle of mass \(3 m \mathrm {~kg}\) is released from rest and falls vertically. After it has fallen 10 m , it explodes into two fragments. Immediately after the explosion, the lower fragment, of mass \(2 m \mathrm {~kg}\), moves vertically downwards with speed \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the upper fragment, of mass \(m \mathrm {~kg}\), moves vertically upwards with speed \(v _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Given that, in the explosion, the kinetic energy of the system increases by \(72 \%\), show that \(2 v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } = 1011.36\).
3. By finding another equation connecting \(v _ { 1 }\) and \(v _ { 2 }\), determine the speeds of the fragments immediately after the explosion.

3 The diagram shows two blocks P and Q of masses 0.5 kg and 2 kg respectively, on a horizontal surface. The points \(\mathrm { A } , \mathrm { B }\) and C lie on the surface in a straight line. There is a wall at C . The surface between B and C is smooth, and the surface between A and B is rough, such that the coefficient of friction between P and AB is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_229_1271_1601_278} P is projected with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards Q , which is at rest. As a result of the collision between P and Q, P changes direction and subsequently comes to rest at A. You may assume that P only collides with Q once.

1. Determine the coefficient of restitution between P and Q .
2. Calculate the impulse exerted on P by Q during their collision. After colliding with P , Q strikes the wall, which is perpendicular to the direction of the motion of Q , and comes to rest exactly halfway between A and B . The collision between Q and the wall is perfectly elastic.
3. Determine the coefficient of friction between Q and AB .

4 Fig. 4.1 shows two spheres, A and B, on a smooth horizontal surface. Their masses are 3 kg and 1 kg respectively. \begin{figure}[h] \end{figure} Initially, sphere A travels at a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line towards B , which is at rest. The spheres collide and the coefficient of restitution between A and B is \(e\).

1. Show that, after the collision, A has a speed of \(\frac { 1 } { 4 } ( 3 - e ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and find an expression for the speed of B in terms of \(e\). During the collision, the kinetic energy of the system decreases by \(21 \%\).
2. Determine the value of \(e\).
3. State why in part (a) it was necessary to assume that A and B have equal radii. Fig. 4.2 shows two spheres, C and D , of equal radii on a smooth horizontal surface. Their masses are 1 kg and 2 kg respectively. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-4_158_1155_1544_244}
   \end{figure} Spheres C and D travel towards each other along the same straight line, C with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and D with a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres collide and during the collision C exerts an impulse on D of magnitude \(\frac { 2 } { 3 } ( u + 1 ) \mathrm { Ns }\).
4. Show that C and D have the same velocity after the collision.
5. Determine the fraction of kinetic energy lost due to the collision between C and D as \(u \rightarrow \infty\).

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 \(e _ { 1 } \leqslant \frac { 1 } { 3 }\).

4. Ryan is playing a game of snooker. The horizontal table is modelled as the horizontal \(x - y\) plane with the point \(O\) as the origin and unit vectors parallel to the \(x\)-axis and the \(y\)-axis denoted by \(\mathbf { i }\) and \(\mathbf { j }\) respectively. All balls on the table have a common mass \(m \mathrm {~kg}\). The table and the four sides, called cushions, are modelled as smooth surfaces. The dimensions of the table, in metres, are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-5_663_1138_667_482} Initially, all balls are stationary. Ryan strikes ball \(A\) so that it collides with ball \(B\). Before the collision, \(A\) has velocity \(( - \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and, after the collision, it has velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Show that the velocity of ball \(B\) after the collision is \(( - 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). After the collision with ball \(A\), ball \(B\) hits the cushion at point \(C\) before rebounding and moving towards the pocket at \(P\). The cushion is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the cushion and ball \(B\) is \(\frac { 5 } { 7 }\).
2. Calculate the velocity of ball \(B\) after impact with the cushion.
3. Find, in terms of \(m\), the magnitude of the impulse exerted on ball \(B\) by the cushion at \(C\), stating your units clearly.
4. Given that \(C\) has position vector \(( x \mathbf { i } + 1 \cdot 75 \mathbf { j } ) \mathrm { m }\),
   1. determine the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\),
   2. find the value of \(x\).
5. Describe one way in which the model used could be refined. Explain how your refinement would affect your answer to (d)(i).

4 Two discs, \(A\) and \(B\), have equal radii and masses 0.8 kg and 0.4 kg respectively. The discs are placed on a horizontal surface. The discs are set in motion when they are 3 metres apart, so that they move directly towards each other, each travelling at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The discs collide directly with each other. After the collision \(A\) moves in the opposite direction with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the two discs is \(e\). 4

1. Assuming that the surface is smooth, show that \(e = 0.8\) 4
2. Describe one way in which the model you have used could be refined. Turn over for the next question

1. A particle \(P\) of mass \(2 m\) and a particle \(Q\) of mass \(5 m\) are moving along the same straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The direction of motion of \(Q\) is reversed by the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\). Given that \(e = \frac { 1 } { 3 }\)
2. show that the kinetic energy lost in the collision is \(\frac { 40 m u ^ { 2 } } { 7 }\).
3. Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if \(e > \frac { 1 } { 3 }\)