Questions (30179 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel M1 Q4
12 marks Moderate -0.3
4. A boy starts at the corner \(O\) of a rectangular playing field and runs across the field with constant velocity vector \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions of two perpendicular sides of the field. After 40 seconds, at the point \(P\) in the field, he changes speed and direction so that his new velocity vector is \(( 2 \cdot 4 \mathbf { i } - 1 \cdot 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and maintains this velocity until he reaches the point \(Q\), where \(P Q = 75 \mathrm {~m}\).
Calculate (a) the distance \(O P\),
(b) the time taken to travel from \(P\) to \(Q\),
(c) the position vector of \(Q\) relative to \(O\). Another boy travels directly from \(O\) to \(Q\) with constant velocity \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), leaving \(O\) and reaching \(Q\) at the same times as the first boy.
(d) Find the values of the constants \(a\) and \(b\). \section*{MECHANICS 1 (A)TEST PAPER 2 Page 2}
Edexcel M1 Q5
12 marks Standard +0.3
  1. Two railway trucks \(A\) and \(B\), of masses 10000 kg and 7000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84000 Ns. Immediately after the collision, the trucks move together with speed \(10 \mathrm {~ms} ^ { - 1 }\). Modelling the trucks as particles,
    1. find the speed of each truck immediately before the collision.
    When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15 . Assuming that no other resisting forces act on the trucks, calculate
  2. the magnitude of the resisting force on the trucks,
  3. the time taken after the collision for the trucks to come to rest.
Edexcel M1 Q6
15 marks Standard +0.3
6. A small package \(P\), of mass 1 kg , is initially at rest on the rough horizontal top surface of a wooden packing case which is 1.5 m long and 1 m high and stands on a horizontal floor. The coefficient of friction between \(P\) and the case is 0.2 . \includegraphics[max width=\textwidth, alt={}, center]{9c9b6087-d5a1-4fb0-b771-5ccc13a04bc4-2_287_517_941_1428} \(P\) is attached by a light inextensible string, which passes over a smooth fixed pulley, to a weight \(Q\) of mass \(M \mathrm {~kg}\) which rests against the smooth vertical side of the case.
The system is released from rest with \(P 0.75 \mathrm {~m}\) from the pulley and \(Q 0.5 \mathrm {~m}\) from the pulley. \(P\) and \(Q\) start to move with acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\). Calculate
  1. the tension in the string, in N ,
  2. the value of \(M\),
  3. the time taken for \(Q\) to hit the floor. Given that \(Q\) does not rebound from the floor,
  4. calculate the distance of \(P\) from the pulley when it comes to rest.
Edexcel M1 Q7
16 marks Standard +0.3
7. A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 \(\mathrm { ms } ^ { - 2 }\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m .
  1. Sketch a graph of velocity against time for the total period of the car's motion.
  2. Find the car's average speed for the whole journey. In reality the car's acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the first 10 seconds is not constant, but increases from 0 to \(4 \mathrm {~ms} ^ { - 2 }\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k \left( m t - t ^ { 2 } \right)\) for \(0 \leq t \leq 10\).
  3. Calculate the values of the constants \(k\) and \(m\).
  4. Find the acceleration of the car when \(t = 2\) according to this model.
Edexcel M1 Q1
7 marks Standard +0.3
  1. A particle \(P\), of mass 2.5 kg , initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively. Find
    1. the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\),
    2. the magnitude and direction of the velocity of \(P\) when \(t = 7\),
    3. the magnitude, in N , of the force acting on \(P\).
    4. An iron bar \(A B\), of length 4 m , is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(P B = 0.85 \mathrm {~m}\). The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(A G = 1.4 \mathrm {~m}\), and the wire is modelled as a light inextensible string. Given that the tension in the wire is 12 N , calculate
    5. the weight of the bar,
    6. the magnitude of the reaction on the bar at \(A\).
    7. State briefly how you have used the given modelling assumption about the bar.
    \includegraphics[max width=\textwidth, alt={}]{f8386a80-e428-43a7-acc8-f7ab11b2a53a-1_201_453_1399_378}
    A small packet, of mass 1.2 kg , is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the packet and the plane is \(\frac { 1 } { 8 }\).
