Questions M1 (2067 questions)

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AQA M1 2015 June Q3
7 marks Moderate -0.3
3 A ship has a mass of 500 tonnes. Two tugs are used to pull the ship using cables that are horizontal. One tug exerts a force of 100000 N at an angle of \(25 ^ { \circ }\) to the centre line of the ship. The other tug exerts a force of \(T \mathrm {~N}\) at an angle of \(20 ^ { \circ }\) to the centre line of the ship. The diagram shows the ship and forces as viewed from above. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-06_279_844_539_664} The ship accelerates in a straight line along its centre line.
  1. \(\quad\) Find \(T\).
  2. A resistance force of magnitude 20000 N directly opposes the motion of the ship. Find the acceleration of the ship.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-06_1419_1714_1288_153}
AQA M1 2015 June Q4
10 marks Moderate -0.8
4 A particle moves with constant acceleration between the points \(A\) and \(B\). At \(A\), it has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(B\), it has velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It takes 10 seconds to move from \(A\) to \(B\).
  1. Find the acceleration of the particle.
  2. Find the distance between \(A\) and \(B\).
  3. Find the average velocity as the particle moves from \(A\) to \(B\).
AQA M1 2015 June Q5
16 marks Standard +0.3
5 A block, of mass \(3 m\), is placed on a horizontal surface at a point \(A\). A light inextensible string is attached to the block and passes over a smooth peg. The string is horizontal between the block and the peg. A particle, of mass \(2 m\), is attached to the other end of the string. The block is released from rest with the string taut and the string between the peg and the particle vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-10_170_726_536_657} Assume that there is no air resistance acting on either the block or the particle, and that the size of the block is negligible. The horizontal surface is smooth between the points \(A\) and \(B\), but rough between the points \(B\) and \(C\). Between \(B\) and \(C\), the coefficient of friction between the block and the surface is 0.8 .
  1. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(A\) and \(B\).
  2. Given that the distance between the points \(A\) and \(B\) is 1.2 metres, find the speed of the block when it reaches \(B\).
  3. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(B\) and \(C\).
  4. Given that the distance between the points \(B\) and \(C\) is 0.9 metres, find the speed of the block when it reaches \(C\).
  5. Explain why it is important to assume that the size of the block is negligible.
    [0pt] [1 mark]
AQA M1 2015 June Q6
12 marks Standard +0.3
6 Emma is in a park with her dog, Roxy. Emma throws a ball and Roxy catches it in her mouth. The ground in the park is horizontal. Emma throws the ball from a point at a height of 1.2 metres above the ground and Roxy catches the ball when it is at a height of 0.5 metres above the ground. Emma throws the ball with an initial velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal.
  1. Find the time that the ball takes to travel from Emma's hand to Roxy's mouth.
  2. Find the horizontal distance travelled by the ball during its flight.
  3. During the flight, the speed of the ball is a maximum when it is at a height of \(h\) metres above the ground. Write down the value of \(h\).
  4. Find the maximum speed of the ball during its flight.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-14_1566_1707_1137_153}
AQA M1 2015 June Q7
11 marks Standard +0.3
7 Two forces, which act in a vertical plane, are applied to a crate. The crate has mass 50 kg , and is initially at rest on a rough horizontal surface. One force has magnitude 80 N and acts at an angle of \(30 ^ { \circ }\) to the horizontal and the other has magnitude 40 N and acts at an angle of \(20 ^ { \circ }\) to the horizontal. The forces are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-16_241_999_493_523} The coefficient of friction between the crate and the surface is 0.6 . Model the crate as a particle.
  1. Draw a diagram to show the forces acting on the crate.
  2. Find the magnitude of the normal reaction force acting on the crate.
  3. Does the crate start to move when the two forces are applied to the crate? Show all your working.
  4. State one aspect of the possible motion of the crate that is ignored by modelling it as a particle.
    [0pt] [1 mark]
AQA M1 2015 June Q8
11 marks Standard +0.3
8 Two trains, \(A\) and \(B\), are moving on straight horizontal tracks which run alongside each other and are parallel. The trains both move with constant acceleration. At time \(t = 0\), the fronts of the trains pass a signal. The velocities of the trains are shown in the graph below. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-18_633_1077_475_424}
  1. Find the distance between the fronts of the two trains when they have the same velocity and state which train has travelled further from the signal.
