Questions - AQA M2 - A-Level Maths

A woman of mass 50 kg stands on the part of the plank which projects over the river. Find the greatest distance from the river bank at which she can safely stand.
The woman wishes to stand safely at the end of the plank which projects over the river. Find the minimum mass which she should place on the other end of the plank so that she can do this.
State how you have used the fact that the plank is uniform in your solution.
State one other modelling assumption which you have made.

AQA M2 2007 June Q5

5 A bead of mass $m$ moves on a smooth circular ring of radius $a$ which is fixed in a vertical plane, as shown in the diagram. Its speed at $A$, the highest point of its path, is $v$ and its speed at $B$, the lowest point of its path, is $7 v$.
\includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-4_419_317_484_842}

Show that $v = \sqrt { \frac { a g } { 12 } }$.
Find the reaction of the ring on the bead, in terms of $m$ and $g$, when the bead is at $A$.

AQA M2 2007 June Q6

6 An elastic string has one end attached to a point $O$, fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point $O$ and it is then released from rest on the table.

Calculate the elastic potential energy when the particle is 2.5 m from the point $O$.
If the table is smooth, show that the speed of the particle when the string becomes slack is $\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.

AQA M2 2007 June Q7

7 A stone of mass $m$ is moving along the smooth horizontal floor of a tank which is filled with a viscous liquid. At time $t$, the stone has speed $v$. As the stone moves, it experiences a resistance force of magnitude $\lambda m v$, where $\lambda$ is a constant.

Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v$$
The initial speed of the stone is $U$. Show that $$v = U \mathrm { e } ^ { - \lambda t }$$

AQA M2 2007 June Q8

8 A particle, $P$, of mass 3 kg is attached to one end of a light inextensible string. The string passes through a smooth fixed ring, $O$, and a second particle, $Q$, of mass 5 kg is attached to the other end of the string. The particle $Q$ hangs at rest vertically below the ring and the particle $P$ moves with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a horizontal circle, as shown in the diagram. The angle between $O P$ and the vertical is $\theta$.
\includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-5_474_476_1425_774}

Explain why the tension in the string is 49 N .
Find $\theta$.
Find the radius of the horizontal circle.

AQA M2 2010 June Q1

1 A particle moves along a straight line through the origin. At time $t$, the displacement, $s$, of the particle from the origin is given by $$s = 5 t ^ { 2 } + 3 \cos 4 t$$ Find the velocity of the particle at time $t$.

\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-03_2484_1709_223_153}

AQA M2 2010 June Q2

2 John is at the top of a cliff, looking out over the sea. He throws a rock, of mass 3 kg , horizontally with a velocity of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The rock falls a vertical distance of 51 metres to reach the surface of the sea.

Calculate the kinetic energy of the rock when it is thrown.
Calculate the potential energy lost by the rock when it reaches the surface of the sea.
1. Find the kinetic energy of the rock when it reaches the surface of the sea.
2. Hence find the speed of the rock when it reaches the surface of the sea.
State one modelling assumption which has been made.
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-05_2484_1709_223_153}

AQA M2 2010 June Q3

3 A uniform circular lamina, of radius 4 cm and mass 0.4 kg , has a centre $O$, and $A B$ is a diameter. To create a medal, a smaller uniform circular lamina, of radius 2 cm and mass 0.1 kg , is attached so that the centre of the smaller lamina is at the point $A$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_671_878_513_598}

Explain why the centre of mass of the medal is on the line $A B$.
Find the distance of the centre of mass of the medal from the point $B$.
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-06_1259_1705_1448_155}
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-07_2484_1709_223_153}

AQA M2 2010 June Q4

4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, $\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)$ newtons, acts on the particle at time $t$ seconds. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively.

Find the acceleration of the particle at time $t$.
When $t = 4$, the velocity of the particle is $( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Find the velocity of the particle at time $t$.
Find $t$ when the particle is moving due west.
Find the speed of the particle when it is moving due west.

\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}

AQA M2 2010 June Q5

5 A particle is moving along a straight line. At time $t$, the velocity of the particle is $v$. The acceleration of the particle throughout the motion is $- \frac { \lambda } { v ^ { \frac { 1 } { 4 } } }$, where $\lambda$ is a positive constant. The velocity of the particle is $u$ when $t = 0$. Find $v$ in terms of $u , \lambda$ and $t$.
(7 marks)

\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-10_2078_1719_632_150}

\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-11_2484_1709_223_153}

AQA M2 2010 June Q6

6 When a car, of mass 1200 kg , travels at a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, it experiences a resistance force of magnitude $30 v$ newtons. The car has a maximum constant speed of $48 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a straight horizontal road.

