Questions M1 (1912 questions)

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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.
AQA M1 2009 January Q2
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
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
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
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
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.
AQA M1 2009 January Q7
7 A boat is travelling in water that is moving north-east at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the boat relative to the water is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due west. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_275_349_415_504} \captionsetup{labelformat=empty} \caption{Velocity of the water}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_81_293_534_1181} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
\end{figure}
  1. Show that the magnitude of the resultant velocity of the boat is \(3.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  2. Find the bearing on which the boat is travelling, giving your answer to the nearest degree.
AQA M1 2009 January Q8
8 A cricket ball is hit at ground level on a horizontal surface. It initially moves at \(28 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal.
  1. Find the maximum height of the ball during its flight.
  2. The ball is caught when it is at a height of 2 metres above ground level, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_332_1070_1601_477} Show that the time that it takes for the ball to travel from the point where it was hit to the point where it was caught is 4.28 seconds, correct to three significant figures.
  3. Find the speed of the ball when it is caught.
AQA M1 2011 January Q1
1 A trolley, of mass 5 kg , is moving in a straight line on a smooth horizontal surface. It has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a stationary trolley, of mass \(m \mathrm {~kg}\). Immediately after the collision, the trolleys move together with velocity \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(m\).
(3 marks)
AQA M1 2011 January Q2
2 The graph shows how the velocity of a train varies as it moves along a straight railway line.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-04_574_1595_402_203}
  1. Find the total distance travelled by the train.
  2. Find the average speed of the train.
  3. Find the acceleration of the train during the first 10 seconds of its motion.
  4. The mass of the train is 200 tonnes. Find the magnitude of the resultant force acting on the train during the first 10 seconds of its motion.
AQA M1 2011 January Q3
3 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal tow bar, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-06_277_901_484_584} Assume that a constant resistance force of magnitude 200 newtons acts on the car and a constant resistance force of magnitude 300 newtons acts on the caravan. A constant driving force of magnitude \(P\) newtons acts on the car in the direction of motion. The car and caravan accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).
    1. Find \(P\).
    2. Find the magnitude of the force in the tow bar that connects the car to the caravan.
    1. Find the time that it takes for the speed of the car and caravan to increase from \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Find the distance that they travel in this time.
  3. Explain why the assumption that the resistance forces are constant is unrealistic.
    (1 mark)
AQA M1 2011 January Q4
4 A canoe is paddled across a river which has a width of 20 metres. The canoe moves from the point \(X\) on one bank of the river to the point \(Y\) on the other bank, so that its path is a straight line at an angle \(\alpha\) to the banks. The velocity of the canoe relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the banks. The water flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-10_469_1333_543_374} Model the canoe as a particle.
  1. Find the magnitude of the resultant velocity of the canoe.
  2. Find the angle \(\alpha\).
  3. Find the time that it takes for the canoe to travel from \(X\) to \(Y\).
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-11_2486_1714_221_153}
AQA M1 2011 January Q5
5 A particle moves with constant acceleration \(( - 0.4 \mathbf { i } + 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially, it has velocity \(( 4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
  1. Find an expression for the velocity of the particle at time \(t\) seconds.
    1. Find the velocity of the particle when \(t = 22.5\).
    2. State the direction in which the particle is travelling at this time.
  3. Find the time when the speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M1 2011 January Q6
6 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes over a smooth peg. Particle \(A\) has mass 2 kg and particle \(B\) has mass 4 kg . Particle \(A\) hangs freely with the string vertical. Particle \(B\) is at rest in equilibrium on a rough horizontal surface with the string at an angle of \(30 ^ { \circ }\) to the vertical. The particles, peg and string are shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-14_419_953_571_541}
  1. By considering particle \(A\), find the tension in the string.
  2. Draw a diagram to show the forces acting on particle \(B\).
  3. Show that the magnitude of the normal reaction force acting on particle \(B\) is 22.2 newtons, correct to three significant figures.
  4. Find the least possible value of the coefficient of friction between particle \(B\) and the surface.
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-16_2486_1714_221_153}
AQA M1 2011 January Q7
7 An arrow is fired from a point at a height of 1.5 metres above horizontal ground. It has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. The arrow hits a target at a height of 1 metre above horizontal ground. The path of the arrow is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-18_341_1260_550_390} Model the arrow as a particle.
  1. Show that the time taken for the arrow to travel to the target is 1.30 seconds, correct to three significant figures.
  2. Find the horizontal distance between the point where the arrow is fired and the target.
  3. Find the speed of the arrow when it hits the target.
  4. Find the angle between the velocity of the arrow and the horizontal when the arrow hits the target.
  5. State one assumption that you have made about the forces acting on the arrow.
    (1 mark)
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-19_2486_1714_221_153}
    \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-20_2486_1714_221_153}
AQA M1 2011 January Q8
8 A van, of mass 2000 kg , is towed up a slope inclined at \(5 ^ { \circ }\) to the horizontal. The tow rope is at an angle of \(12 ^ { \circ }\) to the slope. The motion of the van is opposed by a resistance force of magnitude 500 newtons. The van is accelerating up the slope at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-22_269_991_513_529} Model the van as a particle.
  1. Draw a diagram to show the forces acting on the van.
  2. Show that the tension in the tow rope is 3480 newtons, correct to three significant figures.
AQA M1 2012 January Q1
1 Two particles, \(A\) of mass 7 kg and \(B\) of mass 3 kg , are moving on a smooth horizontal plane when they collide. Just before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). During the collision, the particles coalesce to form a single combined particle. Find the velocity of the single combined particle after the collision.