  2. Find the time when \(A\) has travelled 9 metres further than \(B\).
    \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-20_2288_1707_221_153}
AQA M1 2016 June Q2
3 marks Moderate -0.8
2 Three forces \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { N } , ( p \mathbf { i } + 5 \mathbf { j } ) \mathrm { N }\) and \(( - 8 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) act on a particle of mass 5 kg to produce an acceleration of \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). No other forces act on the particle.
  1. Find the resultant force acting on the particle in terms of \(p\) and \(q\).
  2. \(\quad\) Find \(p\) and \(q\).
  3. Given that the particle is initially at rest, find the displacement of the particle from its initial position when these forces have been acting for 4 seconds.
    [0pt] [3 marks]
AQA M1 2016 June Q3
4 marks Moderate -0.8
3 A toy car is placed at the top of a ramp. After the car has been released from rest, it travels a distance of 1.08 metres down the ramp, in a time of 1.2 seconds. Assume that there is no resistance to the motion of the car.
  1. Find the magnitude of the acceleration of the car while it is moving down the ramp.
  2. Find the speed of the car, when it has travelled 1.08 metres down the ramp.
  3. Find the angle between the ramp and the horizontal, giving your answer to the nearest degree.
    [0pt] [4 marks]
AQA M1 2016 June Q4
3 marks Moderate -0.8
4 An aeroplane is flying in air that is moving due east at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Relative to the air, the aeroplane has a velocity of \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. During a 20 second period, the motion of the air causes the aeroplane to move 240 metres to the east.
  1. \(\quad\) Find \(V\).
  2. Find the magnitude of the resultant velocity of the aeroplane.
  3. Find the direction of the resultant velocity, giving your answer as a three-figure bearing, correct to the nearest degree.
    [0pt] [3 marks]
AQA M1 2016 June Q5
4 marks Moderate -0.3
5 Two particles, of masses 3 kg and 7 kg , are connected by a light inextensible string that passes over a smooth peg. The 3 kg particle is held at ground level with the string above it taut and vertical. The 7 kg particle is at a height of 80 cm above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-10_469_600_486_721} The 3 kg particle is then released from rest.
  1. By forming two equations of motion, show that the magnitude of the acceleration of the particles is \(3.92 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the speed of the 7 kg particle just before it hits the ground.
  3. When the 7 kg particle hits the ground, the string becomes slack and in the subsequent motion the 3 kg particle does not hit the peg. Find the maximum height of the 3 kg particle above the ground.
    [0pt] [4 marks]
AQA M1 2016 June Q6
6 marks Standard +0.3
6 A floor polisher consists of a heavy metal block with a polishing cloth attached to the underside. A light rod is also attached to the block and is used to push the block across the floor that is to be polished. The block has mass 5 kg . Assume that the floor is horizontal. Model the block as a particle. The coefficient of friction between the cloth and the floor is 0.2 .
A person pushes the rod to exert a force on the block. The force is at an angle of \(60 ^ { \circ }\) to the horizontal and the block accelerates at \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The diagram shows the block and the force exerted by the rod in this situation. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-14_309_205_772_1009} The rod exerts a force of magnitude \(T\) newtons on the block.
  1. Find, in terms of \(T\), the magnitude of the normal reaction force acting on the block.
  2. \(\quad\) Find \(T\).
    [0pt] [6 marks]
AQA M1 2016 June Q7
11 marks Moderate -0.3
7 At a school fair, there is a competition in which people try to kick a football so that it lands in a large rectangular box. The height of the top of the box is 1 metre and its width is 3 metres. One student kicks a football so that it initially moves at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. It hits the top front edge of the box, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-16_465_1342_625_351} Model the football as a particle that is not subject to any resistance forces as it moves.
  1. Find the time taken for the football to move from the point where it was kicked to the box.
  2. Find the horizontal distance from the point where the football is kicked to the front of the box.
  3. If the same student kicks the football at the same angle from the same initial position, what is the speed at which the student should kick the football if it is to hit the top back edge of the box?
  4. Explain the significance of modelling the football as a particle in this context.
    [0pt] [1 mark]
    \includegraphics[max width=\textwidth, alt={}]{5dd17095-18a6-470b-a24a-4456c8e3ed31-23_2488_1709_219_153}
    \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
Edexcel M1 Q1
6 marks Standard +0.3
1. \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_490_254_354_347} A vertical pole \(X Y\), of length 2.5 m and mass 0.5 kg , has its lower end \(Y\) free to move in a smooth horizontal groove. Forces of magnitude 0.2 N and 0.14 N are applied to the pole horizontally at the points \(V\) and \(W\) respectively, where \(X V = 1.5 \mathrm {~m}\) and \(V W = 0.5 \mathrm {~m}\).
