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AQA M2 2013 June Q7
6 marks Moderate -0.3
7 A train, of mass 22 tonnes, moves along a straight horizontal track. A constant resistance force of 5000 N acts on the train. The power output of the engine of the train is 240 kW . Find the acceleration of the train when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2013 June Q8
9 marks Standard +0.3
8 A bead, of mass \(m\), moves on a smooth circular ring, of radius \(a\) and centre \(O\), which is fixed in a vertical plane. At \(P\), the highest point on the ring, the speed of the bead is \(2 u\); at \(Q\), the lowest point on the ring, the speed of the bead is \(5 u\).
  1. Show that \(u = \sqrt { \frac { 4 a g } { 21 } }\).
    (4 marks)
  2. \(\quad S\) is a point on the ring so that angle \(P O S\) is \(60 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-4_600_540_657_760} Find, in terms of \(m\) and \(g\), the magnitude of the reaction of the ring on the bead when the bead is at \(S\).
AQA M2 2013 June Q9
14 marks Challenging +1.2
9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.
    1. Show that initially particle \(A\) moves towards the hole in the table.
    2. Show that initially particle \(B\) also moves towards the hole in the table.
  2. Calculate the initial elastic potential energy in the string.
  3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.
AQA M2 2015 June Q1
10 marks Standard +0.3
1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$
    1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
    2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
  2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}
AQA M2 2015 June Q2
4 marks Moderate -0.8
2 A uniform rod \(A B\), of mass 4 kg and length 6 metres, has three masses attached to it. A 3 kg mass is attached at the end \(A\) and a 5 kg mass is attached at the end \(B\). An 8 kg mass is attached at a point \(C\) on the rod. Find the distance \(A C\) if the centre of mass of the system is 4.3 m from point \(A\).
[0pt] [4 marks]
AQA M2 2015 June Q3
9 marks Standard +0.3
3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.
  1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
    1. Find the kinetic energy of Simon when he reaches the point \(R\).
    2. Hence find the speed of Simon when he reaches the point \(R\).
  2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
    Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}
AQA M2 2015 June Q4
10 marks Standard +0.3
4 A particle, \(P\), of mass 5 kg is attached to two light inextensible strings, \(A P\) and \(B P\). The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in a horizontal circle of radius 0.6 metres with centre \(B\). The string \(A P\) is inclined at \(20 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut when the particle is moving. \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}
  1. Find the tension in the string \(A P\).
  2. The speed of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the tension, \(T _ { B P }\), in the string \(B P\) is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
  3. Find \(v\) when the tensions in the two strings are equal.
AQA M2 2015 June Q5
6 marks Standard +0.3
5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]
\includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-10_1883_1709_824_153}
AQA M2 2015 June Q6
9 marks Standard +0.3
6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]
AQA M2 2015 June Q7
9 marks Standard +0.3
7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.
  1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
  2. Find \(v\) in terms of \(t\).
  3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
    [0pt] [2 marks]
AQA M2 2015 June Q8
10 marks Standard +0.3
8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
  2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
  3. Find the maximum value of \(x\).
    1. Find the distance fallen by Carol when her speed is a maximum.
    2. Hence find Carol's maximum speed.
AQA M2 2015 June Q9
8 marks Challenging +1.8
9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks] \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}
AQA M3 Q5
Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-04_330_1411_1902_303}
  3. State two modelling assumptions that you have made.
AQA M3 Q6
Standard +0.8
6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-05_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).
  1. Find the speed of \(B\) immediately after the collision.
  2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.
AQA M3 Q7
Challenging +1.8
7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.
    1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
    2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
  2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-06_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.
AQA M3 2006 June Q1
7 marks Moderate -0.5
1 The time \(T\) taken for a simple pendulum to make a single small oscillation is thought to depend only on its length \(l\), its mass \(m\) and the acceleration due to gravity \(g\). By using dimensional analysis:
  1. show that \(T\) does not depend on \(m\);
  2. express \(T\) in terms of \(l , g\) and \(k\), where \(k\) is a dimensionless constant.
AQA M3 2006 June Q2
12 marks Standard +0.3
2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).
  1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
  2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).
AQA M3 2006 June Q3
9 marks Standard +0.3
3 A ball of mass 0.45 kg is travelling horizontally with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time \(t\) seconds after first coming into contact with the bat, the force exerted on the ball by the bat is \(1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)\) newtons, where \(0 \leqslant t \leqslant 0.1\). In this simple model, ignore the weight of the ball and model the ball as a particle.
  1. Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
  2. Find, to two significant figures, the speed of the ball immediately after the impact.
  3. Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.
AQA M3 2006 June Q4
10 marks Moderate -0.3
4 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
Two cyclists, Aazar and Ben, are cycling on straight horizontal roads with constant velocities of \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and \(( 12 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) respectively. Initially, Aazar and Ben have position vectors \(( 5 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) and \(( 18 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) respectively, relative to a fixed origin.
  1. Find, as a vector in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of Ben relative to Aazar.
  2. The position vector of Ben relative to Aazar at time \(t\) hours after they start is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 13 + 6 t ) \mathbf { i } + ( 6 - 20 t ) \mathbf { j }$$
  3. Find the value of \(t\) when Aazar and Ben are closest together.
  4. Find the closest distance between Aazar and Ben.
AQA M3 2006 June Q5
13 marks Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-3_330_1411_1902_303}
  3. State two modelling assumptions that you have made.
AQA M3 2006 June Q6
11 marks Standard +0.3
6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).
  1. Find the speed of \(B\) immediately after the collision.
  2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.
AQA M3 2006 June Q7
13 marks Challenging +1.2
7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.
    1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
    2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
  2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-5_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.
AQA M3 2007 June Q1
8 marks Moderate -0.5
1 The magnitude of the gravitational force, \(F\), between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(x\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { x ^ { 2 } }$$ where \(G\) is a constant.
  1. By using dimensional analysis, find the dimensions of \(G\).
  2. The lifetime, \(t\), of a planet is thought to depend on its mass, \(m\), its initial radius, \(R\), the constant \(G\) and a dimensionless constant, \(k\), so that $$t = k m ^ { \alpha } R ^ { \beta } G ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
    Find the values of \(\alpha , \beta\) and \(\gamma\).
AQA M3 2007 June Q2
10 marks Standard +0.3
2 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. Two helicopters, \(A\) and \(B\), are flying with constant velocities of \(( 20 \mathbf { i } - 10 \mathbf { j } + 20 \mathbf { k } ) \mathrm { ms } ^ { - 1 }\) and \(( 30 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. At noon, the position vectors of \(A\) and \(B\) relative to a fixed origin, \(O\), are \(( 8000 \mathbf { i } + 1500 \mathbf { j } + 3000 \mathbf { k } ) \mathrm { m }\) and \(( 2000 \mathbf { i } + 500 \mathbf { j } + 1000 \mathbf { k } ) \mathrm { m }\) respectively.
  1. Write down the velocity of \(A\) relative to \(B\).
  2. Find the position vector of \(A\) relative to \(B\) at time \(t\) seconds after noon.
  3. Find the value of \(t\) when \(A\) and \(B\) are closest together.
AQA M3 2007 June Q3
9 marks Moderate -0.3
3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).
  1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
  2. Find the speed of \(P\) when \(t = 3\).
  3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).