Questions M1 (1912 questions)

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AQA M1 2006 June Q3
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
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
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
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
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 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
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
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
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
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
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
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
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 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 \(m\).
AQA M1 2009 June Q1
1 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. During the collision, the two particles coalesce to form a single combined particle. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { r } 6
- 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } - 1
4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the velocity of the combined particle after the collision.
  2. Find the speed of the combined particle after the collision.
AQA M1 2009 June Q2
2 A lift is travelling upwards and accelerating uniformly. During a 5 second period, it travels 16 metres and the speed of the lift increases from \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. \(\quad\) Find \(u\).
  2. Find the acceleration of the lift.
AQA M1 2009 June Q3
3 A car is travelling in a straight line on a horizontal road. A driving force, of magnitude 3000 N , acts in the direction of motion and a resistance force, of magnitude 600 N , opposes the motion of the car. Assume that no other horizontal forces act on the car.
  1. Find the magnitude of the resultant force on the car.
  2. The mass of the car is 1200 kg . Find the acceleration of the car. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_38_118_440_159}
    \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_40_118_529_159}
AQA M1 2009 June Q4
4 A river has parallel banks which are 16 metres apart. The water in the river flows at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. A boat sets off from one bank at the point \(A\) and travels perpendicular to the bank so that it reaches the point \(B\), which is directly opposite the point \(A\). It takes the boat 10 seconds to cross the river. The velocity of the boat relative to the water has magnitude \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at an angle \(\alpha\) to the bank, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-08_400_1011_667_511}
  1. Show that the magnitude of the resultant velocity of the boat is \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. \(\quad\) Find \(V\).
  3. Find \(\alpha\).
  4. State one modelling assumption that you needed to make about the boat.
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-08_72_1689_1617_154}
    .......... \(\_\_\_\_\)
    \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-09_40_118_529_159}
AQA M1 2009 June Q5
5 A block, of mass 14 kg , is held at rest on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.25 . A light inextensible string, which passes over a fixed smooth peg, is attached to the block. The other end of the string is attached to a particle, of mass 6 kg , which is hanging at rest.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-10_264_716_502_708} The block is released and begins to accelerate.
  1. Find the magnitude of the friction force acting on the block.
  2. By forming two equations of motion, one for the block and one for the particle, show that the magnitude of the acceleration of the block and the particle is \(1.225 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
  4. When the block is released, it is 0.8 metres from the peg. Find the speed of the block when it hits the peg.
  5. When the block reaches the peg, the string breaks and the particle falls a further 0.5 metres to the ground. Find the speed of the particle when it hits the ground.
    (3 marks)
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-11_2484_1709_223_153}
AQA M1 2009 June Q6
6 A ball is kicked from the point \(P\) on a horizontal surface. It leaves the surface with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal and hits the surface for the first time at the point \(Q\). Assume that the ball is a particle that moves only under the influence of gravity.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-12_317_1118_513_461}
  1. Show that the time that it takes the ball to travel from \(P\) to \(Q\) is 3.13 s , correct to three significant figures.
  2. Find the distance between the points \(P\) and \(Q\).
  3. If a heavier ball were projected from \(P\) with the same velocity, how would the distance between \(P\) and \(Q\), calculated using the same modelling assumptions, compare with your answer to part (b)? Give a reason for your answer.
  4. Find the maximum height of the ball above the horizontal surface.
  5. State the magnitude and direction of the velocity of the ball as it hits the surface.
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-13_2484_1709_223_153}
AQA M1 2009 June Q7
7 A particle moves on a smooth horizontal plane. It is initially at the point \(A\), with position vector \(( 9 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\), and has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves with a constant acceleration of \(( 0.25 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for 20 seconds until it reaches the point \(B\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find the velocity of the particle at the point \(B\).
  2. Find the velocity of the particle when it is travelling due north.
  3. Find the position vector of the point \(B\).
  4. Find the average velocity of the particle as it moves from \(A\) to \(B\).
    \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-15_2484_1709_223_153}
AQA M1 2009 June Q8
8 The diagram shows a block, of mass 20 kg , being pulled along a rough horizontal surface by a rope inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-16_323_1194_411_424} The coefficient of friction between the block and the surface is \(\mu\). Model the block as a particle which slides on the surface.
  1. If the tension in the rope is 60 newtons, the block moves at a constant speed.
    1. Show that the magnitude of the normal reaction force acting on the block is 166 N .
    2. Find \(\mu\).
  2. If the rope remains at the same angle and the block accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the tension in the rope. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-18_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-19_2488_1719_219_150}
AQA M1 2010 June Q1
1 A bus slows down as it approaches a bus stop. It stops at the bus stop and remains at rest for a short time as the passengers get on. It then accelerates away from the bus stop. The graph shows how the velocity of the bus varies.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-02_627_1296_657_402} Assume that the bus travels in a straight line during the motion described by the graph.
  1. State the length of time for which the bus is at rest.
  2. Find the distance travelled by the bus in the first 40 seconds.
  3. Find the total distance travelled by the bus in the 120 -second period.
  4. Find the average speed of the bus in the 120 -second period.
  5. If the bus had not stopped but had travelled at a constant \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the 120 -second period, how much further would it have travelled?
    \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-03_2484_1709_223_153}
AQA M1 2010 June Q2
2 A block, of mass 10 kg , is at rest on a rough horizontal surface, when a horizontal force, of magnitude \(P\) newtons, is applied to the block, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-04_108_962_461_539} The coefficient of friction between the block and the surface is 0.5 .
  1. Draw and label a diagram to show all the forces acting on the block.
    1. Calculate the magnitude of the normal reaction force acting on the block.
    2. Find the maximum possible magnitude of the friction force between the block and the surface.
    3. Given that \(P = 30\), state the magnitude of the friction force acting on the block.
  3. Given that \(P = 80\), find the acceleration of the block.
    \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-05_2484_1709_223_153}
AQA M1 2010 June Q3
3 Two particles, \(A\) and \(B\), are moving on a smooth horizontal plane when they collide. The mass of \(A\) is 6 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 2
4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } 3
- 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 1
3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } 7
b \end{array} \right] \mathrm { ms } ^ { - 1 }\).
  1. Find \(m\).
  2. \(\quad\) Find \(b\).
    (2 marks)
    .......... \(\_\_\_\_\)
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_40_118_529_159}
    \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_39_117_623_159}