Questions — AQA M2 (163 questions)

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AQA M2 2012 June Q4
4 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular. At time \(t\), the particle's position vector, \(\mathbf { r }\), is given by $$\mathbf { r } = 4 \cos 3 t \mathbf { i } - 4 \sin 3 t \mathbf { j }$$
  1. Prove that the particle is moving on a circle, which has its centre at the origin.
  2. Find an expression for the velocity of the particle at time \(t\).
  3. Find an expression for the acceleration of the particle at time \(t\).
  4. The acceleration of the particle can be written as $$\mathbf { a } = k \mathbf { r }$$ where \(k\) is a constant. Find the value of \(k\).
  5. State the direction of the acceleration of the particle.
AQA M2 2012 June Q5
5 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes through a hole in a smooth horizontal table. The edges of the hole are also smooth. Particle \(A\), of mass 1.4 kg , moves, on the table, with constant speed in a circle of radius 0.3 m around the hole. Particle \(B\), of mass 2.1 kg , hangs in equilibrium under the table, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-4_684_1022_1176_504}
  1. Find the angular speed of particle \(A\).
  2. Find the speed of particle \(A\).
  3. Find the time taken for particle \(A\) to complete one full circle around the hole.
AQA M2 2012 June Q6
6 Simon, a small child of mass 22 kg , is on a swing. He is swinging freely through an angle of \(18 ^ { \circ }\) on both sides of the vertical. Model Simon as a particle, \(P\), of mass 22 kg , attached to a fixed point, \(Q\), by a light inextensible rope of length 2.4 m .
\includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-5_700_310_466_849}
  1. Find Simon's maximum speed as he swings.
  2. Calculate the tension in the rope when Simon's speed is a maximum.
AQA M2 2012 June Q7
7 A stone, of mass 5 kg , is projected vertically downwards, in a viscous liquid, with an initial speed of \(7 \mathrm {~ms} ^ { - 1 }\). At time \(t\) seconds after it is projected, the stone has speed \(v \mathrm {~ms} ^ { - 1 }\) and it experiences a resistance force of magnitude \(9.8 v\) newtons.
  1. When \(t \geqslant 0\), show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.96 ( v - 5 )$$ (2 marks)
  2. Find \(v\) in terms of \(t\).
AQA M2 2012 June Q8
8 Zoë carries out an experiment with a block, which she places on the horizontal surface of an ice rink. She attaches one end of a light elastic string to a fixed point, \(A\), on a vertical wall at the edge of the ice rink at the height of the surface of the ice rink. The block, of mass 0.4 kg , is attached to the other end of the string. The string has natural length 5 m and modulus of elasticity 120 N . The block is modelled as a particle which is placed on the surface of the ice rink at a point \(B\), where \(A B\) is perpendicular to the wall and of length 5.5 m .
\includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-6_499_1429_813_333} The block is set into motion at the point \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) directly towards the point \(A\). The string remains horizontal throughout the motion.
  1. Initially, Zoë assumes that the surface of the ice rink is smooth. Using this assumption, find the speed of the block when it reaches the point \(A\).
  2. Zoë now assumes that friction acts on the block. The coefficient of friction between the block and the surface of the ice rink is \(\mu\).
    1. Find, in terms of \(g\) and \(\mu\), the speed of the block when it reaches the point \(A\).
    2. The block rebounds from the wall in the direction of the point \(B\). The speed of the block immediately after the rebound is half of the speed with which it hit the wall. Find \(\mu\) if the block comes to rest just as it reaches the point \(B\).
AQA M2 2013 June Q1
1 A particle, of mass 3 kg , moves along a straight line. At time \(t\) seconds, the displacement, \(s\) metres, of the particle from the origin is given by $$s = 8 t ^ { 3 } + 15$$
  1. Find the velocity of the particle at time \(t\).
  2. Find the magnitude of the resultant force acting on the particle when \(t = 2\).
AQA M2 2013 June Q2
2 Carol, a circus performer, is on a swing. She jumps off the swing and lands in a safety net. When Carol leaves the swing, she has a speed of \(7 \mathrm {~ms} ^ { - 1 }\) and she is at a height of 8 metres above the safety net. Carol is to be modelled as a particle of mass 52 kg being acted upon only by gravity.
  1. Find the kinetic energy of Carol when she leaves the swing.
  2. Show that the kinetic energy of Carol when she hits the net is 5350 J , correct to three significant figures.
  3. Find the speed of Carol when she hits the net.
AQA M2 2013 June Q3
3 A particle, of mass 10 kg , moves on a smooth horizontal plane. At time \(t\) seconds, the acceleration of the particle is given by $$\left\{ \left( 40 t + 3 t ^ { 2 } \right) \mathbf { i } + 20 \mathrm { e } ^ { - 4 t } \mathbf { j } \right\} \mathrm { m } \mathrm {~s} ^ { - 2 }$$ where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. At time \(t = 1\), the velocity of the particle is \(\left( 6 \mathbf { i } - 5 \mathrm { e } ^ { - 4 } \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  2. Calculate the initial speed of the particle.
AQA M2 2013 June Q4
4 A uniform plank \(A B\), of length 6 m , has mass 25 kg . It is supported in equilibrium in a horizontal position by two vertical inextensible ropes. One of the ropes is attached to the plank at the point \(P\) and the other rope is attached to the plank at the point \(Q\), where \(A P = 1 \mathrm {~m}\) and \(Q B = 0.8 \mathrm {~m}\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-2_227_1187_2252_424}
    1. Find the tension in each rope.
    2. State how you have used the fact that the plank is uniform in your solution. (1 mark)
  2. A particle of mass \(m \mathrm {~kg}\) is attached to the plank at point \(B\), and the tension in each rope is now the same. Find \(m\).