    When a force of magnitude 8.4 N , acting parallel to the plane, is applied to the packet as shown, the packet is just on the point of moving up the plane. Modelling the packet as a particle,
  2. show that \(7 ( \cos \alpha + 8 \sin \alpha ) = 40\). Given that the solution of this equation is \(\alpha = 38 ^ { \circ }\),
  3. find the acceleration with which the packet moves down the plane when it is released from rest with no external force applied.
Edexcel M1 Q4
11 marks Moderate -0.3
4. A car moves in a straight line from \(P\) to \(Q\), a distance of 420 m , with constant acceleration. At \(P\) the speed of the car is \(8 \mathrm {~ms} ^ { - 1 }\). At \(Q\) the speed of the car is \(20 \mathrm {~ms} ^ { - 1 }\). Find
  1. the time taken to travel from \(P\) to \(Q\),
  2. the acceleration of the car,
  3. the time taken for the car to travel 240 m from \(P\). Given that the mass of the car is 1200 kg and the tractive force of the car is 900 N ,
  4. find the magnitude of the resistance to the car's motion. \section*{MECHANICS 1 (A) TEST PAPER 3 Page 2}
Edexcel M1 Q5
11 marks Standard +0.3
  1. Two smooth spheres \(X\) and \(Y\), of masses \(x \mathrm {~kg}\) and \(y \mathrm {~kg}\) respectively, are free to move in a smooth straight groove in a horizontal table. \(X\) is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) towards \(Y\), which is stationary. After the collision \(X\) moves with speed \(2 \mathrm {~ms} ^ { - 1 }\) and \(Y\) moves with speed \(3 \mathrm {~ms} ^ { - 1 }\).
    1. Calculate the two possible values of the ratio \(x : y\).
    2. State a modelling assumption that you have made concerning \(X\) and \(Y\). \(Y\) now strikes a vertical barrier and rebounds along the groove with speed \(k \mathrm {~ms} ^ { - 1 }\), colliding again with \(X\) which is still moving at \(2 \mathrm {~ms} ^ { - 1 }\). Given that in this impact \(Y\) is brought to rest and the direction of motion of \(X\) is reversed,
    3. show that \(k > 1 \cdot 5\).
    4. Two particles \(P\) and \(Q\), of masses 3 kg and 2 kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30 ^ { \circ } , 60 ^ { \circ }\) and \(90 ^ { \circ }\), as shown. \(P\) and \(Q\) are connected by a light \includegraphics[max width=\textwidth, alt={}, center]{f8386a80-e428-43a7-acc8-f7ab11b2a53a-2_255_607_1078_1311}
      string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge.
      The system is released from rest with \(P 0.8 \mathrm {~m}\) from the pulley and \(Q 1 \mathrm {~m}\) from the bottom of the wedge, and \(Q\) starts to move down. Calculate
    5. the acceleration of either particle,
    6. the tension in the string,
    7. the speed with which \(P\) reaches the pulley.
    Two modelling assumptions have been made about the string and the pulley.
  2. State these two assumptions and briefly describe how you have used each one in your solution.
Edexcel M1 Q7
15 marks Standard +0.8
7. Two stones are projected simultaneously from a point \(O\) on horizontal ground. Stone \(A\) is thrown vertically upwards with speed \(98 \mathrm {~ms} ^ { - 1 }\). Stone \(B\) is projected along the smooth ground in a straight line at \(24 \cdot 5 \mathrm {~ms} ^ { - 1 }\).
  1. Find the distances of the two stones from \(O\) after \(t\) seconds, where \(0 \leq t \leq 20\).
  2. Show that the distance \(d \mathrm {~m}\) between the two stones after \(t\) seconds is given by $$d ^ { 2 } = 24 \cdot 01 \left( t ^ { 4 } - 40 t ^ { 3 } + 425 t ^ { 2 } \right) .$$
  3. Hence find the range of values of \(t\) for which the distance between the stones is decreasing.
Edexcel M1 Q1
6 marks Moderate -0.8
  1. A tennis ball, moving horizontally, hits a wall at \(25 \mathrm {~ms} ^ { - 1 }\) and rebounds along the same straight line at \(15 \mathrm {~ms} ^ { - 1 }\). The impulse exerted by the wall on the ball has magnitude 12 Ns .
