Questions — AQA M1 (172 questions)

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AQA M1 2005 January Q1
1 A train travels along a straight horizontal track. It is travelling at a speed of \(12 \mathrm {~ms} ^ { - 1 }\) when it begins to accelerate uniformly. It reaches a speed of \(40 \mathrm {~ms} ^ { - 1 }\) after accelerating for 100 seconds.
    1. Show that the acceleration of the train is \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Find the distance that the train travelled in the 100 seconds.
  2. The mass of the train is 200 tonnes and a resistance force of 40000 N acts on the train. Find the magnitude of the driving force produced by the engine that acts on the train as it accelerates.
AQA M1 2005 January Q2
2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4
7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .
  1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2
    3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
  2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1
    4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.
AQA M1 2005 January Q3
3 The diagram shows a rope that is attached to a box of mass 25 kg , which is being pulled along rough horizontal ground. The rope is at an angle of \(30 ^ { \circ }\) to the ground. The tension in the rope is 40 N . The box accelerates at \(0.1 \mathrm {~ms} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_214_729_504_644}
  1. Draw a diagram to show all of the forces acting on the box.
  2. Show that the magnitude of the friction force acting on the box is 32.1 N , correct to three significant figures.
  3. Show that the magnitude of the normal reaction force that the ground exerts on the box is 225 N .
  4. Find the coefficient of friction between the box and the ground.
  5. State what would happen to the magnitude of the friction force if the angle between the rope and the horizontal were increased. Give a reason for your answer.
AQA M1 2005 January Q4
4 Two particles are connected by a string, which passes over a pulley. Model the string as light and inextensible. The particles have masses of 2 kg and 5 kg . The particles are released from rest.
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_392_209_1685_909}
  1. State one modelling assumption that you should make about the pulley in order to determine the acceleration of the particles.
  2. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
AQA M1 2005 January Q5
5 Two ropes are attached to a load of mass 500 kg . The ropes make angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) to the vertical, as shown in the diagram. The tensions in these ropes are \(T _ { 1 }\) and \(T _ { 2 }\) newtons. The load is also supported by a vertical spring.
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_533_565_507_744} The system is in equilibrium and \(T _ { 1 } = 200\).
  1. Show that \(T _ { 2 } = 141\), correct to three significant figures.
  2. Find the force that the spring exerts on the load.
AQA M1 2005 January Q6
6 A motor boat can travel at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water. It is used to cross a river in which the current flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the boat makes an angle of \(60 ^ { \circ }\) to the river bank, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_561_1339_1692_350} The angle between the direction in which the boat is travelling relative to the water and the resultant velocity is \(\alpha\).
  1. Show that \(\alpha = 16.8 ^ { \circ }\), correct to three significant figures.
  2. Find the magnitude of the resultant velocity.
AQA M1 2005 January Q7
7 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A yacht moves with a constant acceleration. At time \(t\) seconds the position vector of the yacht is \(\mathbf { r }\) metres. When \(t = 0\) the velocity of the yacht is \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), and when \(t = 10\) the velocity of the yacht is \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the acceleration of the yacht.
  2. When \(t = 0\) the yacht is 20 metres due east of the origin. Find an expression for \(\mathbf { r }\) in terms of \(t\).
    1. Show that when \(t = 20\) the yacht is due north of the origin.
    2. Find the speed of the yacht when \(t = 20\).
AQA M1 2005 January Q8
8 A football is placed on a horizontal surface. It is then kicked, so that it has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal.
  1. State two modelling assumptions that it would be appropriate to make when considering the motion of the football.
    1. Find the time that it takes for the ball to reach its maximum height.
    2. Hence show that the maximum height of the ball is 3.04 metres, correct to three significant figures.
  3. After the ball has reached its maximum height, it hits the bar of a goal at a height of 2.44 metres. Find the horizontal distance of the goal from the point where the ball was kicked.
AQA M1 2007 January Q1
1 Two particles \(A\) and \(B\) have masses of 3 kg and 2 kg respectively. They are moving along a straight horizontal line towards each other. Each particle is moving with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when they collide.
\includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-2_225_579_676_660}
  1. If the particles coalesce during the collision to form a single particle, find the speed of the combined particle after the collision.
  2. If, after the collision, \(A\) moves in the same direction as before the collision with speed \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the speed of \(B\) after the collision.
