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AQA M1 2010 January Q8
10 marks Standard +0.3
8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, \(T\) newtons, is the same in both parts of the rope.
  1. Draw and label a diagram to show the forces acting on the crate.
  2. Express the normal reaction between the surface and the crate in terms of \(T\).
  3. Find \(T\).
AQA M1 2007 June Q1
7 marks Easy -1.2
1 A ball is released from rest at a height \(h\) metres above ground level. The ball hits the ground 1.5 seconds after it is released. Assume that the ball is a particle that does not experience any air resistance.
  1. Show that the speed of the ball is \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the ground.
  2. Find \(h\).
  3. Find the distance that the ball has fallen when its speed is \(5 \mathrm {~ms} ^ { - 1 }\).
AQA M1 2007 June Q2
5 marks Moderate -0.8
2 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. Particle \(A\) has mass 2 kg and velocity \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). Particle \(B\) has mass 3 kg and velocity \(\left[ \begin{array} { r } - 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). The two particles collide, and they coalesce during the collision.
  1. Find the velocity of the combined particles after the collision.
  2. Find the speed of the combined particles after the collision.
AQA M1 2007 June Q3
10 marks Moderate -0.8
3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of \(35 ^ { \circ }\) to the vertical, as shown in the diagram. Model the sign as a particle. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}
  1. By resolving forces horizontally, show that the tension is the same in each string.
  2. Find the tension in each string.
  3. If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.
AQA M1 2007 June Q4
9 marks Moderate -0.3
4 A car, of mass 1200 kg , is connected by a tow rope to a truck, of mass 2800 kg . The truck tows the car in a straight line along a horizontal road. Assume that the tow rope is horizontal. A horizontal driving force of magnitude 3000 N acts on the truck. A horizontal resistance force of magnitude 800 N acts on the car. The car and truck accelerate at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-3_177_1002_580_513}
  1. Find the tension in the tow rope.
  2. Show that the magnitude of the horizontal resistance force acting on the truck is 600 N .
  3. In fact, the tow rope is not horizontal. Assume that the resistance forces and the driving force are unchanged. Is the tension in the tow rope greater or less than in part (a)? Explain why.
AQA M1 2007 June Q5
5 marks Moderate -0.3
5 An aeroplane flies in air that is moving due east at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the aeroplane relative to the air is \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The aeroplane actually travels on a bearing of \(030 ^ { \circ }\).
  1. Show that \(V = 86.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  2. Find the magnitude of the resultant velocity of the aeroplane.
AQA M1 2007 June Q6
15 marks Moderate -0.8
6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.
  1. A simple model assumes that the slope is smooth.
    1. Draw a diagram to show the forces acting on the box.
    2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
  2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
    1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Find the magnitude of the friction force acting on the box.
    3. Find the coefficient of friction between the box and the slope.
    4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.
AQA M1 2007 June Q7
12 marks Moderate -0.3
7 An arrow is fired from a point \(A\) with a velocity of \(25 \mathrm {~ms} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. The arrow hits a target at the point \(B\) which is at the same level as the point \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-4_195_1093_1594_511}
  1. State two assumptions that you should make in order to model the motion of the arrow.
    (2 marks)
  2. Show that the time that it takes for the arrow to travel from \(A\) to \(B\) is 3.28 seconds, correct to three significant figures.
  3. Find the distance between the points \(A\) and \(B\).
  4. State the magnitude and direction of the velocity of the arrow when it hits the target.
  5. Find the minimum speed of the arrow during its flight.
AQA M1 2007 June Q8
12 marks Moderate -0.8
8 A boat is initially at the origin, heading due east at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then experiences a constant acceleration of \(( - 0.2 \mathbf { i } + 0.25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. State the initial velocity of the boat as a vector.
  2. Find an expression for the velocity of the boat \(t\) seconds after it has started to accelerate.
  3. Find the value of \(t\) when the boat is travelling due north.
  4. Find the bearing of the boat from the origin when the boat is travelling due north.
AQA M2 Q1
Moderate -0.3
1 A uniform beam, \(A B\), has mass 20 kg and length 7 metres. A rope is attached to the beam at \(A\). A second rope is attached to the beam at the point \(C\), which is 2 metres from \(B\). Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.
AQA M2 Q2
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]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_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 Q3
Moderate -0.8
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 \(t\).
    (4 marks)
AQA M2 Q4
Standard +0.3
4 A car has a maximum speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is moving on a horizontal road. When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons.
  1. Show that the maximum power of the car is 52920 W .
  2. The car has mass 1200 kg . It travels, from rest, up a slope inclined at \(5 ^ { \circ }\) to the horizontal.
    1. Show that, when the car is travelling at its maximum speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
    2. Hence find \(V\).
AQA M2 Q5
Standard +0.3
5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude \(40 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car after it has been freewheeling for \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
AQA M2 Q6
Standard +0.3
6 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]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}
  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 Q7
Standard +0.3
7 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 Q8
Standard +0.3
8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).
