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
CAIE M1 2002 June Q4
7 marks Standard +0.2
4 A box of mass 4.5 kg is pulled at a constant speed of \(2 \mathrm {~ms} ^ { - 1 }\) along a rough horizontal floor by a horizontal force of magnitude 15 N .
  1. Find the coefficient of friction between the box and the floor. The horizontal pulling force is now removed. Find
  2. the deceleration of the box in the subsequent motion,
  3. the distance travelled by the box from the instant the horizontal force is removed until the box comes to rest.
  4. A cyclist travels in a straight line from \(A\) to \(B\) with constant acceleration \(0.06 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). His speed at \(A\) is \(3 \mathrm {~ms} ^ { - 1 }\) and his speed at \(B\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find
    (a) the time taken by the cyclist to travel from \(A\) to \(B\),
    (b) the distance \(A B\).
  5. A car leaves \(A\) at the same instant as the cyclist. The car starts from rest and travels in a straight line to \(B\). The car reaches \(B\) at the same instant as the cyclist. At time \(t \mathrm {~s}\) after leaving \(A\) the speed of the car is \(k t ^ { 2 } \mathrm {~ms} ^ { - 1 }\), where \(k\) is a constant. Find
    (a) the value of \(k\),
    (b) the speed of the car at \(B\).
  6. A lorry \(P\) of mass 15000 kg climbs a straight hill of length 800 m at a steady speed. The hill is inclined at \(2 ^ { \circ }\) to the horizontal. For \(P\) 's journey from the bottom of the hill to the top, find
    (a) the gain in gravitational potential energy,
    (b) the work done by the driving force, which has magnitude 7000 N ,
    (c) the work done against the force resisting the motion.
  7. A second lorry, \(Q\), also has mass 15000 kg and climbs the same hill as \(P\). The motion of \(Q\) is subject to a constant resisting force of magnitude 900 N , and \(Q\) s speed falls from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(10 \mathrm {~ms} ^ { - 1 }\) at the top. Find the work done by the driving force as \(Q\) climbs from the bottom of the hill to the top. \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-3_483_231_1537_973} Particles \(A\) and \(B\), of masses 0.15 kg and 0.25 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The system is held at rest with the string taut and with \(A\) and \(B\) at the same horizontal level, as shown in the diagram. The system is then released.
  8. Find the downward acceleration of \(B\). After \(2 \mathrm {~s} B\) hits the floor and comes to rest without rebounding. The string becomes slack and \(A\) moves freely under gravity.
  9. Find the time that elapses until the string becomes taut again.
  10. Sketch on a single diagram the velocity-time graphs for both particles, for the period from their release until the instant that \(B\) starts to move upwards.
CAIE M1 2003 June Q1
4 marks Easy -1.2
1 A crate of mass 800 kg is lifted vertically, at constant speed, by the cable of a crane. Find
  1. the tension in the cable,
  2. the power applied to the crate in increasing the height by 20 m in 50 s .
CAIE M1 2003 June Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_405_384_550_884} Three coplanar forces of magnitudes \(10 \mathrm {~N} , 10 \mathrm {~N}\) and 6 N act at a point \(P\) in the directions shown in the diagram. \(P Q\) is the bisector of the angle between the two forces of magnitude 10 N .
  1. Find the component of the resultant of the three forces
    (a) in the direction of \(P Q\),
    (b) in the direction perpendicular to \(P Q\).
  2. Find the magnitude of the resultant of the three forces.
CAIE M1 2003 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_556_974_1548_587} The diagram shows the velocity-time graphs for the motion of two cyclists \(P\) and \(Q\), who travel in the same direction along a straight path. Both cyclists start from rest at the same point \(O\) and both accelerate at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Both then continue at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(Q\) starts his journey \(T\) seconds after \(P\).
  1. Show in a sketch of the diagram the region whose area represents the displacement of \(P\), from \(O\), at the instant when \(Q\) starts. Given that \(P\) has travelled 16 m at the instant when \(Q\) starts, find
  2. the value of \(T\),
  3. the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(10 \mathrm {~ms} ^ { - 1 }\).
CAIE M1 2003 June Q4
6 marks Moderate -0.8
4 A particle moves in a straight line. Its displacement \(t\) seconds after leaving the fixed point \(O\) is \(x\) metres, where \(x = \frac { 1 } { 2 } t ^ { 2 } + \frac { 1 } { 30 } t ^ { 3 }\). Find
  1. the speed of the particle when \(t = 10\),
  2. the value of \(t\) for which the acceleration of the particle is twice its initial acceleration.
CAIE M1 2003 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881} \(S _ { 1 }\) and \(S _ { 2 }\) are light inextensible strings, and \(A\) and \(B\) are particles each of mass 0.2 kg . Particle \(A\) is suspended from a fixed point \(O\) by the string \(S _ { 1 }\), and particle \(B\) is suspended from \(A\) by the string \(S _ { 2 }\). The particles hang in equilibrium as shown in the diagram.
  1. Find the tensions in \(S _ { 1 }\) and \(S _ { 2 }\). The string \(S _ { 1 }\) is cut and the particles fall. The air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.2 N .