AQA M2 2013 June Q5
5 Tom is travelling on a train which is moving at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal track. Tom has placed his mobile phone on a rough horizontal table. The coefficient of friction between the phone and the table is 0.2 . The train moves round a bend of constant radius. The phone does not slide as the train travels round the bend. Model the phone as a particle moving round part of a circle, with centre \(O\) and radius \(r\) metres. Find the least possible value of \(r\).
AQA M2 2013 June Q6
6 A car accelerates from rest along a straight horizontal road. The car's engine produces a constant horizontal force of magnitude 4000 N .
At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and a resistance force of magnitude \(40 v\) newtons acts upon the car. The mass of the car is 1600 kg .
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 100 - v } { 40 }\).
  2. Find the velocity of the car at time \(t\).
AQA M2 2013 June Q7
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
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
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 2014 June Q1
1 An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it drops the salmon. The salmon falls a vertical distance of 13 metres back into the sea. The salmon is to be modelled as a particle. The salmon's weight is the only force that acts on it as it falls to the sea.
  1. Calculate the kinetic energy of the salmon when it is dropped by the eagle.
  2. Calculate the potential energy lost by the salmon as it falls to the sea.
    1. Find the kinetic energy of the salmon when it reaches the sea.
    2. Hence find the speed of the salmon when it reaches the sea.
      \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-02_1291_1709_1416_153}
      \(2 \quad\) A particle has mass 6 kg . A single force \(\left( 24 \mathrm { e } ^ { - 2 t } \mathbf { i } - 12 t ^ { 3 } \mathbf { j } \right)\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  4. Find the acceleration of the particle at time \(t\).
  5. At time \(t = 0\), the velocity of the particle is \(( - 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  6. Find the speed of the particle when \(t = 0.5\).
AQA M2 2014 June Q3
5 marks
3 Four tools are attached to a board.
The board is to be modelled as a uniform lamina and the four tools as four particles.
The diagram shows the lamina, the four particles \(A , B , C\) and \(D\), and the \(x\) and \(y\) axes.
\includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-06_597_960_550_532} The lamina has mass 5 kg and its centre of mass is at the point \(( 7,6 )\).
Particle \(A\) has mass 4 kg and is at the point ( 11,2 ).
Particle \(B\) has mass 3 kg and is at the point \(( 3,6 )\).
Particle \(C\) has mass 7 kg and is at the point ( 5,9 ).
Particle \(D\) has mass 1 kg and is at the point ( 1,4 ).
Find the coordinates of the centre of mass of the system of board and tools.
[0pt] [5 marks]
AQA M2 2014 June Q4
4 A particle, of mass 0.8 kg , is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(35 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-08_808_600_520_721} The particle moves in a horizontal circle and completes 20 revolutions each minute.
  1. Find the angular speed of the particle in radians per second.
  2. Find the tension in the string.
  3. Find the radius of the horizontal circle.
AQA M2 2014 June Q5
5 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 of the string. The particle is moving in a vertical circle with centre \(O\). The point \(Q\) is the highest point of the particle's path. When the particle is at \(P\), vertically below \(O\), the string is taut and the particle is moving with speed \(7 \sqrt { a g }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-10_887_812_525_628}
  1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\).
  2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\).
AQA M2 2014 June Q6
6 A puck, of mass \(m \mathrm {~kg}\), is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the puck moves, it experiences an air resistance force of magnitude \(0.3 m v ^ { \frac { 1 } { 3 } }\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Model the puck as a particle.
  1. Show that $$v = ( 4 - 0.2 t ) ^ { \frac { 3 } { 2 } }$$
  2. Find the value of \(t\) when the puck comes to rest.
  3. Find the distance travelled by the puck as its speed decreases from \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to zero.
    \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-12_1479_1709_1228_153}
AQA M2 2014 June Q7
6 marks
7 A uniform ladder \(A B\), of length 6 metres and mass 22 kg , rests with its foot, \(A\), on rough horizontal ground. The ladder rests against the top of a smooth vertical wall at the point \(C\), where the length \(A C\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60 ^ { \circ }\). A man, of mass 88 kg , is standing on the ladder. The man may be modelled as a particle at the point \(D\), where the length of \(A D\) is 4 metres. The ladder is on the point of slipping.
\includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-14_864_808_758_616}
  1. Draw a diagram to show the forces acting on the ladder.
  2. Find the coefficient of friction between the ladder and the horizontal ground.
    [0pt] [6 marks]
    \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-15_2484_1707_221_153}
AQA M2 2014 June Q8
8 An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg . The coefficient of friction between the particle and the plane is 0.8 . The three points, \(A , B\) and \(C\), lie on a line of greatest slope.
The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane.
\includegraphics[max width=\textwidth, alt={}, center]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-16_250_615_703_717}
  1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N , correct to three significant figures.
  2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
    1. Find \(x\).
    2. Find the acceleration of the particle as it starts to move back towards \(A\).
      \includegraphics[max width=\textwidth, alt={}]{8ca9234e-1a01-4e35-8b1b-17c6039cf8d7-17_2484_1707_221_153}
AQA M2 2015 June Q1
5 marks
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
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
4 marks
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
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.