Questions M3 (796 questions)

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AQA M3 2011 June Q7
13 marks Standard +0.8
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]
\captionsetup{labelformat=empty} \caption{Before collision} \includegraphics[alt={},max width=\textwidth]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_362_933_664_450}
\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}
AQA M3 2012 June Q1
7 marks Standard +0.3
1 An ice-hockey player has mass 60 kg . He slides in a straight line at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on the horizontal smooth surface of an ice rink towards the vertical perimeter wall of the rink, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-02_476_594_769_715} The player collides directly with the wall, and remains in contact with the wall for 0.5 seconds. At time \(t\) seconds after coming into contact with the wall, the force exerted by the wall on the player is \(4 \times 10 ^ { 4 } t ^ { 2 } ( 1 - 2 t )\) newtons, where \(0 \leqslant t \leqslant 0.5\).
  1. Find the magnitude of the impulse exerted by the wall on the player.
  2. The player rebounds from the wall. Find the player's speed immediately after the collision.
AQA M3 2012 June Q2
4 marks Moderate -0.5
2 A pile driver of mass \(m _ { 1 }\) falls from a height \(h\) onto a pile of mass \(m _ { 2 }\), driving the pile a distance \(s\) into the ground. The pile driver remains in contact with the pile after the impact. A resistance force \(R\) opposes the motion of the pile into the ground. Elizabeth finds an expression for \(R\) as $$R = \frac { g } { s } \left[ s \left( m _ { 1 } + m _ { 2 } \right) + \frac { h \left( m _ { 1 } \right) ^ { 2 } } { m _ { 1 } + m _ { 2 } } \right]$$ where \(g\) is the acceleration due to gravity.
Determine whether the expression is dimensionally consistent.
AQA M3 2012 June Q3
10 marks Standard +0.8
3 (In this question, take \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).)
A projectile is fired from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal so as to pass through a point \(P\). The projectile travels in a vertical plane through \(O\) and \(P\). The point \(P\) is at a horizontal distance \(2 k\) from \(O\) and at a vertical distance \(k\) above \(O\).
  1. Show that \(\alpha\) satisfies the equation $$20 k \tan ^ { 2 } \alpha - 2 u ^ { 2 } \tan \alpha + u ^ { 2 } + 20 k = 0$$
  2. Deduce that $$u ^ { 4 } - 20 k u ^ { 2 } - 400 k ^ { 2 } \geqslant 0$$
AQA M3 2012 June Q4
13 marks Standard +0.3
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.
AQA M3 2012 June Q5
12 marks Standard +0.8
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)
AQA M3 2012 June Q6
14 marks Standard +0.3
6 At noon, two ships, \(A\) and \(B\), are a distance of 12 km apart, with \(B\) on a bearing of \(065 ^ { \circ }\) from \(A\). The ship \(B\) travels due north at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The ship \(A\) travels at a constant speed of \(18 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-16_492_585_445_738}
  1. Find the direction in which \(A\) should travel in order to intercept \(B\). Give your answer as a bearing.
  2. In fact, the ship \(A\) actually travels on a bearing of \(065 ^ { \circ }\).
    1. Find the distance between the ships when they are closest together.
    2. Find the time when the ships are closest together.
AQA M3 2012 June Q7
15 marks Challenging +1.2
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.
AQA M3 2013 June Q1
6 marks Standard +0.3
1 A stone, of mass 2 kg , is moving in a straight line on a smooth horizontal sheet of ice under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude \(( 3 t + 1 )\) newtons, \(0 \leqslant t \leqslant 3\). When \(t = 0\), the stone has velocity \(1 \mathrm {~ms} ^ { - 1 }\).
When \(t = T\), the stone has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(T\).
(6 marks)
AQA M3 2013 June Q2
6 marks Moderate -0.5
2 A car has mass \(m\) and travels up a slope which is inclined at an angle \(\theta\) to the horizontal. The car reaches a maximum speed \(v\) at a height \(h\) above its initial position. A constant resistance force \(R\) opposes the motion of the car, which has a maximum engine power output \(P\). Neda finds a formula for \(P\) as $$P = m g v \sin \theta + R v + \frac { 1 } { 2 } m v ^ { 3 } \frac { \sin \theta } { h }$$ where \(g\) is the acceleration due to gravity.
