Questions — AQA M2 (163 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA M2 2006 January Q1
1 A stone, of mass 0.4 kg , is thrown vertically upwards with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 6 metres above ground level.
  1. Calculate the initial kinetic energy of the stone.
    1. Show that the kinetic energy of the stone when it hits the ground is 36.3 J , correct to three significant figures.
    2. Hence find the speed at which the stone hits the ground.
    3. State one assumption that you have made.
AQA M2 2006 January Q2
2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-2_344_340_1418_842}
  1. Show that the tension in the string is 22.6 N , correct to three significant figures.
  2. Find the speed of the particle.
AQA M2 2006 January Q3
3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. State the range of values of the acceleration of the particle.
  2. 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 2006 January Q4
4 The diagram shows a uniform lamina \(A B C D E F G H\).
\includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-3_346_933_1123_577}
  1. Explain why the centre of mass is 25 cm from \(A H\).
  2. Show that the centre of mass is 4.375 cm from \(H G\).
  3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
  4. Explain, briefly, how you have used the fact that the lamina is uniform.
AQA M2 2006 January Q5
5 A particle moves such that at time \(t\) seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$
  1. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 0\).
  2. When \(t = 0\), the velocity of the particle is \(( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find an expression for the velocity of the particle at time \(t\).
AQA M2 2006 January Q6
6 A student is modelling the motion of a small boat as it moves on a lake. When the speed of the boat is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. At time \(t\) seconds later, it has a velocity of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and experiences a resistance force of magnitude \(20 v\) newtons. The mass of the boat is 80 kg . To set up a simple model for the motion of the boat, the student assumes that the water in the lake is still and that the boat travels in a straight line.
  1. Explain how these two assumptions allow the student to create a simple model.
  2. State one other assumption that the student should make.
    1. Express \(\frac { \mathrm { d } v } { \mathrm {~d} t }\) in terms of \(v\).
    2. Find an expression for \(v\) in terms of \(t\).
AQA M2 2006 January Q7
7 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-4_415_1007_1573_513}
  1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
  2. Find the value of \(\theta\) when the particle leaves the hemisphere.
AQA M2 2006 January Q8
8 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).
  1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
  2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
  3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
    1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
    2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.
AQA M2 2008 January Q1
1 A ball is thrown vertically upwards from ground level with an initial speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has a mass of 0.6 kg . Assume that the only force acting on the ball after it is thrown is its weight.
  1. Calculate the initial kinetic energy of the ball.
  2. By using conservation of energy, find the maximum height above ground level reached by the ball.
  3. By using conservation of energy, find the kinetic energy and the speed of the ball when it is at a height of 3 m above ground level.
  4. State one modelling assumption which has been made.
AQA M2 2008 January Q2
2 A particle moves in a straight line and at time \(t\) it has velocity \(v\), where $$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. When \(t = \frac { \pi } { 3 }\), show that the acceleration of the particle is \(2 \pi + 6\).
  2. 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 2008 January Q3
3 A uniform ladder of length 4 metres and mass 20 kg rests in equilibrium with its foot, \(A\), on a rough horizontal floor and its top leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the angle between the ladder and the floor is \(60 ^ { \circ }\). A man of mass 80 kg is standing at point \(C\) on the ladder. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the floor is 0.4 . The man may be modelled as a particle at \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-3_567_448_708_788}
  1. Draw a diagram to show the forces acting on the ladder.
  2. Show that the magnitude of the frictional force between the ladder and the ground is 392 N .
  3. Find the distance \(A C\).
AQA M2 2008 January Q4
4 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 position vector, \(\mathbf { r }\) metres, of the particle is given by $$\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } + 4 \right) \mathbf { i } + \left( 4 t + t ^ { 2 } \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. The mass of the particle is 3 kg .