Show that the maximum power of the car is 69120 watts.
The car is travelling along a straight horizontal road. Find the maximum possible acceleration of the car when it is travelling at a speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The car starts to descend a hill on a straight road which is inclined at an angle of $3 ^ { \circ }$ to the horizontal. Find the maximum possible constant speed of the car as it travels on this road down the hill. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-13_2484_1709_223_153}
$7 \quad$ A uniform rod $A B$, of length 4 m and mass 6 kg , rests in equilibrium with one end, $A$, on smooth horizontal ground. The rod rests on a rough horizontal peg at the point $C$, where $A C$ is 3 m . The rod is inclined at an angle of $20 ^ { \circ }$ to the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-14_422_984_447_529}

AQA M2 2010 June Q8

8 A particle is attached to one end of a light inextensible string of length 3 metres. The other end of the string is attached to a fixed point $O$. The particle is set into motion horizontally at point $P$ with speed $v$, so that it describes part of a vertical circle whose centre is $O$. The point $P$ is vertically below $O$.
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-16_510_334_493_861} The particle first comes momentarily to rest at the point $Q$, where $O Q$ makes an angle of $15 ^ { \circ }$ to the vertical.

Find the value of $v$.
When the particle is at rest at the point $Q$, the tension in the string is 22 newtons. Find the mass of the particle.

\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-17_2484_1709_223_153}

AQA M2 2010 June Q9

9 A particle, of mass 8 kg , is attached to one end of a length of elastic string. The particle is placed on a smooth horizontal surface. The other end of the elastic string is attached to a point $O$ fixed on the horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 192 N .
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-18_165_789_571_630} The particle is set in motion on the horizontal surface so that it moves in a circle, centre $O$, with constant speed $3 \mathrm {~ms} ^ { - 1 }$. Find the radius of the circle. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-19_2349_1691_221_153}
\includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-20_2505_1730_212_139}

AQA M2 2011 June Q1

1 In an Olympic diving competition, Kim, who has mass 58 kg , dives from a fixed platform, 10 metres above the surface of the pool. She leaves the platform with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Assume that Kim's weight is the only force that acts on her after she leaves the platform. Kim is to be modelled as a particle which is initially 1 metre above the platform.

Calculate Kim's initial kinetic energy.
By using conservation of energy, find Kim's speed when she is 6 metres below the platform.

AQA M2 2011 June Q2

2 The diagram shows four particles, $A , B , C$ and $D$, which are fixed in a horizontal plane which contains the $x$ - and $y$-axes, as shown. Particle $A$ has mass 2 kg and is attached at the point ( 9,6 ).
Particle $B$ has mass 3 kg and is attached at the point ( 2,4 ).
Particle $C$ has mass 8 kg and is attached at the point $( 3,8 )$.
Particle $D$ has mass 7 kg and is attached at the point $( 6,11 )$.
\includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-2_748_774_1402_625} Find the coordinates of the centre of mass of the four particles.

AQA M2 2011 June Q3

3 A particle moves in a horizontal plane under the action of a single force, $\mathbf { F }$ newtons. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively. At time $t$ seconds, the velocity of the particle, $\mathbf { v } \mathrm { ms } ^ { - 1 }$, is given by $$\mathbf { v } = 4 \mathrm { e } ^ { - 2 t } \mathbf { i } + \left( 6 t - 3 t ^ { 2 } \right) \mathbf { j }$$

Find an expression for the acceleration of the particle at time $t$.
The mass of the particle is 5 kg .
1. Find an expression for the force $\mathbf { F }$ acting on the particle at time $t$.
2. Find the magnitude of $\mathbf { F }$ when $t = 0$.
Find the value of $t$ when $\mathbf { F }$ acts due west.
When $t = 0$, the particle is at the point with position vector $( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }$. Find the position vector, $\mathbf { r }$ metres, of the particle at time $t$.

AQA M2 2011 June Q4

4 Ken is trying to cross a river of width 4 m . He has a uniform plank, $A B$, of length 8 m and mass 17 kg . The ground on both edges of the river bank is horizontal. The plank rests at two points, $C$ and $D$, on fixed supports which are on opposite sides of the river. The plank is at right angles to both river banks and is horizontal. The distance $A C$ is 1 m , and the point $C$ is at a horizontal distance of 0.6 m from the river bank. Ken, who has mass 65 kg , stands on the plank directly above the middle of the river, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-3_468_1086_1710_479}

Draw a diagram to show the forces acting on the plank.
Given that the reaction on the plank at the point $D$ is $44 g \mathrm {~N}$, find the horizontal distance of the point $D$ from the nearest river bank.
State how you have used the fact that the plank is uniform in your solution.

AQA M2 2011 June Q5

5 A train consists of an engine and five carriages. A constant resistance force of 3000 N acts on the engine, and a constant resistance force of 400 N acts on each of the five carriages. The maximum speed of the train on a horizontal track is $90 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.