Find, to the nearest cm , the distance from \(X\) at which an opposing horizontal force must be applied to keep the pole at rest in equilibrium, and state the magnitude of this force.
Edexcel M1 Q2
7 marks Standard +0.3
2. A particle passes through a point \(O\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) and moves in a straight line with constant acceleration \(3.6 \mathrm {~ms} ^ { - 2 }\) for \(t\) seconds until it reaches the point \(P\). The acceleration is then reduced to \(2 \mathrm {~ms} ^ { - 2 }\) and this is maintained for another \(t\) seconds until the particle passes the point \(Q\) with speed \(16 \mathrm {~ms} ^ { - 1 }\). Calculate
  1. the time taken by the particle to travel from \(O\) to \(Q\),
  2. the distance \(O Q\).
Edexcel M1 Q3
9 marks Standard +0.3
3. A lump of clay, of mass 0.8 kg , is attached to the end \(A\) of a light \(\operatorname { rod } A B\), which is pivoted at its other end \(B\) so that it can rotate smoothly in a vertical plane. A force is applied to \(A\) at an angle of \(60 ^ { \circ }\) to the vertical, as shown, the magnitude \(F \mathrm {~N}\) of this force being just enough to hold the lump of clay in equilibrium with \(A B\) inclined \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_309_335_1453_1590}
at an angle of \(30 ^ { \circ }\) to the upward vertical.
  1. Find the value of \(F\),
  2. Find the magnitude of the force in the \(\operatorname { rod } A B\).
  3. State the modelling assumption that you have made about the lump of clay.
    (6 marks)
    (2 marks)
    (1 mark)
Edexcel M1 Q4
10 marks Standard +0.3
4. Two particles \(A\) and \(B\), of masses 50 grams and \(y\) grams, are moving in the same straight line, in opposite directions, with speeds \(7 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively, and collide. \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_218_508_2143_1382}
In each of the following separate cases, find the value of \(y\) and the magnitude of the impulse exerted by each particle on the other:
  1. after impact the particles move together with speed \(2.25 \mathrm {~ms} ^ { - 1 }\);
  2. after impact the particles move in opposite directions with speed \(5 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 1 (A) TEST PAPER 6 Page 2}
Edexcel M1 Q5
12 marks Standard +0.3
5.
\includegraphics[max width=\textwidth, alt={}]{31efa627-5114-4797-9d46-7f1311c18ff8-2_262_597_276_356}
A small stone is projected with speed \(7 \mathrm {~ms} ^ { - 1 }\) from \(P\), the bottom of a rough plane inclined at \(25 ^ { \circ }\) to the horizontal, and moves up a line of greatest slope of the plane until it comes to instantaneous rest at \(Q\), where \(P Q = 4 \mathrm {~m}\).
  1. Show that the deceleration of the stone as it moves up the plane has magnitude \(\frac { 49 } { 8 } \mathrm {~ms} ^ { - 2 }\).
  2. Find the coefficient of friction between the stone and the plane,
  3. Find the speed with which the stone returns to \(P\).
  4. Name one force which you have ignored in your mathematical model, and state whether the answer to part (c) would be larger or smaller if that force were taken into account.
Edexcel M1 Q6
14 marks Standard +0.3
6. The points \(A\) and \(B\) have position vectors \(( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }\) and \(( - 20 \mathbf { i } + 60 \mathbf { j } ) \mathrm { m }\) respectively relative to an origin \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. A cyclist, Chris, starts at \(A\) and cycles towards \(B\) with constant speed \(2.6 \mathrm {~ms} ^ { - 1 }\). Another cyclist, Doug, starts at \(O\) and cycles towards \(B\) with constant speed \(k \sqrt { } 10 \mathrm {~ms} ^ { - 1 }\).
  1. Show that Chris's velocity vector is \(( - \mathbf { i } + 2 \cdot 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  2. Find Doug's velocity vector in the form \(k ( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Given that Chris and Doug arrive at \(B\) at the same time,
  3. find the value of \(k\).
Edexcel M1 Q7
17 marks Standard +0.3
7.
