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AQA M1 2005 January Q6
7 marks Standard +0.3
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
12 marks Moderate -0.3
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
16 marks Moderate -0.8
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
6 marks Moderate -0.8
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
10 marks Standard +0.3
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
6 marks Moderate -0.8
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
13 marks Moderate -0.3
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
9 marks Moderate -0.8
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
9 marks Moderate -0.8
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
10 marks Moderate -0.8
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
12 marks Standard +0.3
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
6 marks Moderate -0.8
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
4 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.3
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
16 marks Standard +0.3
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
10 marks Standard +0.3
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
12 marks Moderate -0.3
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
14 marks Standard +0.3
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
3 marks Moderate -0.8
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.
AQA M1 2009 January Q2
10 marks Moderate -0.8
2 The graph shows how the velocity of a particle varies during a 50 -second period as it moves forwards and then backwards on a straight line.
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-2_615_1312_1007_301}
  1. State the times at which the velocity of the particle is zero.
  2. Show that the particle travels a distance of 75 metres during the first 30 seconds of its motion.
  3. Find the total distance travelled by the particle during the 50 seconds.
  4. Find the distance of the particle from its initial position at the end of the 50 -second period.
AQA M1 2009 January Q3
7 marks Moderate -0.8
3 A box of mass 4 kg is held at rest on a plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal. The box is then released and slides down the plane.
  1. A simple model assumes that the only forces acting on the box are its weight and the normal reaction from the plane. Show that, according to this simple model, the acceleration of the box would be \(6.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
  2. In fact, the box moves down the plane with constant acceleration and travels 0.9 metres in 0.6 seconds. By using this information, find the acceleration of the box.
  3. Explain why the answer to part (b) is less than the answer to part (a).
AQA M1 2009 January Q4
14 marks Moderate -0.3
4 Two particles, \(A\) and \(B\), are connected by a string that passes over a fixed peg, as shown in the diagram. The mass of \(A\) is 9 kg and the mass of \(B\) is 11 kg .
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-3_320_538_1117_806} The particles are released from rest in the position shown, where \(B\) is \(d\) metres higher than \(A\). The motion of the particles is to be modelled using simple assumptions.
  1. State one assumption that should be made about the peg.
  2. State two assumptions that should be made about the string.
  3. By forming an equation of motion for each of the particles \(A\) and \(B\), show that the acceleration of each particle has magnitude \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  4. When the particles have been moving for 0.5 seconds, they are at the same level.
    1. Find the speed of the particles at this time.
    2. Find \(d\).
AQA M1 2009 January Q5
9 marks Moderate -0.8
5 A sledge of mass 8 kg is at rest on a rough horizontal surface. A child tries to move the sledge by pushing it with a pole, as shown in the diagram, but the sledge does not move. The pole is at an angle of \(30 ^ { \circ }\) to the horizontal and exerts a force of 40 newtons on the sledge.
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-4_221_922_513_552} Model the sledge as a particle.
  1. Draw a diagram to show the four forces acting on the sledge.
  2. Show that the normal reaction force between the sledge and the surface has magnitude 98.4 N .
  3. Find the magnitude of the friction force that acts on the sledge.
  4. Find the least possible value of the coefficient of friction between the sledge and the surface.
AQA M1 2009 January Q6
10 marks Moderate -0.3
6 Two forces, \(\mathbf { P } = ( 6 \mathbf { i } - 3 \mathbf { j } )\) newtons and \(\mathbf { Q } = ( 3 \mathbf { i } + 15 \mathbf { j } )\) newtons, act on a particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.
  1. Find the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Calculate the magnitude of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  3. When these two forces act on the particle, it has an acceleration of \(( 1.5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find the mass of the particle.
  4. The particle was initially at rest at the origin.
    1. Find an expression for the position vector of the particle when the forces have been applied to the particle for \(t\) seconds.
    2. Find the distance of the particle from the origin when the forces have been applied to the particle for 2 seconds.