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AQA M1 2013 January Q8
10 marks Moderate -0.3
8 A golf ball is hit from a point on a horizontal surface, so that it has an initial velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The ball travels through the air and after 2.4 seconds hits a vertical wall at a height of 3 metres. The wall is at a horizontal distance of 38.4 metres from the point where the ball was hit. The path of the ball is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-18_300_1000_566_520} Assume that the weight of the ball is the only force that acts on it as it travels through the air.
  1. Find the horizontal component of the velocity of the ball.
  2. \(\quad\) Find \(V\).
  3. \(\quad\) Find \(\alpha\).
AQA M1 2005 June Q1
7 marks Moderate -0.3
1 A particle of mass \(m\) has velocity \(\left[ \begin{array} { l } 4 \\ 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides with a particle of mass 3 kg which has velocity \(\left[ \begin{array} { l } - 1 \\ - 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision the particles coalesce and move with velocity \(\left[ \begin{array} { l } 1 \\ V \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that \(m = 2\).
  2. Find \(V\).
AQA M1 2005 June Q2
10 marks Moderate -0.8
2 A train travels along a straight horizontal track between two points \(A\) and \(B\).
Initially the train is at \(A\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Due to a problem, the train has to slow down and stop. At time \(t = 40\) seconds it begins to move again. At time \(t = 120\) seconds the train is at \(B\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) again. The graph below shows how the velocity of the train varies as it moves from \(A\) to \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-2_408_1086_1505_434}
  1. Use the graph to find the total distance between the points \(A\) and \(B\).
  2. The train should have travelled between \(A\) and \(B\) at a constant velocity of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Calculate the time that the train would take to travel between \(A\) and \(B\) at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Calculate the time by which the train was delayed.
  3. The train has mass 500 tonnes. Find the resultant force acting on the train when \(40 < t < 120\).
    (4 marks)
AQA M1 2005 June Q3
7 marks Moderate -0.3
3 A boat can travel at a speed of \(2 \mathrm {~ms} ^ { - 1 }\) in still water. The boat is to cross a river in which a current flows at a speed of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angle between the direction in which the boat is pointing and the bank is \(\alpha\). The boat travels so that the resultant velocity of the boat is perpendicular to the bank.
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_264_1040_575_493}
  1. Show that \(\alpha = 66.4 ^ { \circ }\) correct to three significant figures.
    1. Find the magnitude of the resultant velocity of the boat.
    2. The width of the river is 14 metres. Find the time that it takes for the boat to cross the river.
AQA M1 2005 June Q4
11 marks Standard +0.3
4 Two particles, \(A\) of mass 5 kg and \(B\) of mass 9 kg , are attached to the ends of a light inextensible string. The string passes over a light smooth pulley as shown in the diagram. The particles are released from rest at the same height.
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_378_287_1580_872}
  1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the tension in the string.
  3. When \(B\) has been moving for 0.5 seconds it hits the floor. Find the height of \(A\), above the floor, at this time. Assume that \(A\) is still below the pulley when \(B\) hits the floor.
    (4 marks)
AQA M1 2005 June Q5
7 marks Moderate -0.5
5 A sphere of mass 200 grams is released from rest and allowed to fall vertically.
  1. A student states that the acceleration of the sphere is \(9.8 \mathrm {~ms} ^ { - 2 }\) while it is falling. What modelling assumption is this student making?
  2. The student conducts an experiment and finds that the acceleration of the ball is in fact \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). He formulates a model for the motion that assumes a constant resistance force acts on the ball as it is falling.
    1. Calculate the magnitude of this resistance force based on this assumption.
    2. Describe how the resistance force would vary in reality.
  3. In a revised model the resistance force is assumed to be proportional to the speed of the sphere.
    1. State the initial acceleration of the sphere.
    2. State what would happen to the acceleration of the sphere if it were able to fall for a long period of time.
AQA M1 2005 June Q6
12 marks Moderate -0.8
6 A ball is hit from horizontal ground with velocity \(( 10 \mathbf { i } + 24.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively.
  1. State two assumptions that you should make about the ball in order to make predictions about its motion.
  2. The path of the ball is shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_771_705_625}
    1. Show that the time of flight of the ball is 5 seconds.
    2. Find the range of the ball.
  3. In fact the ball hits a vertical wall that is 20 metres from the initial position of the ball.
    \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_403_1466_769} Find the height of the ball when it hits the wall.
  4. If a heavier ball were projected in the same way, would your answers to part (b) of this question change? Explain why.
AQA M1 2005 June Q7
8 marks Moderate -0.3
7 A particle moves on a smooth horizontal surface with acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). Initially the velocity of the particle is \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find an expression for the velocity of the particle at time \(t\) seconds.