[diagram]
A particle \(P\), of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle \(Q\) of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially \(Q\) is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.
  1. Find the acceleration of the two particles.
  2. Find the speed with which \(Q\) hits the ground. Assuming that \(Q\) does not rebound from the ground while the string is slack,
  3. show that \(P\) does not reach the pulley before \(Q\) starts to move again.
  4. Find the speed with which \(Q\) leaves the ground when the string again becomes taut.
    (3 marks)
Edexcel M1 Q1
5 marks Easy -1.2
A golf ball and a table tennis ball are dropped together from the top of a building. The golf ball hits the ground after 1.7 seconds.
  1. Calculate the height of the top of the building above the ground. According to a simple model, the two balls hit the ground at the same time.
  2. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution.
Edexcel M1 Q2
5 marks Standard +0.3
2. A plank of wood \(X Y\) has length \(5 a\) m and mass 5 kg . It rests on a support at \(Q\), where \(X Q = 3 a\) m . When a kitten of mass 8 kg sits on the plank at \(P\), where \(P Y = a \mathrm {~m}\), the plank just remains horizontal. By modelling the plank as a non-uniform rod and the kitten as a particle, find
  1. the magnitude of the reaction at the support,
  2. the distance from \(X\) to the centre of mass of the plank, in terms of \(a\).
Edexcel M1 Q3
9 marks Moderate -0.3
3. A particle is in equilibrium under the action of three forces \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) acting in the same horizontal plane. \(P\) has magnitude 9 N and acts on a bearing of \(030 ^ { \circ } . Q\) has magnitude 12 N . and acts on a bearing of \(225 ^ { \circ }\).
  1. Find the values of \(a\) and \(b\) such that \(\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively.
  2. Calculate the magnitude and direction of \(\mathbf { R }\)
Edexcel M1 Q4
13 marks Standard +0.3
4. \(X\) and \(Y\) are two points 1 m apart on a line of greatest slope of a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 1 kg is released from rest at \(X\).
  1. Find the speed with which \(P\) reaches \(Y\). \(P\) is now connected to another particle \(Q\), of mass \(M \mathrm {~kg}\), by a light inextensible string. The system is placed with \(P\) at \(Y\) on the plane and \(Q\) hanging vertically at the other end of the string, which passes over a fixed pulley at the top of the plane.
    The system is released from rest and \(P\) moves up the plane with acceleration \(\frac { g } { 5 }\). \includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-1_358_321_2024_1597}
  2. Show that \(M = \frac { 5 \sqrt { } 3 + 2 } { 8 }\).
  3. State a modelling assumption that you have made about the pulley. Briefly state what would be implied if this assumption were not made. \section*{MECHANICS 1 (A) TEST PAPER 8 Page 2}
Edexcel M1 Q5
13 marks Standard +0.3
Two model cars \(A\) and \(B\) have masses 200 grams and \(k\) grams respectively. They move towards each other in a straight line and collide directly when their speeds are \(5 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively. As a result the speed of \(A\) is reduced to \(2 \mathrm {~ms} ^ { - 1 }\), in the same direction as before. The direction of \(B\) 's motion is reversed and its speed immediately after the impact is \(5 \mathrm {~ms} ^ { - 1 }\).
  1. Find the magnitude of the impulse exerted by \(A\) on \(B\) in the impact. State the units of your answer.
  2. Find the value of \(k\). The surface on which the cars move is rough, and \(B\) comes to rest 3 seconds after the impact. The coefficient of friction between both cars and the surface is \(\mu\).
  3. Find the value of \(\mu\).
  4. Find the distance travelled by \(A\) after the impact before it comes to rest.
Edexcel M1 Q6
15 marks Standard +0.3
6. A small ring, of mass \(m \mathrm {~kg}\), can slide along a straight wire which is fixed at an angle of \(45 ^ { \circ }\) to the horizontal as shown. The coefficient of friction between the ring and the wire is \(\frac { 2 } { 7 }\).
The ring rests in equilibrium on the wire and is just prevented from \includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-2_273_296_1192_1617}
sliding down the wire when a horizontal string is attached to it, as shown
  1. Show that the tension in the string has magnitude \(\frac { 5 m g } { 9 } \mathrm {~N}\). The string is now removed and the ring starts to slide down the wire.
  2. Find the time that elapses before the ring has moved 10 cm along the wire.