    1. Calculate the mass of the ball.
    2. State any modelling assumptions that you have made.
    \includegraphics[max width=\textwidth, alt={}]{977c24cc-8280-4881-8a62-65b7efd336ac-1_278_337_751_434}
    Forces of magnitude \(4 \mathrm {~N} , 5 \mathrm {~N}\) and 8 N act on a particle in directions whose bearings are \(000 ^ { \circ } , 090 ^ { \circ }\) and \(210 ^ { \circ }\) respectively. Find the magnitude of the resultant force and the bearing of the direction in which it acts.
Edexcel M1 Q3
9 marks Moderate -0.3
3. A packing-case, of mass 60 kg , is standing on the floor of a lift. The mass of the lift-cage is 200 kg . The lift-cage is raised and lowered by means of a cable attached to its roof. In each of the following cases, find the magnitude of the force exerted by the floor of the liftcage on the packing-case and the tension in the cable supporting the lift:
  1. The lift is descending with constant speed.
  2. The lift is ascending and accelerating at \(1.2 \mathrm {~ms} ^ { - 2 }\).
  3. State any modelling assumptions you have made.
Edexcel M1 Q4
12 marks Standard +0.3
4. \(A B\) is a light rod. Forces \(\mathbf { F } , \mathbf { G }\) and \(\mathbf { H }\), of magnitudes \(3 \mathrm {~N} , 2 \mathrm {~N}\) and 6 N respectively, act upwards at right angles to the rod in a vertical plane at points dividing \includegraphics[max width=\textwidth, alt={}, center]{977c24cc-8280-4881-8a62-65b7efd336ac-1_186_586_1741_1329} \(A B\) in the ratio \(1 : 4 : 2 : 4\), as shown.
A single force \(\mathbf { P }\) is applied downwards at the point \(C\) to keep the rod horizontal in equilibrium.
  1. State the magnitude of \(\mathbf { P }\).
  2. Show that \(A C : C B = 5 : 6\). Two particles, of weights 3 N and \(k \mathrm {~N}\), are now placed on the rod at \(A\) and \(B\) respectively, while the same upward forces \(\mathbf { F } , \mathbf { G }\) and \(\mathbf { H }\) act as before. It is found that a single downward force at the same point \(C\) as before keeps \(A B\) horizontal under gravity.
  3. Find the value of \(k\). \section*{MECHANICS 1 (A) TEST PAPER 4 Page 2}
Edexcel M1 Q5
13 marks Standard +0.3
  1. Two smooth spheres \(A\) and \(B\), of masses \(2 m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\) \includegraphics[max width=\textwidth, alt={}, center]{977c24cc-8280-4881-8a62-65b7efd336ac-2_253_323_255_1622}
    and the table being \(\frac { 2 } { 7 }\). \(B\) hangs freely on the end of the vertical portion of the string. \(A\) is now given an impulse, directed away from the pulley, of magnitude 5 m Ns.
    1. Show that the system starts to move with speed \(2.5 \mathrm {~ms} ^ { - 1 }\).
    2. State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal.
    Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,
  2. find the time taken for the system to come to rest.
  3. Find the distance travelled by \(A\) before it first comes to rest.
Edexcel M1 Q6
14 marks Standard +0.3
6. The diagram shows the velocity-time graph for a cyclist's journey. Each section has constant acceleration or deceleration and the three sections are of equal duration \(x\) seconds each.
Given that the total distance travelled is 792 m , \includegraphics[max width=\textwidth, alt={}, center]{977c24cc-8280-4881-8a62-65b7efd336ac-2_337_647_1128_1284}
  1. find the value of \(x\) and the acceleration for the first section of the journey. Another cyclist covers the same journey in three sections of equal duration, accelerating at \(\frac { 1 } { 11 } \mathrm {~ms} ^ { - 2 }\) for the first section, travelling at constant speed for the second section and decelerating at \(\frac { 1 } { 11 } \mathrm {~ms} ^ { - 2 }\) for the third section.