Show that this speed is $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Hence find the maximum power output of the engine. Give your answer in kilowatts.
(3 marks)

AQA M2 2011 June Q6

6 A car, of mass $m \mathrm {~kg}$, is moving along a straight horizontal road. At time $t$ seconds, the car has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. As the car moves, it experiences a resistance force of magnitude $2 m v ^ { \frac { 5 } { 4 } }$ newtons. No other horizontal force acts on the car.

Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2 v ^ { \frac { 5 } { 4 } }$$ (1 mark)
The initial speed of the car is $16 \mathrm {~ms} ^ { - 1 }$. Show that $$v = \left( \frac { 2 } { t + 1 } \right) ^ { 4 }$$ (5 marks)

AQA M2 2011 June Q7

7 Two light inextensible strings each have one end attached to a particle, $P$, of mass 4 kg . 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 in a horizontal circle. The centre, $C$, of this circle is directly below the point $B$. The two strings are inclined at $30 ^ { \circ }$ and $50 ^ { \circ }$ to the vertical, as shown in the diagram. Both strings are taut. As the particle moves in the horizontal circle, the tension in the string $B P$ is 20 N .
\includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-5_750_469_742_781}

Find the tension in the string $A P$.
The speed of the particle is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the length of $C P$, the radius of the horizontal circle.

AQA M2 2011 June Q8

8 A smooth wire is fixed in a vertical plane so that it forms a circle of radius $a$ metres and centre $O$. A bead, $B$, of mass 0.3 kg , is threaded on the wire and is set in motion with a speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the lowest point of its circular path, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-6_364_378_466_845}

Show that, if the bead is going to make complete revolutions around the wire, $$u > 2 \sqrt { a g }$$
At time $t$ seconds, the angle between $O B$ and the horizontal is $\theta$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-6_330_328_1231_858} It is given that $u = \sqrt { \frac { 9 } { 2 } a g }$.
1. Find the reaction of the bead on the wire, giving your answer in terms of $g$ and $\theta$.
2. Find $\theta$ when this reaction is zero.

AQA M2 2011 June Q9

9 At a theme park, a light elastic rope is used to bring a carriage to rest at the end of a ride. The carriage has mass 200 kg and is travelling at $8 \mathrm {~ms} ^ { - 1 }$ when the elastic rope is attached to the carriage as it passes over a point $O$. The other end of the elastic rope is fixed to the point $O$. The carriage then moves along a horizontal surface until it is brought to rest. The elastic rope is then detached so that the carriage remains at rest. The elastic rope has natural length 6 m and modulus of elasticity 1800 N . The rope, once taut, remains horizontal throughout the motion.

Calculate the elastic potential energy of the rope when the carriage is 10 m from $O$.
(3 marks)
A student's simple model assumes that there are no resistance forces acting on the carriage so that it is brought to rest by the elastic rope alone. Find the distance of the carriage from $O$ when it is brought to rest.
The student improves the model by also including a constant resistance force of 800 N which acts while the carriage is in motion. Find the distance of the carriage from $O$ when it is brought to rest.
(8 marks)

AQA M2 2012 June Q1

1 Alan, of mass 76 kg , performed a ski jump. He took off at the point $A$ at the end of the ski run with a speed of $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and landed at the point $B$. The level of the point $B$ is 31 metres vertically below the level of the point $A$, as shown in the diagram. Assume that his weight is the only force that acted on Alan during the jump.
\includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-2_581_914_664_571}

Calculate the kinetic energy of Alan when he was at the point $A$.
Calculate the potential energy lost by Alan during the jump as he moved from the point $A$ to the point $B$.
1. Find the kinetic energy of Alan when he reached the point $B$.
2. Hence find the speed of Alan when he reached the point $B$.

AQA M2 2012 June Q2

2 A particle moves in a straight line. At time $t$ seconds, it has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $$v = 6 t ^ { 2 } - 2 \mathrm { e } ^ { - 4 t } + 8$$ and $t \geqslant 0$.

1. Find an expression for the acceleration of the particle at time $t$.
2. Find the acceleration of the particle when $t = 0.5$.
The particle has mass 4 kg . Find the magnitude of the force acting on the particle when $t = 0.5$.
When $t = 0$, the particle is at the origin. Find an expression for the displacement of the particle from the origin at time $t$.

AQA M2 2012 June Q3

3 A uniform rectangular lamina $A B C D$, of mass 1.6 kg , has side $A B$ of length 12 cm and side $B C$ of length 8 cm . To create a logo, a uniform circular lamina, of mass 0.4 kg , is attached. The centre of the circular lamina is at the point $C$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-3_520_780_593_630}

Find the distance of the centre of mass of the logo:
1. from the line $A B$;
2. from the line $A D$.
The logo is suspended in equilibrium, with $A B$ horizontal, by two vertical strings. One string is attached at the point $A$ and the other string is attached at the point $B$. Find the tension in each of the two strings.

Questions — AQA M2 (163 questions)

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