  2. Find the time when the particle is travelling in the \(\mathbf { i }\) direction.
  3. Show that when \(t = 4\) the speed of the particle is \(20 \mathrm {~ms} ^ { - 1 }\).
AQA M1 2005 June Q8
11 marks Standard +0.3
8 A rough slope is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. A particle of mass 6 kg is on the slope. A string is attached to the particle and is at an angle of \(30 ^ { \circ }\) to the slope. The tension in the string is 20 N . The diagram shows the slope, the particle and the string.
\includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-6_259_684_518_676} The particle moves up the slope with an acceleration of \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Draw a diagram to show the forces acting on the particle.
  2. Show that the magnitude of the normal reaction force is 47.9 N , correct to three significant figures.
  3. Find the coefficient of friction between the particle and the slope.
AQA M1 2006 June Q1
6 marks Easy -1.2
1 A stone is dropped from a high bridge and falls vertically.
  1. Find the distance that the stone falls during the first 4 seconds of its motion.
  2. Find the average speed of the stone during the first 4 seconds of its motion.
  3. State one modelling assumption that you have made about the forces acting on the stone during the motion.
AQA M1 2006 June Q2
5 marks Moderate -0.8
2 A particle is in equilibrium under the action of four horizontal forces of magnitudes 5 newtons, 8 newtons, \(P\) newtons and \(Q\) newtons, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-2_355_357_1146_852}
  1. Show that \(P = 9\).
  2. Find the value of \(Q\).
AQA M1 2006 June Q3
10 marks Easy -1.2
3 A car travels along a straight horizontal road. The motion of the car can be modelled as three separate stages. During the first stage, the car accelerates uniformly from rest to a velocity of \(10 \mathrm {~ms} ^ { - 1 }\) in 6 seconds. During the second stage, the car travels with a constant velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 seconds. During the third stage of the motion, the car travels with a uniform retardation of magnitude \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.
  1. Show that the time taken for the third stage of the motion is 12.5 seconds.
  2. Sketch a velocity-time graph for the car during the three stages of the motion.
  3. Find the total distance travelled by the car during the motion.
  4. State one criticism of the model of the motion.
AQA M1 2006 June Q4
10 marks Moderate -0.3
4 A block is being pulled up a rough plane inclined at an angle of \(22 ^ { \circ }\) to the horizontal by a rope parallel to the plane, as shown in the diagram. The mass of the block is 0.7 kg , and the tension in the rope is \(T\) newtons.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-3_264_460_1649_779}
  1. Draw a diagram to show the forces acting on the block.
  2. Show that the normal reaction force between the block and the plane has magnitude 6.36 newtons, correct to three significant figures.
  3. The coefficient of friction between the block and the plane is 0.25 . Find the magnitude of the frictional force acting on the block during its motion.
  4. The tension in the rope is 5.6 newtons. Find the acceleration of the block.
AQA M1 2006 June Q5
14 marks Standard +0.3
5 A small block \(P\) is attached to another small block \(Q\) by a light inextensible string. The block \(P\) rests on a rough horizontal surface and the string hangs over a smooth peg so that \(Q\) hangs freely, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-4_222_426_507_810} The block \(P\) has mass 0.4 kg and the coefficient of friction between \(P\) and the surface is 0.5 . The block \(Q\) has mass 0.3 kg . The system is released from rest and \(Q\) moves vertically downwards.
    1. Draw a diagram to show the forces acting on \(P\).
    2. Show that the frictional force between \(P\) and the surface has magnitude 1.96 newtons.
  2. By forming an equation of motion for each block, show that the magnitude of the acceleration of each block is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the speed of the blocks after 3 seconds of motion.
  4. After 3 seconds of motion, the string breaks. The blocks continue to move. Comment on how the speed of each block will change in the subsequent motion. For each block, give a reason for your answer.
AQA M1 2006 June Q6
9 marks Moderate -0.8
6 The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 2 \mathbf { j } )\) metres and \(( 6 \mathbf { i } - 4 \mathbf { j } )\) metres respectively. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.
  1. A particle moves from \(A\) to \(B\) with constant velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Calculate the time that the particle takes to move from \(A\) to \(B\).
  2. The particle then moves from \(B\) to a point \(C\) with a constant acceleration of \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It takes 4 seconds to move from \(B\) to \(C\).
    1. Find the position vector of \(C\).
    2. Find the distance \(A C\).
AQA M1 2006 June Q7
13 marks Standard +0.3
7 A golf ball is struck from a point \(O\) with velocity \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) to the horizontal. The ball first hits the ground at a point \(P\), which is at a height \(h\) metres above the level of \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-5_318_990_484_543} The horizontal distance between \(O\) and \(P\) is 57 metres.
  1. Show that the time that the ball takes to travel from \(O\) to \(P\) is 3.10 seconds, correct to three significant figures.