    1. Find an expression for \(\mathbf { F }\) at time \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
  3. Find the value of \(t\) when \(\mathbf { F }\) acts due north.
AQA M2 2008 January Q5
5 Two light inextensible strings, of lengths 0.4 m and 0.2 m , each have one end attached to a particle, \(P\), of mass 4 kg . The other ends of the strings are attached to the points \(A\) and \(B\) respectively. The point \(A\) is vertically above the point \(B\). The particle moves in a horizontal circle, centre \(B\) and radius 0.2 m , at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle and strings are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_396_558_587_735} $$\text { ← } 0.2 \mathrm {~m} \longrightarrow$$
  1. Calculate the magnitude of the acceleration of the particle.
  2. Show that the tension in string \(P A\) is 45.3 N , correct to three significant figures.
  3. Find the tension in string \(P B\).
AQA M2 2008 January Q6
6 A light elastic string has one end attached to a point \(A\) fixed on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle of mass 6 kg . The elastic string has natural length 4 metres and modulus of elasticity 300 newtons. The particle is pulled down the plane in the direction of the line of greatest slope through \(A\). The particle is released from rest when it is 5.5 metres from \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_314_713_1900_660}
  1. Calculate the elastic potential energy of the string when the particle is 5.5 metres from the point \(A\).
  2. Show that the speed of the particle when the string becomes slack is \(3.66 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  3. Show that the particle will not reach point \(A\) in the subsequent motion.
AQA M2 2008 January Q7
7 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end. The particle is moving in a vertical circle, centre \(O\). When the particle is at \(B\), vertically above \(O\), the string is taut and the particle is moving with speed \(3 \sqrt { a g }\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-5_422_399_497_778}
  1. Find, in terms of \(g\) and \(a\), the speed of the particle at the lowest point, \(A\), of its path.
  2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(A\).
AQA M2 2008 January Q8
8 A car of mass 600 kg is driven along a straight horizontal road. The resistance to motion of the car is \(k v ^ { 2 }\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t\) seconds and \(k\) is a constant.
  1. When the engine of the car has power 8 kW , show that the equation of motion of the car is $$600 \frac { \mathrm {~d} v } { \mathrm {~d} t } - \frac { 8000 } { v } + k v ^ { 2 } = 0$$
  2. When the velocity of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is turned off.
    1. Show that the equation of motion of the car now becomes $$600 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - k v ^ { 2 }$$
    2. Find, in terms of \(k\), the time taken for the velocity of the car to drop to \(10 \mathrm {~ms} ^ { - 1 }\).
AQA M2 2011 January Q1
1 The velocity of a particle at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 4 + 3 t ^ { 2 } \right) \mathbf { i } + ( 12 - 8 t ) \mathbf { j }$$
  1. When \(t = 0\), the particle is at the point with position vector \(( 5 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
  2. Find the acceleration of the particle at time \(t\).
  3. The particle has mass 2 kg . Find the magnitude of the force acting on the particle when \(t = 1\).
AQA M2 2011 January Q2
2 A particle is placed on a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The particle, of mass 4 kg , is released from rest at a point \(A\) and travels down the plane, passing through a point \(B\). The distance \(A B\) is 5 m .
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-04_371_693_500_680}
  1. Find the potential energy lost as the particle moves from point \(A\) to point \(B\).
  2. Hence write down the kinetic energy of the particle when it reaches point \(B\).
  3. Hence find the speed of the particle when it reaches point \(B\).
AQA M2 2011 January Q3
3 A pump is being used to empty a flooded basement.
In one minute, 400 litres of water are pumped out of the basement.
The water is raised 8 metres and is ejected through a pipe at a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
The mass of 400 litres of water is 400 kg .
  1. Calculate the gain in potential energy of the 400 litres of water.
  2. Calculate the gain in kinetic energy of the 400 litres of water.
  3. Hence calculate the power of the pump, giving your answer in watts.
AQA M2 2011 January Q4
4 A uniform rectangular lamina \(A B C D\) has a mass of 5 kg . The side \(A B\) has length 60 cm and the side \(B C\) has length 30 cm . The points \(P , Q , R\) and \(S\) are the mid-points of the sides, as shown in the diagram below. A uniform triangular lamina \(S R D\), of mass 4 kg , is fixed to the rectangular lamina to form a shop sign. The centre of mass of the triangular lamina \(S R D\) is 10 cm from the side \(A D\) and 5 cm from the side \(D C\).
