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AQA M1 2006 January Q8
16 marks Standard +0.3
8 A rough slope is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude \(T\) newtons. The diagram shows the slope, the box and the rope. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-7_307_469_500_840}
  1. The box is held in equilibrium by the rope.
    1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
    2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
    3. Find the least possible tension in the rope to prevent the box from moving down the slope.
    4. Find the greatest possible tension in the rope.
    5. Show that the mass of the box is approximately 8.16 kg .
  2. The rope is now released and the box slides down the slope. Find the acceleration of the box.
AQA M1 2010 January Q1
3 marks Easy -1.2
1 Two particles, \(A\) and \(B\), are travelling in the same direction along a straight line on a smooth horizontal surface. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Particle \(A\) has a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle \(A\) and particle \(B\) collide and coalesce to form a single particle. Find the speed of this single particle after the collision.
AQA M1 2010 January Q2
10 marks Easy -1.2
2 A sprinter accelerates from rest at a constant rate for the first 10 metres of a 100 -metre race. He takes 2.5 seconds to run the first 10 metres.
  1. Find the acceleration of the sprinter during the first 2.5 seconds of the race.
  2. Show that the speed of the sprinter at the end of the first 2.5 seconds of the race is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The sprinter completes the 100 -metre race, travelling the remaining 90 metres at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the total time taken for the sprinter to travel the 100 metres.
  4. Calculate the average speed of the sprinter during the 100 -metre race.
AQA M1 2010 January Q3
5 marks Easy -1.2
3 A particle of mass 3 kg is on a smooth slope inclined at \(60 ^ { \circ }\) to the horizontal. The particle is held at rest by a force of \(T\) newtons parallel to the slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_337_284_2023_879}
  1. Draw a diagram to show all the forces acting on the particle.
  2. Show that the magnitude of the normal reaction acting on the particle is 14.7 newtons.
  3. Find \(T\).
AQA M1 2010 January Q4
10 marks Moderate -0.3
4 A ball is released from rest at a height of 15 metres above ground level.
  1. Find the speed of the ball when it hits the ground, assuming that no air resistance acts on the ball.
  2. In fact, air resistance does act on the ball. Assume that the air resistance force has a constant magnitude of 0.9 newtons. The ball has a mass of 0.5 kg .
    1. Draw a diagram to show the forces acting on the ball, including the magnitudes of the forces acting.
    2. Show that the acceleration of the ball is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    3. Find the speed at which the ball hits the ground.
    4. Explain why the assumption that the air resistance force is constant may not be valid.
AQA M1 2010 January Q5
14 marks Moderate -0.8
5 The constant forces \(\mathbf { F } _ { 1 } = ( 8 \mathbf { i } + 12 \mathbf { j } )\) newtons and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } - 4 \mathbf { j } )\) newtons act on a particle. No other forces act on the particle.
  1. Find the resultant force acting on the particle.
  2. Given that the mass of the particle is 4 kg , show that the acceleration of the particle is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  3. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. When \(t = 20 , \mathbf { v } = 40 \mathbf { i } + 32 \mathbf { j }\). Show that \(\mathbf { v } = - 20 \mathbf { i } - 8 \mathbf { j }\) when \(t = 0\).
    2. Write down an expression for \(\mathbf { v }\) at time \(t\).
    3. Find the times when the speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M1 2010 January Q6
9 marks Moderate -0.8
6 A small train at an amusement park consists of an engine and two carriages connected to each other by light horizontal rods, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_190_1038_420_493} The engine has mass 2000 kg and each carriage has mass 500 kg . The train moves along a straight horizontal track. A resistance force of magnitude 400 newtons acts on the engine, and resistance forces of magnitude 300 newtons act on each carriage. The train is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Draw a diagram to show the horizontal forces acting on Carriage 2.
  2. Show that the magnitude of the force that the rod exerts on Carriage 2 is 550 newtons.
  3. Find the magnitude of the force that the rod attached to the engine exerts on Carriage 1.
  4. A forward driving force of magnitude \(P\) newtons acts on the engine. Find \(P\).
AQA M1 2010 January Q7
14 marks Moderate -0.8
7 A ball is projected horizontally with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at a height of 5 metres above horizontal ground. When the ball has travelled a horizontal distance of 15 metres, it hits the ground. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_433_1296_1674_338}
  1. Show that the time it takes for the ball to travel to the point where it hits the ground is 1.01 seconds, correct to three significant figures.
  2. Find \(V\).
  3. Find the speed of the ball when it hits the ground.
  4. Find the angle between the velocity of the ball and the horizontal when the ball hits the ground. Give your answer to the nearest degree.
  5. State two assumptions that you have made about the ball while it is moving.
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 \(x\).