  1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
  2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
  3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
    1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
    2. Find the value of \(x\).
AQA M2 2007 January Q1
8 marks Moderate -0.8
1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point \(A\) to the point \(B\), which is 10 metres vertically below the level of \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}
  1. In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at \(B\).
  2. State one resistance force that has been ignored in answering part (a).
  3. In fact, when the child slides down the slide, she reaches \(B\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of \(F\) newtons, use an energy method to find the value of \(F\).
    (4 marks)
AQA M2 2007 January Q2
6 marks Moderate -0.8
2 A hotel sign consists of a uniform rectangular lamina of weight \(W\). The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points \(A\) and \(B\), as shown in the diagram. The edge containing \(A\) and \(B\) is horizontal. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively.
  1. Draw a diagram to show the forces acting on the sign.
  2. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(W\).
  3. Explain how you have used the fact that the lamina is uniform in answering part (b).
AQA M2 2007 January Q3
6 marks Moderate -0.3
3 A light inextensible string has length \(2 a\). One end of the string is attached to a fixed point \(O\) and a particle of mass \(m\) is attached to the other end. Initially, the particle is held at the point \(A\) with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point \(B\), which is directly below \(O\). The points \(O , A\) and \(B\) are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}
  1. Show that the speed of the particle at \(B\) is \(2 \sqrt { a g }\).
  2. Find the tension in the string as the particle passes through \(B\). Give your answer in terms of \(m\) and \(g\).
AQA M2 2007 January Q4
9 marks Standard +0.3
4 A uniform T-shaped lamina is formed by rigidly joining two rectangles \(A B C H\) and \(D E F G\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}
  1. Show that the centre of mass of the lamina is 26 cm from the edge \(A B\).
  2. Explain why the centre of mass of the lamina is 5 cm from the edge \(G F\).
  3. The point \(X\) is on the edge \(A B\) and is 7 cm from \(A\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from \(X\) and hangs in equilibrium.
    Find the angle between the edge \(A B\) and the vertical, giving your answer to the nearest degree.
    (4 marks)
AQA M2 2007 January Q5
12 marks Moderate -0.3
5 Tom is on a fairground ride.
Tom's position vector, \(\mathbf { r }\) metres, at time \(t\) seconds is given by $$\mathbf { r } = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j } + ( 10 - 0.4 t ) \mathbf { k }$$ The perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane and the unit vector \(\mathbf { k }\) is directed vertically upwards.
    1. Find Tom's position vector when \(t = 0\).
    2. Find Tom's position vector when \(t = 2 \pi\).
    3. Write down the first two values of \(t\) for which Tom is directly below his starting point.
  2. Find an expression for Tom's velocity at time \(t\).
  3. Tom has mass 25 kg . Show that the resultant force acting on Tom during the motion has constant magnitude. State the magnitude of the resultant force.
    (5 marks)
AQA M2 2007 January Q6
11 marks Moderate -0.8
6 A particle is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set into motion, so that it describes a horizontal circle whose centre is vertically below \(O\). The angle between the string and the vertical is \(\theta\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-6_506_442_534_794}
  1. The particle completes 40 revolutions every minute. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
  2. The radius of the circle is 0.2 metres. Find, in terms of \(\pi\), the magnitude of the acceleration of the particle.
  3. The mass of the particle is \(m \mathrm {~kg}\) and the tension in the string is \(T\) newtons.
    1. Draw a diagram showing the forces acting on the particle.
    2. Explain why \(T \cos \theta = m g\).
    3. Find the value of \(\theta\), giving your answer to the nearest degree.
AQA M2 2007 January Q7
11 marks Moderate -0.3
7 A motorcycle has a maximum power of 72 kilowatts. The motorcycle and its rider are travelling along a straight horizontal road. When they are moving at a speed of \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\), they experience a total resistance force of magnitude \(k V\) newtons, where \(k\) is a constant.
  1. The maximum speed of the motorcycle and its rider is \(60 \mathrm {~ms} ^ { - 1 }\). Show that \(k = 20\).
  2. When the motorcycle is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rider allows the motorcycle to freewheel so that the only horizontal force acting is the resistance force. When the motorcycle has been freewheeling for \(t\) seconds, its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the resistance force is \(20 v\) newtons. The mass of the motorcycle and its rider is 500 kg .
    1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - \frac { v } { 25 }\).
    2. Hence find the time that it takes for the speed of the motorcycle to reduce from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      (6 marks)
AQA M2 2007 January Q8
12 marks Standard +0.3
8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).
  1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-8_385_239_669_881} Find the extension of the spring.
  2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
  3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
    1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
    2. Find the value of \(x\).