  2. Find the acceleration of the particles and the tension in \(S _ { 2 }\).
CAIE M1 2003 June Q6
10 marks Standard +0.3
6 A small block of mass 0.15 kg moves on a horizontal surface. The coefficient of friction between the block and the surface is 0.025 .
  1. Find the frictional force acting on the block.
  2. Show that the deceleration of the block is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The block is struck from a point \(A\) on the surface and, 4 s later, it hits a boundary board at a point \(B\). The initial speed of the block is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the distance \(A B\). The block rebounds from the board with a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves along the line \(B A\). Find
  4. the speed with which the block passes through \(A\),
  5. the total distance moved by the block, from the instant when it was struck at \(A\) until the instant when it comes to rest.
CAIE M1 2003 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-4_257_988_267_580} The diagram shows a vertical cross-section \(A B C D\) of a surface. The parts \(A B\) and \(C D\) are straight and have lengths 2.5 m and 5.2 m respectively. \(A D\) is horizontal, and \(A B\) is inclined at \(60 ^ { \circ }\) to the horizontal. The points \(B\) and \(C\) are at the same height above \(A D\). The parts of the surface containing \(A B\) and \(B C\) are smooth. A particle \(P\) is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction \(A B\), and it subsequently reaches \(D\). The particle does not lose contact with the surface during this motion.
  1. Find the speed of \(P\) at \(B\).
  2. Show that the maximum height of the cross-section, above \(A D\), is less than 3.2 m .
  3. State briefly why \(P\) 's speed at \(C\) is the same as its speed at \(B\).
  4. The frictional force acting on the particle as it travels from \(C\) to \(D\) is 1.4 N . Given that the mass of \(P\) is 0.4 kg , find the speed with which \(P\) reaches \(D\).
CAIE M1 2004 June Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_200_588_267_781} A ring of mass 1.1 kg is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of magnitude 13 N at an angle \(\alpha\) below the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) (see diagram). The ring is in equilibrium.
  1. Find the frictional component of the contact force on the ring.
  2. Find the normal component of the contact force on the ring.
  3. Given that the equilibrium of the ring is limiting, find the coefficient of friction between the ring and the rod.
CAIE M1 2004 June Q2
6 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_684_257_1114_945} Coplanar forces of magnitudes \(250 \mathrm {~N} , 100 \mathrm {~N}\) and 300 N act at a point in the directions shown in the diagram. The resultant of the three forces has magnitude \(R \mathrm {~N}\), and acts at an angle \(\alpha ^ { \circ }\) anticlockwise from the force of magnitude 100 N . Find \(R\) and \(\alpha\).
[0pt] [6]
CAIE M1 2004 June Q3
6 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-3_727_899_267_625} A boy runs from a point \(A\) to a point \(C\). He pauses at \(C\) and then walks back towards \(A\) until reaching the point \(B\), where he stops. The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the boy's velocity at time \(t\) seconds after leaving \(A\). The boy runs and walks in the same straight line throughout.
  1. Find the distances \(A C\) and \(A B\).
  2. Sketch the graph of \(x\) against \(t\), where \(x\) metres is the boy's displacement from \(A\). Show clearly the values of \(t\) and \(x\) when the boy arrives at \(C\), when he leaves \(C\), and when he arrives at \(B\). [3]
CAIE M1 2004 June Q4
7 marks Moderate -0.8
4 The top of an inclined plane is at a height of 0.7 m above the bottom. A block of mass 0.2 kg is released from rest at the top of the plane and slides a distance of 2.5 m to the bottom. Find the kinetic energy of the block when it reaches the bottom of the plane in each of the following cases:
  1. the plane is smooth,
  2. the coefficient of friction between the plane and the block is 0.15 .
CAIE M1 2004 June Q5
7 marks Moderate -0.3
5 A particle \(P\) moves in a straight line that passes through the origin \(O\). The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 20 t - t ^ { 3 }\). At time \(t = 0\) the particle is at rest at a point whose displacement from \(O\) is - 36 m .
  1. Find an expression for the displacement of \(P\) from \(O\) in terms of \(t\).
  2. Find the displacement of \(P\) from \(O\) when \(t = 4\).
  3. Find the values of \(t\) for which the particle is at \(O\).
CAIE M1 2004 June Q6
8 marks Moderate -0.3
6 A car of mass 1200 kg travels along a horizontal straight road. The power of the car's engine is 20 kW . The resistance to the car's motion is 400 N .
  1. Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Show that the maximum possible speed of the car is \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done by the car's engine as the car travels from a point \(A\) to a point \(B\) is 1500 kJ .
  3. Given that the car is travelling at its maximum possible speed between \(A\) and \(B\), find the time taken to travel from \(A\) to \(B\).
CAIE M1 2004 June Q7
11 marks Standard +0.3
7 A particle \(P _ { 1 }\) is projected vertically upwards, from horizontal ground, with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle \(P _ { 2 }\) is projected vertically upwards from the top of a tower of height 25 m , with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the time for which \(P _ { 1 }\) is higher than the top of the tower,
  2. the velocities of the particles at the instant when the particles are at the same height,
  3. the time for which \(P _ { 1 }\) is higher than \(P _ { 2 }\) and is moving upwards.