AQA M1 2007 January Q2
2 A lift rises vertically from rest with a constant acceleration.
After 4 seconds, it is moving upwards with a velocity of \(2 \mathrm {~ms} ^ { - 1 }\).
It then moves with a constant velocity for 5 seconds.
The lift then slows down uniformly, coming to rest after it has been moving for a total of 12 seconds.
  1. Sketch a velocity-time graph for the motion of the lift.
  2. Calculate the total distance travelled by the lift.
  3. The lift is raised by a single vertical cable. The mass of the lift is 300 kg . Find the maximum tension in the cable during this motion.
AQA M1 2007 January Q3
3 The diagram shows three forces which act in the same plane and are in equilibrium.
\includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_419_516_383_761}
  1. Find \(F\).
  2. Find \(\alpha\).
AQA M1 2007 January Q4
4 The diagram shows a block, of mass 13 kg , on a rough horizontal surface. It is attached by a string that passes over a smooth peg to a sphere of mass 7 kg , as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_323_974_1256_575} The system is released from rest, and after 4 seconds the block and the sphere both have speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the block has not reached the peg.
  1. State two assumptions that you should make about the string in order to model the motion of the sphere and the block.
  2. Show that the acceleration of the sphere is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Find the tension in the string.
  4. Find the coefficient of friction between the block and the surface.
AQA M1 2007 January Q5
5 A girl in a boat is rowing across a river, in which the water is flowing at \(0.1 \mathrm {~ms} ^ { - 1 }\). The velocity of the boat relative to the water is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is perpendicular to the bank, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-4_314_1152_468_450}
  1. Find the magnitude of the resultant velocity of the boat.
  2. Find the acute angle between the resultant velocity and the bank.
  3. The width of the river is 15 metres.
    1. Find the time that it takes the boat to cross the river.
    2. Find the total distance travelled by the boat as it crosses the river.
AQA M1 2007 January Q6
6 A trolley, of mass 100 kg , rolls at a constant speed along a straight line down a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. Assume that a constant resistance force, of magnitude \(P\) newtons, acts on the trolley as it moves. Model the trolley as a particle.
  1. Draw a diagram to show the forces acting on the trolley.
  2. Show that \(P = 68.4 \mathrm {~N}\), correct to three significant figures.
    1. Find the acceleration of the trolley if it rolls down a slope inclined at \(5 ^ { \circ }\) to the horizontal and experiences the same constant force of magnitude \(P\) that you found in part (b).
    2. Make one criticism of the assumption that the resistance force on the trolley is constant.
AQA M1 2007 January Q7
7 A golf ball is struck from a point on horizontal ground so that it has an initial velocity of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. Assume that the golf ball is a particle and its weight is the only force that acts on it once it is moving.
  1. Find the maximum height of the golf ball.
  2. After it has reached its maximum height, the golf ball descends but hits a tree at a point which is at a height of 6 metres above ground level.
    \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-5_289_1358_813_335} \begin{displayquote} Find the time that it takes for the ball to travel from the point where it was struck to the tree. \end{displayquote}
AQA M1 2007 January Q8
8 A particle is initially at the origin, where it has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It moves with a constant acceleration \(\mathbf { a } \mathrm { ms } ^ { - 2 }\) for 10 seconds to the point with position vector \(75 \mathbf { i }\) metres.
  1. Show that \(\mathbf { a } = 0.5 \mathbf { i } + 0.4 \mathbf { j }\).
  2. Find the position vector of the particle 8 seconds after it has left the origin.
  3. Find the position vector of the particle when it is travelling parallel to the unit vector \(\mathbf { i }\).
AQA M1 2008 January Q1
1 A crane is used to lift a crate, of mass 70 kg , vertically upwards. As the crate is lifted, it accelerates uniformly from rest, rising 8 metres in 5 seconds.
  1. Show that the acceleration of the crate is \(0.64 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. The crate is attached to the crane by a single cable. Assume that there is no resistance to the motion of the crate. Find the tension in the cable.
  3. Calculate the average speed of the crate during these 5 seconds.
AQA M1 2008 January Q2
2 The velocity of a ship, relative to the water in which it is moving, is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The water is moving due east with a speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the ship has magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find \(U\).
