Questions M1 (1912 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 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
CAIE M1 2008 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_478_1041_269_552}
\(O A B C\) is a vertical cross-section of a smooth surface. The straight part \(O A\) has length 2.4 m and makes an angle of \(50 ^ { \circ }\) with the horizontal. \(A\) and \(C\) are at the same horizontal level and \(B\) is the lowest point of the cross-section (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(O\) and moves on the surface. \(P\) remains in contact with the surface until it leaves the surface at \(C\). Find
  1. the kinetic energy of \(P\) at \(A\),
  2. the speed of \(P\) at \(C\). The greatest speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the depth of \(B\) below the horizontal through \(A\) and \(C\).
CAIE M1 2008 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_314_867_1457_639} A block \(B\) of mass 0.6 kg and a particle \(A\) of mass 0.4 kg are attached to opposite ends of a light inextensible string. The block is held at rest on a rough horizontal table, and the coefficient of friction between the block and the table is 0.5 . The string passes over a small smooth pulley \(C\) at the edge of the table and \(A\) hangs in equilibrium vertically below \(C\). The part of the string between \(B\) and \(C\) is horizontal and the distance \(B C\) is 3 m (see diagram). \(B\) is released and the system starts to move.
  1. Find the acceleration of \(B\) and the tension in the string.
  2. Find the time taken for \(B\) to reach the pulley.
CAIE M1 2008 June Q6
6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find
  1. the greatest height above the ground reached by \(P\),
  2. the speed with which \(P\) reaches the ground,
  3. the total work done on \(P\) by the air resistance.
CAIE M1 2008 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543} An object \(P\) travels from \(A\) to \(B\) in a time of 80 s . The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\). The graph consists of straight line segments for the intervals \(0 \leqslant t \leqslant 10\) and \(30 \leqslant t \leqslant 80\), and a curved section whose equation is \(v = - 0.01 t ^ { 2 } + 0.5 t - 1\) for \(10 \leqslant t \leqslant 30\). Find
  1. the maximum velocity of \(P\),
  2. the distance \(A B\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2009 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_388_565_264_790} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground.
CAIE M1 2009 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_349_1139_1011_502} A crate \(C\) is pulled at constant speed up a straight inclined path by a constant force of magnitude \(F \mathrm {~N}\), acting upwards at an angle of \(15 ^ { \circ }\) to the path. \(C\) passes through points \(P\) and \(Q\) which are 100 m apart (see diagram). As \(C\) travels from \(P\) to \(Q\) the work done against the resistance to \(C\) 's motion is 900 J , and the gain in \(C\) 's potential energy is 2100 J . Write down the work done by the pulling force as \(C\) travels from \(P\) to \(Q\), and hence find the value of \(F\).
CAIE M1 2009 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_492_606_1763_772} Forces of magnitudes \(7 \mathrm {~N} , 10 \mathrm {~N}\) and 15 N act on a particle in the directions shown in the diagram.
  1. Find the component of the resultant of the three forces
    (a) in the \(x\)-direction,
    (b) in the \(y\)-direction.
  2. Hence find the direction of the resultant.
    \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_414_833_267_657} A block of mass 8 kg is at rest on a plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).
  3. Given that the tension in the string is 13 N , find the frictional and normal components of the force exerted on the block by the plane. The string is cut; the block remains at rest, but is on the point of slipping down the plane.
  4. Find the coefficient of friction between the block and the plane.
CAIE M1 2009 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_165_1417_1320_365} A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest at the top \(A\) of a straight path \(A B\), and freewheels (moves without pedalling or braking) down the path to \(B\). The path \(A B\) is inclined at \(2.6 ^ { \circ }\) to the horizontal and is of length 250 m (see diagram).
  1. Given that the cyclist passes through \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\), find the gain in kinetic energy and the loss in potential energy of the cyclist and his machine. Hence find the work done against the resistance to motion of the cyclist and his machine. The cyclist continues to freewheel along a horizontal straight path \(B D\) until he reaches the point \(C\), where the distance \(B C\) is \(d \mathrm {~m}\). His speed at \(C\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant, and is the same on \(B D\) as on \(A B\).
  2. Find the value of \(d\). The cyclist starts to pedal at \(C\), generating 425 W of power.
  3. Find the acceleration of the cyclist immediately after passing through \(C\).
CAIE M1 2009 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-4_712_526_264_813} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s .
  1. Find the acceleration of \(A\) as it falls. The mass of \(A\) is 0.45 kg . Find
  2. the tension in the string while \(A\) is falling,
  3. the mass of \(B\),
  4. the maximum height above the floor reached by \(B\).
CAIE M1 2009 June Q7
7 A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(A B\) the velocity of the particle is \(\left( 0.5 t - 0.01 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(t \mathrm {~s}\) is the time after leaving \(A\).
  1. Given that the acceleration of \(P\) at \(B\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the time taken for \(P\) to travel from \(A\) to \(B\). The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that \(P\) reaches \(C\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken for \(P\) to travel from \(B\) to \(C\).
    \(P\) travels with constant deceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(C\) to \(D\). Given that the distance \(C D\) is 300 m , find
  3. the speed with which \(P\) reaches \(D\),
  4. the distance \(A D\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2010 June Q1
1 A car of mass 1150 kg travels up a straight hill inclined at \(1.2 ^ { \circ }\) to the horizontal. The resistance to motion of the car is 975 N . Find the acceleration of the car at an instant when it is moving with speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at a power of 35 kW .
