Questions — CAIE M1 (732 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 2018 November Q4
4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang freely below it. The system is released from rest, with both particles 0.8 m above horizontal ground. Particle \(A\) reaches the ground with a speed of \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the tension in the string during the motion before \(A\) reaches the ground.
  2. Find the value of \(m\).
CAIE M1 2018 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{98a5537b-d503-4a42-bbfe-0bd221084ee0-06_449_654_260_742} Coplanar forces, of magnitudes \(15 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , act at a point \(B\) on the line \(A B C\) in the directions shown in the diagram.
  1. Find the magnitude and direction of the resultant force.
  2. The force of magnitude 15 N is now replaced by a force of magnitude \(F \mathrm {~N}\) acting in the same direction. The new resultant force has zero component in the direction \(B C\). Find the value of \(F\), and find also the magnitude and direction of the new resultant force.
CAIE M1 2018 November Q6
6 A particle is projected from a point \(P\) with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope \(P Q R\) of a rough inclined plane. The distances \(P Q\) and \(Q R\) are both equal to 0.8 m . The particle takes 0.6 s to travel from \(P\) to \(Q\) and 1 s to travel from \(Q\) to \(R\).
  1. Show that the deceleration of the particle is \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and hence find \(u\), giving your answer as an exact fraction.
  2. Given that the plane is inclined at \(3 ^ { \circ }\) to the horizontal, find the value of the coefficient of friction between the particle and the plane.
CAIE M1 2018 November Q7
7 A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 5.4 - 1.62 t$$
  1. Find the positive value of \(t\) at which the velocity of the particle is zero, giving your answer as an exact fraction.
  2. Find the velocity of the particle at \(t = 10\) and sketch the velocity-time graph for the first ten seconds of the motion.
  3. Find the total distance travelled during the first ten seconds of the motion.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2018 November Q2
2 A block of mass 5 kg is being pulled by a rope up a rough plane inclined at \(6 ^ { \circ }\) to the horizontal. The rope is parallel to a line of greatest slope of the plane and the block is moving at constant speed. The coefficient of friction between the block and the plane is 0.3 . Find the tension in the rope.
CAIE M1 2018 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{26c1e840-1eed-46d2-b007-1ec94d7b7c4a-04_789_1151_260_497} The velocity of a particle moving in a straight line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds. The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = T\). The graph consists of four straight line segments. The particle reaches its maximum velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(t = 10\).
  1. Find the acceleration of the particle during the first 2 seconds.
  2. Find the value of \(V\).
    At \(t = 6\), the particle is instantaneously at rest at the point \(A\). At \(t = T\), the particle comes to rest at the point \(B\). At \(t = 0\) the particle starts from rest at a point one third of the way from \(A\) to \(B\).
  3. Find the distance \(A B\) and hence find the value of \(T\).
    \includegraphics[max width=\textwidth, alt={}, center]{26c1e840-1eed-46d2-b007-1ec94d7b7c4a-06_392_625_260_758} Two particles \(P\) and \(Q\), of masses 0.4 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The coefficient of friction between \(P\) and the plane is 0.5 . The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). Particle \(P\) lies on the plane and particle \(Q\) hangs vertically. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude \(X \mathrm {~N}\), acting directly down the plane, is applied to \(P\).
  4. Show that the greatest value of \(X\) for which \(P\) remains stationary is 6.2.
  5. Given instead that \(X = 0.8\), find the acceleration of \(P\).
CAIE M1 2018 November Q5
5 A particle moves in a straight line starting from a point \(O\) with initial velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$a = 1.2 t ^ { \frac { 1 } { 2 } } - 0.6 t$$
  1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\).
  2. Find the velocity of the particle when its acceleration is maximum (you do not need to verify that the acceleration is a maximum rather than a minimum).
CAIE M1 2018 November Q6
6 A car of mass 1200 kg is driving along a straight horizontal road at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant resistance to motion of 350 N .
  1. Find the power of the car's engine.
    The car comes to a hill inclined at \(1 ^ { \circ }\) to the horizontal, still travelling at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. The car starts to descend the hill with reduced power and with an acceleration of \(0.12 \mathrm {~ms} ^ { - 2 }\). Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill.