CAIE M1 2005 June Q1
3 marks Moderate -0.5
1 A small block is pulled along a rough horizontal floor at a constant speed of \(1.5 \mathrm {~ms} ^ { - 1 }\) by a constant force of magnitude 30 N acting at an angle of \(\theta ^ { \circ }\) upwards from the horizontal. Given that the work done by the force in 20 s is 720 J , calculate the value of \(\theta\).
CAIE M1 2005 June Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{9bb53600-e7ba-4228-84ae-d8ddf7649387-2_350_688_493_731} Three coplanar forces act at a point. The magnitudes of the forces are \(5 \mathrm {~N} , 6 \mathrm {~N}\) and 7 N , and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the three forces. \(3 A\) and \(B\) are points on the same line of greatest slope of a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. \(A\) is higher up the plane than \(B\) and the distance \(A B\) is 2.25 m . A particle \(P\), of mass \(m \mathrm {~kg}\), is released from rest at \(A\) and reaches \(B 1.5\) s later. Find the coefficient of friction between \(P\) and the plane. [6]
CAIE M1 2005 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{9bb53600-e7ba-4228-84ae-d8ddf7649387-2_478_597_1398_776} Particles \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. Particle \(A\) hangs freely and particle \(B\) is in contact with the table (see diagram).
  1. The system is in limiting equilibrium with the string taut and \(A\) about to move downwards. Find the coefficient of friction between \(B\) and the table. A force now acts on particle \(B\). This force has a vertical component of 1.8 N upwards and a horizontal component of \(X\) N directed away from the pulley.
  2. The system is now in limiting equilibrium with the string taut and \(A\) about to move upwards. Find \(X\).
CAIE M1 2005 June Q5
7 marks Moderate -0.8
5 A particle \(P\) moves along the \(x\)-axis in the positive direction. The velocity of \(P\) at time \(t \mathrm {~s}\) is \(0.03 t ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 5\) the displacement of \(P\) from the origin \(O\) is 2.5 m .
  1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\).
  2. Find the velocity of \(P\) when its displacement from \(O\) is 11.25 m .
CAIE M1 2005 June Q6
9 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{9bb53600-e7ba-4228-84ae-d8ddf7649387-3_735_1484_625_333} The diagram shows the velocity-time graph for a lift moving between floors in a building. The graph consists of straight line segments. In the first stage the lift travels downwards from the ground floor for 5 s , coming to rest at the basement after travelling 10 m .
  1. Find the greatest speed reached during this stage. The second stage consists of a 10 s wait at the basement. In the third stage, the lift travels upwards until it comes to rest at a floor 34.5 m above the basement, arriving 24.5 s after the start of the first stage. The lift accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 3 s of the third stage, reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  2. the value of \(V\),
  3. the time during the third stage for which the lift is moving at constant speed,
  4. the deceleration of the lift in the final part of the third stage.
CAIE M1 2005 June Q7
12 marks Standard +0.3
7 A car of mass 1200 kg travels along a horizontal straight road. The power provided by the car's engine is constant and equal to 20 kW . The resistance to the car's motion is constant and equal to 500 N . The car passes through the points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The car takes 30.5 s to travel from \(A\) to \(B\).
  1. Find the acceleration of the car at \(A\).
  2. By considering work and energy, find the distance \(A B\).
CAIE M1 2006 June Q1
4 marks Moderate -0.3
1 A car of mass 1200 kg travels on a horizontal straight road with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Given that the car's speed increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while travelling a distance of 525 m , find the value of \(a\). The car's engine exerts a constant driving force of 900 N . The resistance to motion of the car is constant and equal to \(R \mathrm {~N}\).
  2. Find \(R\).
CAIE M1 2006 June Q2
5 marks Moderate -0.8
2 A motorcyclist starts from rest at \(A\) and travels in a straight line until he comes to rest again at \(B\). The velocity of the motorcyclist \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t - 0.01 t ^ { 2 }\). Find
  1. the time taken for the motorcyclist to travel from \(A\) to \(B\),
  2. the distance \(A B\).
CAIE M1 2006 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-2_508_1011_1096_568} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of horizontal forces of magnitudes \(F\) N, \(F\) N, \(G\) N and 12 N acting in the directions shown. Find the values of \(F\) and \(G\). [6]
CAIE M1 2006 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_568_1084_269_532} The diagram shows the velocity-time graph for the motion of a small stone which falls vertically from rest at a point \(A\) above the surface of liquid in a container. The downward velocity of the stone \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone hits the surface of the liquid with velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0.7\). It reaches the bottom of the container with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 1.2\).
  1. Find
    (a) the height of \(A\) above the surface of the liquid,
    (b) the depth of liquid in the container.
  2. Find the deceleration of the stone while it is moving in the liquid.
  3. Given that the resistance to motion of the stone while it is moving in the liquid has magnitude 0.7 N , find the mass of the stone.