CAIE M1 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-2_661_1351_479_397} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes 3 s to return to its starting position. Find
  1. the acceleration of the tool during the first 2 s of the motion,
  2. the distance the tool moves forward while cutting,
  3. the greatest speed of the tool during the return to its starting position.
CAIE M1 2010 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-2_241_511_1676_817} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at \(45 ^ { \circ }\) to the horizontal (see diagram).
  1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N , correct to 3 significant figures.
  2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.
CAIE M1 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-3_755_561_248_790} Coplanar forces of magnitudes \(250 \mathrm {~N} , 160 \mathrm {~N}\) and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction.
\(5 P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(P Q\).
  1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\).
  2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J . Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\).
CAIE M1 2010 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.
  1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.
CAIE M1 2010 June Q7
7 A vehicle is moving in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after the vehicle starts is given by $$\begin{aligned} & v = A \left( t - 0.05 t ^ { 2 } \right) \quad \text { for } 0 \leqslant t \leqslant 15 ,
& v = \frac { B } { t ^ { 2 } } \quad \text { for } t \geqslant 15 , \end{aligned}$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is 225 m .
  1. Find the value of \(A\) and show that \(B = 3375\).
  2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geqslant 15\).
  3. Find the speed of the vehicle when it has travelled a total distance of 315 m . \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-2_661_1351_479_397} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes 3 s to return to its starting position. Find
  1. the acceleration of the tool during the first 2 s of the motion,
  2. the distance the tool moves forward while cutting,
  3. the greatest speed of the tool during the return to its starting position.
CAIE M1 2010 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-2_241_511_1676_817} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at \(45 ^ { \circ }\) to the horizontal (see diagram).
  1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N , correct to 3 significant figures.
  2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.
CAIE M1 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-3_755_561_248_790} Coplanar forces of magnitudes \(250 \mathrm {~N} , 160 \mathrm {~N}\) and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction.
\(5 P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(P Q\).
  1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\).
  2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J . Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\).
CAIE M1 2010 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.
  1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.
CAIE M1 2010 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-2_582_751_255_696} Three coplanar forces act at a point. The magnitudes of the forces are \(5.5 \mathrm {~N} , 6.8 \mathrm {~N}\) and 7.3 N , and the directions in which the forces act are as shown in the diagram. Given that the resultant of the three forces is in the same direction as the force of magnitude 6.8 N , find the value of \(\alpha\) and the magnitude of the resultant.
CAIE M1 2010 June Q2
2 A particle starts at a point \(O\) and moves along a straight line. Its velocity \(t\) s after leaving \(O\) is \(\left( 1.2 t - 0.12 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the displacement of the particle from \(O\) when its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2010 June Q3
3 A load is pulled along a horizontal straight track, from \(A\) to \(B\), by a force of magnitude \(P \mathrm {~N}\) which acts at an angle of \(30 ^ { \circ }\) upwards from the horizontal. The distance \(A B\) is 80 m . The speed of the load is constant and equal to \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(A\) to the mid-point \(M\) of \(A B\).
  1. For the motion from \(A\) to \(M\) the value of \(P\) is 25 . Calculate the work done by the force as the load moves from \(A\) to \(M\). The speed of the load increases from \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(M\) towards \(B\). For the motion from \(M\) to \(B\) the value of \(P\) is 50 and the work done against resistance is the same as that for the motion from \(A\) to \(M\). The mass of the load is 35 kg .
  2. Find the gain in kinetic energy of the load as it moves from \(M\) to \(B\) and hence find the speed with which it reaches \(B\).
CAIE M1 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-3_499_567_260_788} The diagram shows a vertical cross-section of a triangular prism which is fixed so that two of its faces are inclined at \(60 ^ { \circ }\) to the horizontal. One of these faces is smooth and one is rough. Particles \(A\) and \(B\), of masses 0.36 kg and 0.24 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the highest point of the cross-section. \(B\) is held at rest at a point of the cross-section on the rough face and \(A\) hangs freely in contact with the smooth face (see diagram). \(B\) is released and starts to move up the face with acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. By considering the motion of \(A\), show that the tension in the string is 3.03 N , correct to 3 significant figures.
  2. Find the coefficient of friction between \(B\) and the rough face, correct to 2 significant figures.
CAIE M1 2010 June Q5
5 A ball moves on the horizontal surface of a billiards table with deceleration of constant magnitude \(d \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The ball starts at \(A\) with speed \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and reaches the edge of the table at \(B , 1.2 \mathrm {~s}\) later, with speed \(1.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance \(A B\) and the value of \(d\).
    \(A B\) is at right angles to the edge of the table containing \(B\). The table has a low wall along each of its edges and the ball rebounds from the wall at \(B\) and moves directly towards \(A\). The ball comes to rest at \(C\) where the distance \(B C\) is 2 m .
  2. Find the speed with which the ball starts to move towards \(A\) and the time taken for the ball to travel from \(B\) to \(C\).
  3. Sketch a velocity-time graph for the motion of the ball, from the time the ball leaves \(A\) until it comes to rest at \(C\), showing on the axes the values of the velocity and the time when the ball is at \(A\), at \(B\) and at \(C\).