  3. When the car is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains 350 N, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2018 November Q7
7 A particle of mass 0.3 kg is released from rest above a tank containing water. The particle falls vertically, taking 0.8 s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is 1.25 m . The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is 1.2 J .
  1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank.
    When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the particle rises through the water, it experiences a constant resistance force of 1.8 N . The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.
  2. Find the value of \(t\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2018 November Q3
3 A particle of mass 1.2 kg moves in a straight line \(A B\). It is projected with speed \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) towards \(B\) and experiences a resistance force. The work done against this resistance force in moving from \(A\) to \(B\) is 25 J .
  1. Given that \(A B\) is horizontal, find the speed of the particle at \(B\).
  2. It is given instead that \(A B\) is inclined at \(30 ^ { \circ }\) below the horizontal and that the speed of the particle at \(B\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance force remains the same. Find the distance \(A B\).
CAIE M1 2018 November Q4
4 A runner sets off from a point \(P\) at time \(t = 0\), where \(t\) is in seconds. The runner starts from rest and accelerates at \(1.2 \mathrm {~ms} ^ { - 2 }\) for 5 s . For the next 12 s the runner moves at constant speed before decelerating uniformly over a period of 3 s , coming to rest at \(Q\). A cyclist sets off from \(P\) at time \(t = 10\) and accelerates uniformly for 10 s , before immediately decelerating uniformly to rest at \(Q\) at time \(t = 30\).
  1. Sketch the velocity-time graph for the runner and show that the distance \(P Q\) is 96 m .
    \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-06_821_1451_708_388}
  2. Find the magnitude of the acceleration of the cyclist.
CAIE M1 2018 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-08_538_414_260_868} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley with the particles hanging freely below it. \(Q\) is held at rest with the string taut at a height of \(h \mathrm {~m}\) above a horizontal floor (see diagram). \(Q\) is now released and both particles start to move. The pulley is sufficiently high so that \(P\) does not reach it at any stage. The time taken for \(Q\) to reach the floor is 0.6 s .
  1. Find the acceleration of \(Q\) before it reaches the floor and hence find the value of \(h\).
    \(Q\) remains at rest when it reaches the floor, and \(P\) continues to move upwards.
  2. Find the velocity of \(P\) at the instant when \(Q\) reaches the floor and the total time taken from the instant at which \(Q\) is released until the string becomes taut again.
CAIE M1 2018 November Q6
6 A van of mass 3200 kg travels along a horizontal road. The power of the van's engine is constant and equal to 36 kW , and there is a constant resistance to motion acting on the van.
  1. When the speed of the van is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the resistance force.
    When the van is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it begins to ascend a hill inclined at \(1.5 ^ { \circ }\) to the horizontal. The power is increased and the resistance force is still equal to the value found in part (i).
  2. Find the power required to maintain this speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. The engine is now stopped, with the van still travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the van decelerates to rest. Find the distance the van moves up the hill from the point at which the engine is stopped until it comes to rest.
CAIE M1 2018 November Q7
7 A particle moves in a straight line. The particle is initially at rest at a point \(O\) on the line. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle is given by \(a = 25 - t ^ { 2 }\) for \(0 \leqslant t \leqslant 9\).
  1. Find the maximum velocity of the particle in this time period.
  2. Find the total distance travelled until the maximum velocity is reached.
    The acceleration of the particle for \(t > 9\) is given by \(a = - 3 t ^ { - \frac { 1 } { 2 } }\).
  3. Find the velocity of the particle when \(t = 25\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2019 November Q1
1
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value for the acceleration due to gravity is needed, use \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. 1 A crane is lifting a load of 1250 kg vertically at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the power of the crane is a constant 20 kW , find the value of \(V\).
CAIE M1 2019 November Q2
2 The total mass of a cyclist and her bicycle is 75 kg . The cyclist ascends a straight hill of length 0.7 km inclined at \(1.5 ^ { \circ }\) to the horizontal. Her speed at the bottom of the hill is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at the top it is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J . The cyclist exerts a constant force of magnitude \(F \mathrm {~N}\) in the direction of motion. Find \(F\).
