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OCR MEI M1 Q3
8 marks Moderate -0.8
3 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$
  1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
  2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
  3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.
OCR MEI M1 Q4
8 marks Moderate -0.3
4 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).
  1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
  2. Find \(\mathbf { G }\) and \(\mathbf { H }\).
OCR MEI M1 Q5
6 marks Moderate -0.3
5 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).
OCR MEI M1 Q6
7 marks Moderate -0.5
6 The force acting on a particle of mass 1.5 kg is given by the vector \(\binom { 6 } { 9 } \mathrm {~N}\).
  1. Give the acceleration of the particle as a vector.
  2. Calculate the angle that the acceleration makes with the direction \(\binom { 1 } { 0 }\).
  3. At a certain point of its motion, the particle has a velocity of \(\binom { - 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the displacement of the particle over the next two seconds.
OCR MEI M1 Q7
7 marks Moderate -0.3
7 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.
  1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
  2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } ) \mathrm { N }\). Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
  3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).
AQA M2 2006 January Q1
8 marks Moderate -0.8
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
7 marks Moderate -0.8
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
9 marks Moderate -0.3
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
10 marks Standard +0.3
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
8 marks Moderate -0.3
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
10 marks Standard +0.3
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
9 marks Standard +0.3
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
14 marks Standard +0.3
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
10 marks Moderate -0.8
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
8 marks Moderate -0.8
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
11 marks Standard +0.3
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
9 marks Standard +0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
8 marks Standard +0.3
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
10 marks Standard +0.3
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
10 marks Moderate -0.3
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
5 marks Moderate -0.8
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
4 marks Moderate -0.8
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
14 marks Standard +0.3
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}