Given that the engine power output may be measured in newton metres per second, determine whether the formula is dimensionally consistent.
AQA M3 2013 June Q3
16 marks Standard +0.3
3 A player projects a basketball with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The basketball travels in a vertical plane through the point of projection and goes into the basket. During the motion, the horizontal and upward vertical displacements of the basketball from the point of projection are \(x\) metres and \(y\) metres respectively. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-06_737_937_513_550}
  1. Find an expression for \(y\) in terms of \(x , u , g\) and \(\tan \theta\).
  2. The player projects the basketball with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 0.5 metres vertically below and 5 metres horizontally from the basket.
    1. Show that the two possible values of \(\theta\) are approximately \(63.1 ^ { \circ }\) and \(32.6 ^ { \circ }\), correct to three significant figures.
    2. Given that the player projects the basketball at \(63.1 ^ { \circ }\) to the horizontal, find the direction of the motion of the basketball as it enters the basket. Give your answer to the nearest degree.
  3. State a modelling assumption needed for answering parts (a) and (b) of this question.
    (1 mark)
AQA M3 2013 June Q4
11 marks Standard +0.3
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.
AQA M3 2013 June Q5
10 marks Standard +0.8
5 A particle is projected from a point \(O\) on a plane which is inclined at an angle \(\theta\) to the horizontal. The particle is projected down the plane with velocity \(u\) at an angle \(\alpha\) above the plane. The particle first strikes the plane at a point \(P\), as shown in the diagram. 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]{3a1726d9-1b0c-41de-8b43-56019e18aac1-12_389_789_557_639}
  1. Given that the time of flight from \(O\) to \(P\) is \(T\), find an expression for \(u\) in terms of \(\theta , \alpha , T\) and \(g\).
  2. Using the identity \(\cos ( X - Y ) = \cos X \cos Y + \sin X \sin Y\), show that the distance \(O P\) is given by \(\frac { 2 u ^ { 2 } \sin \alpha \cos ( \alpha - \theta ) } { g \cos ^ { 2 } \theta }\).
    (6 marks)
AQA M3 2013 June Q6
12 marks Standard +0.3
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.
AQA M3 2013 June Q7
14 marks Challenging +1.2
7 From an aircraft \(A\), a helicopter \(H\) is observed 20 km away on a bearing of \(120 ^ { \circ }\). The helicopter \(H\) is travelling horizontally with a constant speed \(240 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(340 ^ { \circ }\). The aircraft \(A\) is travelling with constant speed \(v _ { A } \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a straight line and at the same altitude as \(H\). \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-18_774_801_504_607}
  1. Given that \(v _ { A } = 200\) :
    1. find a bearing, to one decimal place, on which \(A\) could travel in order to intercept \(H\);
    2. find the time, in minutes, that it would take \(A\) to intercept \(H\) on this bearing.
  2. Given that \(v _ { A } = 150\), find the bearing on which \(A\) should travel in order to approach \(H\) as closely as possible. Give your answer to one decimal place.
    \includegraphics[max width=\textwidth, alt={}]{3a1726d9-1b0c-41de-8b43-56019e18aac1-20_2253_1691_221_153}
AQA M3 2014 June Q2
6 marks Standard +0.3
2 A rod, of length \(x \mathrm {~m}\) and moment of inertia \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end. When the rod is hanging at rest, its lower end receives an impulse of magnitude \(J\) Ns, which is just sufficient for the rod to complete full revolutions. It is thought that there is a relationship between \(J , x , I\), the acceleration due to gravity \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and a dimensionless constant \(k\), such that $$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
Find the values of \(\alpha , \beta\) and \(\gamma\) for which this relationship is dimensionally consistent.
[0pt] [6 marks]
AQA M3 2014 June Q3
9 marks Moderate -0.3
3 A particle of mass 0.5 kg is moving in a straight line on a smooth horizontal surface.
The particle is then acted on by a horizontal force for 3 seconds. This force acts in the direction of motion of the particle and at time \(t\) seconds has magnitude \(( 3 t + 1 ) \mathrm { N }\). When \(t = 0\), the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse of the force on the particle between the times \(t = 0\) and \(t = 3\).