  2. Find the value of \(h\).
    1. Find the speed with which the ball hits the ground at \(P\).
    2. Find the angle between the direction of motion and the horizontal as the ball hits the ground at \(P\).
AQA M1 2006 June Q8
8 marks Moderate -0.3
8 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface.
The particle \(A\) has mass \(m \mathrm {~kg}\) and is moving with velocity \(\left[ \begin{array} { r } 5 \\ - 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\). The particle \(B\) has mass 0.2 kg and is moving with velocity \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 }\).
  1. Find, in terms of \(m\), an expression for the total momentum of the particles.
  2. The particles \(A\) and \(B\) collide and form a single particle \(C\), which moves with velocity \(\left[ \begin{array} { c } k \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
    1. Show that \(m = 0.1\).
    2. Find the value of \(k\).
AQA M1 2008 June Q1
7 marks Moderate -0.8
1 The diagram shows a velocity-time graph for a lift.
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_337_917_552_557}
  1. Find the distance travelled by the lift.
  2. Find the acceleration of the lift during the first 4 seconds of the motion.
  3. The lift is raised by a single vertical cable. The mass of the lift is 400 kg . Find the tension in the cable during the first 4 seconds of the motion.
AQA M1 2008 June Q2
7 marks Moderate -0.8
2 The diagram shows three forces and the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\), which all lie in the same plane.
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_415_398_1507_605}
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_172_166_1567_1217}
  1. Express the resultant of the three forces in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
  2. Find the magnitude of the resultant force.
  3. Draw a diagram to show the direction of the resultant force, and find the angle that it makes with the unit vector \(\mathbf { i }\).
AQA M1 2008 June Q3
9 marks Moderate -0.3
3 Two particles, \(A\) and \(B\), have masses 4 kg and 6 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg. A second light inextensible string is attached to \(A\). The other end of this string is attached to the ground directly below \(A\). The system remains at rest, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-3_457_711_523_845}
    1. Write down the tension in the string connecting \(A\) and \(B\).
    2. Find the tension in the string connecting \(A\) to the ground.
  2. The string connecting particle \(A\) to the ground is cut. Find the acceleration of \(A\) after the string has been cut.
AQA M1 2008 June Q4
8 marks Moderate -0.3
4 An aeroplane is travelling due north at \(180 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the air. The air is moving north-west at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the resultant velocity of the aeroplane.
  2. Find the direction of the resultant velocity, giving your answer as a three-figure bearing to the nearest degree.
AQA M1 2008 June Q5
13 marks Moderate -0.8
5 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A helicopter moves horizontally with a constant acceleration of \(( - 0.4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the helicopter is at the origin and has velocity \(20 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Write down an expression for the velocity of the helicopter at time \(t\) seconds.
  2. Find the time when the helicopter is travelling due north.
  3. Find an expression for the position vector of the helicopter at time \(t\) seconds.
  4. When \(t = 100\) :
    1. show that the helicopter is due north of the origin;
    2. find the speed of the helicopter.
AQA M1 2008 June Q6
11 marks Moderate -0.3
6 A block, of mass 5 kg , slides down a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. When modelling the motion of the block, assume that there is no air resistance acting on it.
  1. Draw and label a diagram to show the forces acting on the block.
  2. Show that the magnitude of the normal reaction force acting on the block is 37.5 N , correct to three significant figures.
  3. Given that the acceleration of the block is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the coefficient of friction between the block and the plane.
  4. In reality, air resistance does act on the block. State how this would change your value for the coefficient of friction and explain why.
AQA M1 2008 June Q7
12 marks Moderate -0.3
7 A ball is hit by a bat so that, when it leaves the bat, its velocity is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(35 ^ { \circ }\) above the horizontal. Assume that the ball is a particle and that its weight is the only force that acts on the ball after it has left the bat.
  1. A simple model assumes that the ball is hit from the point \(A\) and lands for the first time at the point \(B\), which is at the same level as \(A\), as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-4_321_1063_1370_484}
    1. Show that the time that it takes for the ball to travel from \(A\) to \(B\) is 4.68 seconds, correct to three significant figures.
    2. Find the horizontal distance from \(A\) to \(B\).
  2. A revised model assumes that the ball is hit from the point \(C\), which is 1 metre above \(A\). The ball lands at the point \(D\), which is at the same level as \(A\), as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-4_431_1177_2181_420} Find the time that it takes for the ball to travel from \(C\) to \(D\).
AQA M1 2008 June Q8
8 marks Standard +0.3
8 Two particles, \(A\) and \(B\), are travelling towards each other along a straight horizontal line.
Particle \(A\) has velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and mass \(m \mathrm {~kg}\).
Particle \(B\) has velocity \(- 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and mass 3 kg .
\includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-5_220_1157_516_440} The particles collide.
  1. If the particles move in opposite directions after the collision, each with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(m\).
  2. If the particles coalesce during the collision, forming a single particle which moves with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the two possible values of \(m\).