  2. Find the time taken by this cyclist to complete the journey.
  3. Show that the maximum speeds of both cyclists are the same.
Edexcel M1 Q7
14 marks Moderate -0.3
7. Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m }\) and \(( 12 \mathbf { i } + \mathbf { j } ) \mathbf { m }\) respectively, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the distance \(X Y\). A particle \(P\) of mass 2 kg moves from \(X\) to \(Y\) in 4 seconds, in a straight line at a constant speed.
  2. Show that the velocity vector of \(P\) is \(( 2 \mathbf { i } + 1 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). The particle continues beyond \(Y\) with the same constant velocity.
  3. Write down an expression for the position vector of \(P t\) seconds after leaving \(X\).
  4. Find the value of \(t\) when \(P\) is at the point with position vector \(( 16 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass 4 kg , which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.
  5. Find the velocity vector of the combined particle after the collision.
    (5 marks)
Edexcel M1 Q1
5 marks Moderate -0.8
  1. Two forces, both of magnitude 5 N , act on a particle in the directions with bearings \(000 ^ { \circ }\) and \(070 ^ { \circ }\), as shown. Calculate
    1. the magnitude of the resultant force on the particle,
    2. the bearing on which this resultant force acts.
    3. A uniform plank \(X Y\) has length 7 m and mass 2 kg . It is placed with the portion \(Z Y\) in contact with a \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-1_149_616_843_1334}
      horizontal surface, where \(Z Y = 2.8 \mathrm {~m}\). To prevent the \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-1_207_253_404_1505}
    \section*{MECHANICS 1 (A) TEST PAPER 5 Page 2}
Edexcel M1 Q5
15 marks Standard +0.3
  1. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. The point \(A\) has position vector \(6 \mathbf { j } \mathrm {~m}\) relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity ( \(5 \mathbf { i } + 2 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4 \mathrm { ims } ^ { - 1 }\).
    1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds.
    2. Show that the distance \(d \mathrm {~m}\) between \(P\) and \(Q\) at time \(t\) seconds is such that
    $$d ^ { 2 } = 5 t ^ { 2 } - 24 t + 36 .$$
  2. Find the value of \(t\) for which \(d ^ { 2 }\) is a minimum.
  3. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together.
Edexcel M1 Q6
15 marks Standard +0.3
6. \(A , B\) and \(C\) are three small spheres of equal radii and masses \(2 m , m\) and \(5 m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude 8 m Ns on \(B\).
  1. Find the speed with which \(B\) starts to move.
  2. Show that the speed of \(A\) after it collides with \(B\) is \(2 \mathrm {~ms} ^ { - 1 }\). After travelling \(3 \mathrm {~m} , B\) hits \(C\), which is then travelling towards \(B\) at \(2 \cdot 2 \mathrm {~ms} ^ { - 1 } . C\) is brought to rest by this impact.
  3. Show that the direction of \(B\) 's motion is reversed and find its new speed.
  4. Find how far \(B\) now travels before it collides with \(A\) again.
  5. State a modelling assumption that you have made about the spheres.
Edexcel M1 Q7
16 marks Standard +0.8
7. A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal as shown. A light string is attached to \(P\) and makes an angle of \(30 ^ { \circ }\) with the plane. When the tension in this string has magnitude \(k m g , P\) is \includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-2_268_474_1759_1407}
just on the point of moving up the plane.
  1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac { k \sqrt { } 3 - 1 } { \sqrt { } 3 - k }\).
  2. Given further that \(k = \frac { 3 \sqrt { } 3 } { 7 }\), deduce that \(\mu = \frac { \sqrt { } 3 } { 6 }\). The string is now removed.
  3. Determine whether \(P\) will move down the plane and, if it does, find its acceleration.
  4. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made.
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. \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-2_456_380_1862_395} 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)