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-08_613_1086_660_518}
  1. Find the distance of the centre of mass of the shop sign from \(A D\).
  2. Find the distance of the centre of mass of the shop sign from \(A B\).
  3. The shop sign is freely suspended from \(P\). Find the angle between \(A B\) and the horizontal when the shop sign is in equilibrium.
  4. To ensure that the side \(A B\) is horizontal when the shop sign is freely suspended from point \(P\), a particle of mass \(m \mathrm {~kg}\) is attached to the shop sign at point \(B\). Calculate \(m\).
  5. Explain how you have used the fact that the rectangular lamina \(A B C D\) is uniform in your solution to this question.
    (1 mark)
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-10_2486_1714_221_153}
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-11_2486_1714_221_153}
AQA M2 2011 January Q5
5
  1. A shiny coin is on a rough horizontal turntable at a distance 0.8 m from its centre. The turntable rotates at a constant angular speed. The coefficient of friction between the shiny coin and the turntable is 0.3 . Find the maximum angular speed, in radians per second, at which the turntable can rotate if the shiny coin is not going to slide.
  2. The turntable is stopped and the shiny coin is removed. An old coin is placed on the turntable at a distance 0.15 m from its centre. The turntable is made to rotate at a constant angular speed of 45 revolutions per minute.
    1. Find the angular speed of the turntable in radians per second.
    2. The old coin remains in the same position on the turntable. Find the least value of the coefficient of friction between the old coin and the turntable needed to prevent the old coin from sliding.
AQA M2 2011 January Q6
6 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A small bead, of mass \(m\), is attached to the other end of the string. The bead is moving in a vertical circle, centre \(O\). When the bead is at \(B\), vertically below \(O\), the string is taut and the bead is moving with speed \(5 v\).
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_536_554_502_774}
  1. The speed of the bead at the highest point of its path is \(3 v\). Find \(v\) in terms of \(a\) and \(g\).
  2. Find the ratio of the greatest tension to the least tension in the string, as the bead travels around its circular path.
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_1261_1709_1446_153}
AQA M2 2011 January Q7
7
  1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
  2. A block, of mass 4 kg , is attached to one end of a light elastic string. The string has natural length 2 m and modulus of elasticity 196 N . The other end of the string is attached to a fixed point \(O\).
    1. A second block, of mass 3 kg , is attached to the 4 kg block and the system hangs in equilibrium, as shown in the diagram.
      \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-16_374_291_890_877} Find the extension in the string.
    2. The block of mass 3 kg becomes detached from the 4 kg block and falls to the ground. The 4 kg block now begins to move vertically upwards. Find the extension of the string when the 4 kg block is next at rest.
    3. Find the extension of the string when the speed of the 4 kg block is a maximum.
      (3 marks)
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-18_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-19_2486_1714_221_153}
AQA M2 2011 January Q8
8 Vicky has mass 65 kg and is skydiving. She steps out of a helicopter and falls vertically. She then waits a short period of time before opening her parachute. The parachute opens at time \(t = 0\) when her speed is \(19.6 \mathrm {~ms} ^ { - 1 }\), and she then experiences an air resistance force of magnitude \(260 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is her speed at time \(t\) seconds.
  1. When \(t > 0\) :
    1. show that the resultant downward force acting on Vicky is 65(9.8-4v) newtons
    2. show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 4 ( v - 2.45 )\).
  2. By showing that \(\int \frac { 1 } { v - 2.45 } \mathrm {~d} v = - \int 4 \mathrm {~d} t\), find \(v\) in terms of \(t\).
    \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-21_2349_1707_221_153}
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AQA M2 2012 January Q1
1 A plane is dropping packets of aid as it flies over a flooded village. The speed of a packet when it leaves the plane is \(60 \mathrm {~ms} ^ { - 1 }\). The packet has mass 25 kg . The packet falls a vertical distance of 34 metres to reach the ground.
  1. Calculate the kinetic energy of the packet when it leaves the plane.
  2. Calculate the potential energy lost by the packet as it falls to the ground.
  3. Assume that the effect of air resistance on the packet as it falls can be neglected.
    1. Find the kinetic energy of the packet when it reaches the ground.
    2. Hence find the speed of the packet when it reaches the ground.