CAIE M1 2019 November Q3
3 A block of mass 3 kg is at rest on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.
  1. Find the coefficient of friction between the block and the plane.
    The force of magnitude 15 N is now replaced by a force of magnitude \(X \mathrm {~N}\) acting up the line of greatest slope.
  2. Find the greatest value of \(X\) for which the block does not move.
    \includegraphics[max width=\textwidth, alt={}, center]{dd1828e1-5b90-4584-92de-f00f9c4f9657-06_332_967_260_589} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).
CAIE M1 2019 November Q6
6 A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.
  1. Find the velocity of the particle at the instant when it reaches the surface of the water.
  2. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank.
  3. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water.
CAIE M1 2019 November Q7
7 A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 0.6 t ^ { 2 } - 0.12 t ^ { 3 }$$
  1. Show that the distance \(O P\) is 6.25 m .
    On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(s \mathrm {~m}\), where $$s = k t ^ { 3 } + c t ^ { 5 }$$ It is given that the particle passes point \(P\) with velocity \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 5\).
  2. Find the values of the constants \(k\) and \(c\).
  3. Find the acceleration of the particle at time \(t = 5\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2019 November Q1
1 A particle moves in a straight line. The displacement of the particle at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$s = t ^ { 3 } - 6 t ^ { 2 } + 4 t$$ Find the velocity of the particle at the instant when its acceleration is zero.
CAIE M1 2019 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{cb1cc219-608f-4f11-ab2c-97cc8f0798c7-04_602_1249_260_447} The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes a point \(O\) at time \(t = 0\) with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The tractor accelerates to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 5 s , and then travels at this speed for a further 25 s . The tractor then accelerates to a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 5 s . The tractor then decelerates to rest over a period of 15 s .
  1. Given that the acceleration of the tractor between \(t = 30\) and \(t = 35\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the value of \(V\).
  2. Given also that the total distance covered by the tractor in the 50 seconds of motion is 375 m , find the value of \(U\).
    \includegraphics[max width=\textwidth, alt={}, center]{cb1cc219-608f-4f11-ab2c-97cc8f0798c7-05_465_611_264_767} A particle \(P\) of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N , acting vertically upwards. The particle is attached to two strings \(P A\) and \(P B\) of lengths 0.9 m and 1.2 m respectively. The points \(A\) and \(B\) lie on the plane and angle \(A P B = 90 ^ { \circ }\) (see diagram). Find the tension in each of the strings.
CAIE M1 2019 November Q4
4 A lorry of mass 25000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.
  1. Find the power required to maintain a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The lorry comes to a straight hill inclined at \(2 ^ { \circ }\) to the horizontal. The driver switches off the engine of the lorry at the point \(A\) which is at the foot of the hill. Point \(B\) is further up the hill. The speeds of the lorry at \(A\) and \(B\) are \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The resistance force is still 3000 N .
  2. Use an energy method to find the height of \(B\) above the level of \(A\).
CAIE M1 2019 November Q5
5 Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). One second later particle \(B\) is dropped from rest from a height of 40 m .
  1. Find the height above the ground at which the two particles collide.
  2. Find the difference in the speeds of the two particles at the instant when the collision occurs.
CAIE M1 2019 November Q6
6 A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac { 24 } { 25 }\). The force is applied for a period of 5 s , during which time the block moves a distance of 4.5 m .
  1. Find the magnitude of the frictional force on the block.
  2. Show that the coefficient of friction between the block and the plane is 0.165 , correct to 3 significant figures.
  3. When the block has moved a distance of 4.5 m , the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion.
    \includegraphics[max width=\textwidth, alt={}, center]{cb1cc219-608f-4f11-ab2c-97cc8f0798c7-12_512_1097_258_523} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 } . P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.
  4. Find the tension in the string and the magnitude of the acceleration of the particles.
    \(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.
  5. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor.
  6. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2019 November Q1
1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.