  2. Hence find the velocity of the particle when \(t = 3\).
  3. Find the value of \(t\) when the velocity of the particle is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M3 2014 June Q4
14 marks Standard +0.3
4 Two boats, \(A\) and \(B\), are moving on straight courses with constant speeds. At noon, \(A\) and \(B\) have position vectors \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }\) and \(( - \mathbf { i } + \mathbf { j } ) \mathrm { km }\) respectively relative to a lighthouse. Thirty minutes later, the position vectors of \(A\) and \(B\) are ( \(- \mathbf { i } + 3 \mathbf { j }\) ) km and \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) respectively relative to the lighthouse.
  1. Find the velocity of \(A\) relative to \(B\) in the form \(( m \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\), where \(m\) and \(n\) are integers.
  2. The position vector of \(A\) relative to \(B\) at time \(t\) hours after noon is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 2 - 10 t ) \mathbf { i } + ( 1 + 6 t ) \mathbf { j }$$
  3. Determine the value of \(t\) when \(A\) and \(B\) are closest together.
  4. Find the shortest distance between \(A\) and \(B\).
AQA M3 2014 June Q5
12 marks Standard +0.3
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 $$4 h e ( e + 1 ) \sin \theta$$
AQA M3 2014 June Q6
12 marks Challenging +1.2
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.
AQA M3 2014 June Q7
15 marks Standard +0.3
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}
AQA M3 2015 June Q1
6 marks Standard +0.3
1 A formula for calculating the lift force acting on the wings of an aircraft moving through the air is of the form $$F = k v ^ { \alpha } A ^ { \beta } \rho ^ { \gamma }$$ where \(F\) is the lift force in newtons, \(k\) is a dimensionless constant, \(v\) is the air velocity in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), \(A\) is the surface area of the aircraft's wings in \(\mathrm { m } ^ { 2 }\), and \(\rho\) is the density of the air in \(\mathrm { kg } \mathrm { m } ^ { - 3 }\).
By using dimensional analysis, find the values of the constants \(\alpha , \beta\) and \(\gamma\).
[0pt] [6 marks]
AQA M3 2015 June Q2
5 marks Standard +0.3
2 A projectile is launched from a point \(O\) on top of a cliff with initial velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\) and moves in a vertical plane. During the motion, the position vector of the projectile relative to the point \(O\) is \(( x \mathbf { i } + y \mathbf { j } )\) metres where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively.
  1. Show that, during the motion, the equation of the trajectory of the projectile is given by $$y = x \tan \alpha - \frac { 4.9 x ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$$
  2. When \(u = 21\) and \(\alpha = 55 ^ { \circ }\), the projectile hits a small buoy \(B\). The buoy is at a distance \(s\) metres vertically below \(O\) and at a distance \(s\) metres horizontally from \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-04_601_935_964_548}
    1. Find the value of \(s\).
    2. Find the acute angle between the velocity of the projectile and the horizontal just before the projectile hits \(B\), giving your answer to the nearest degree.
      [0pt] [5 marks]
AQA M3 2015 June Q3
4 marks Moderate -0.3
3 A disc of mass 0.5 kg is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface when it receives a horizontal impulse in a direction perpendicular to its direction of motion. Immediately after the impulse, the disc has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse received by the disc.
  2. Before the impulse, the disc is moving parallel to a smooth vertical wall, as shown in the diagram. \section*{11/1/1/1/1/1/1/1/1/1/1/1/ Wall} $$\overbrace { 3 \mathrm {~ms} ^ { - 1 } } ^ { \underset { < } { \bigcirc } } \text { Disc }$$ After the impulse, the disc hits the wall and rebounds with speed \(3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Find the coefficient of restitution between the disc and the wall.
    [0pt] [4 marks]
AQA M3 2015 June Q4
2 marks Standard +0.3
4 Three uniform smooth spheres, \(A , B\) and \(C\), have equal radii and masses \(m , 2 m\) and \(6 m\) 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 \(u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).
    1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 } { 9 } u\).
    2. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
  2. Subsequently, \(B\) collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). Show that \(B\) will collide with \(A\) again if \(e > k\), where \(k\) is a constant to be determined.
  3. Explain why it is not necessary to model the spheres as particles in this question.
    [0pt] [2 marks]