  2. Find the direction of the resultant velocity of the ship. Give your answer as a bearing to the nearest degree.
AQA M1 2008 January Q3
3 A particle, of mass 4 kg , is suspended in equilibrium by two light strings, \(A P\) and \(B P\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) to the horizontal and the other string, \(B P\), is horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-2_231_757_1841_639}
  1. Draw and label a diagram to show the forces acting on the particle.
  2. Show that the tension in the string \(A P\) is 78.4 N .
  3. Find the tension in the horizontal string \(B P\).
AQA M1 2008 January Q4
4 Two particles, \(A\) and \(B\), are moving on a horizontal plane when they collide and coalesce to form a single particle. The mass of \(A\) is 5 kg and the mass of \(B\) is 15 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 2 U
U \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { c } V
- 1 \end{array} \right] \mathrm { ms } ^ { - 1 }\). After the collision, the velocity of the combined particle is \(\left[ \begin{array} { l } V
0 \end{array} \right] \mathrm { ms } ^ { - 1 }\).
  1. Find:
    1. \(U\);
    2. \(V\).
  2. Find the speed of \(A\) before the collision.
AQA M1 2008 January Q5
5 A puck, of mass 0.2 kg , is placed on a slope inclined at \(20 ^ { \circ }\) above the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-3_280_773_1249_623} The puck is hit so that initially it moves with a velocity of \(4 \mathrm {~ms} ^ { - 1 }\) directly up the slope.
  1. A simple model assumes that the surface of the slope is smooth.
    1. Show that the acceleration of the puck up the slope is \(- 3.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
    2. Find the distance that the puck will travel before it comes to rest.
    3. What will happen to the puck after it comes to rest? Explain why.
  2. A revised model assumes that the surface is rough and that the coefficient of friction between the puck and the surface is 0.5 .
    1. Show that the magnitude of the friction force acting on the puck during this motion is 0.921 N , correct to three significant figures.
    2. Find the acceleration of the puck up the slope.
    3. What will happen to the puck after it comes to rest in this case? Explain why.
AQA M1 2008 January Q6
6 A tractor, of mass 4000 kg , is used to pull a skip, of mass 1000 kg , over a rough horizontal surface. The tractor is connected to the skip by a rope, which remains taut and horizontal throughout the motion, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-4_243_880_477_571} Assume that only two horizontal forces act on the tractor. One is a driving force, which has magnitude \(P\) newtons and acts in the direction of motion. The other is the tension in the rope. The coefficient of friction between the skip and the ground is 0.4 .
The tractor and the skip accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).
  1. Show that the magnitude of the friction force acting on the skip is 3920 N .
  2. Show that \(P = 7920\).
  3. Find the tension in the rope.
  4. Suppose that, during the motion, the rope is not horizontal, but inclined at a small angle to the horizontal, with the higher end of the rope attached to the tractor, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-4_241_880_1665_571} How would the magnitude of the friction force acting on the skip differ from that found in part (a)? Explain why.
AQA M1 2008 January Q7
7 A golfer hits a ball which is on horizontal ground. The ball initially moves with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. There is a pond further along the horizontal ground. The diagram below shows the initial position of the ball and the position of the pond.
\includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-5_387_1230_502_395}
  1. State two assumptions that you should make in order to model the motion of the ball.
    (2 marks)
  2. Show that the horizontal distance, in metres, travelled by the ball when it returns to ground level is $$\frac { V ^ { 2 } \sin 40 ^ { \circ } \cos 40 ^ { \circ } } { 4.9 }$$
  3. Find the range of values of \(V\) for which the ball lands in the pond.
AQA M1 2008 January Q8
8 A Jet Ski is at the origin and is travelling due north at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it begins to accelerate uniformly. After accelerating for 40 seconds, it is travelling due east at \(4 \mathrm {~ms} ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Show that the acceleration of the Jet Ski is \(( 0.1 \mathbf { i } - 0.125 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find the position vector of the Jet Ski at the end of the 40 second period.
  3. The Jet Ski is travelling southeast \(t\) seconds after it leaves the origin.
    1. Find \(t\).
    2. Find the velocity of the Jet Ski at this time.
AQA M1 2009 January Q1
1 Two particles, \(A\) and \(B\), are travelling in the same direction with constant speeds along a straight line when they collide. Particle \(A\) has mass 2.5 kg and speed \(12 \mathrm {~ms} ^ { - 1 }\). Particle \(B\) has mass 1.5 kg and speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision, the two particles move together at the same speed. Find the speed of the particles after the collision.