Questions M1 (2067 questions)

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CAIE M1 2024 November Q6
10 marks Challenging +1.2
6 A particle, \(P\), travels in a straight line, starting from a point \(O\) with velocity \(6 \mathrm {~ms} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = - 1.5 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 1 , \\ a = 1.5 t ^ { \frac { 1 } { 2 } } - 3 t ^ { - \frac { 1 } { 2 } } & \text { for } t > 1 . \end{array}$$
  1. Find the velocity of \(P\) at \(t = 1\).
  2. Given that there is no change in the velocity of \(P\) when \(t = 1\), find an expression for the velocity of \(P\) for \(t > 1\). \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-11_2725_35_99_20}
  3. Given that the velocity of \(P\) is positive for \(t \leqslant 4\), find the total distance travelled between \(t = 0\) and \(t = 4\). \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_723_762_248_653} Two particles, \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the bottom of a rough plane inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = 0.6\). Particle \(A\) lies on the plane, and particle \(B\) hangs vertically below the pulley, 0.25 m above horizontal ground. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between \(A\) and the plane is 1.125 . Particle \(A\) is released from rest.
    1. Find the tension in the string and the magnitude of the acceleration of the particles. \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_2716_38_109_2012}
    2. When \(B\) reaches the ground, it comes to rest. Find the total distance that \(A\) travels down the plane from when it is released until it comes to rest. You may assume that \(A\) does not reach the pulley.
      If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-14_2715_31_106_2016}
CAIE M1 2020 Specimen Q1
4 marks Easy -1.3
1 A particle \(P\) is projected vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground.
  1. Find the greatest height above the ground reached by \(P\).
  2. Find the total time from projection until \(P\) returns to the ground.
CAIE M1 2020 Specimen Q2
5 marks Moderate -0.8
2 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg .
  1. The car is moving along a straight level road at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
  2. The car travels at a constant speed down a hill inclined at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta ^ { \circ } = \frac { 1 } { 20 }\), with the engine working at 31.5 kW . Find the speed of the car.
CAIE M1 2020 Specimen Q3
6 marks Standard +0.3
3 Three small smooth spheres \(A , B\) and \(C\) of equal radii and of masses \(4 \mathrm {~kg} , 2 \mathrm {~kg}\) and 3 kg respectively, lie in that order in a straight line on a smooth horizontal plane. Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\). After the collison with \(B\), sphere \(A\) continues to move in the same direction but with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(B\) after this collison.
    Sphere \(B\) collides with \(C\). In this collison these two spheres coalesce to form an object \(D\).
  2. Find the speed of \(D\) after this collision.
  3. Show that the total loss of kinetic energy in the system due to the two collisions is 38.4 J .
CAIE M1 2020 Specimen Q4
6 marks Standard +0.3
4 A particle of mass 20 kg is on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. A force of magnitude 25 N , acting at an angle of \(20 ^ { \circ }\) above a line of greatest slope of the plane, is used to prevent the particle from sliding down the plane. The coefficient of friction between the particle and the plane is \(\mu\).
  1. Complete the diagram below to show all the forces acting on the particle. \includegraphics[max width=\textwidth, alt={}, center]{87b42689-791c-4f4e-a36e-bfae3191ca11-06_495_615_543_726}
  2. Find the least possible value of \(\mu\).
CAIE M1 2020 Specimen Q5
9 marks Standard +0.3
5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle of \(\sin ^ { - 1 } ( 0.1 )\) to the horizontal. The car and the trailer are connected by a light rigid tow-bar which is parallel to the road. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 300 N and 100 N respectively.
  1. Find the acceleration of the system and the tension in the tow-bar.
  2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. Find the time, in seconds, before the system comes to rest and the force in the tow-bar during this time.
CAIE M1 2020 Specimen Q6
11 marks Moderate -0.3
6 A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\ v = k & \text { for } 4 \leqslant t \leqslant 14 \\ v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.
  1. Find \(k\).
  2. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
  3. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
  4. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).
CAIE M1 2020 Specimen Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{87b42689-791c-4f4e-a36e-bfae3191ca11-12_244_668_264_701} Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m (see diagram). The particles are released from rest with both sections of the string taut.
  1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
  2. It is given instead that the surface is rough and that the speed of \(A\) immediately before it reaches the pulley is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction as \(A\) moves from rest to the pulley is 2 J . Use an energy method to find \(v\).
CAIE M1 2002 June Q1
3 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-2_300_748_274_708} One end of a light inextensible string is attached to a ring which is threaded on a fixed horizontal bar. The string is used to pull the ring along the bar at a constant speed of \(0.4 \mathrm {~ms} ^ { - 1 }\). The string makes a constant angle of \(30 ^ { \circ }\) with the bar and the tension in the string is 5 N (see diagram). Find the work done by the tension in 10 s .
CAIE M1 2002 June Q2
5 marks Moderate -0.8
2 A basket of mass 5 kg slides down a slope inclined at \(12 ^ { \circ }\) to the horizontal. The coefficient of friction between the basket and the slope is 0.2 .
  1. Find the frictional force acting on the basket.
  2. Determine whether the speed of the basket is increasing or decreasing.
CAIE M1 2002 June Q3
5 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-2_368_584_1302_794} Two forces, each of magnitude 10 N , act at a point \(O\) in the directions of \(O A\) and \(O B\), as shown in the diagram. The angle between the forces is \(\theta\). The resultant of these two forces has magnitude 12 N .
  1. Find \(\theta\).
  2. Find the component of the resultant force in the direction of \(O A\).
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
    1. the time taken by the cyclist to travel from \(A\) to \(B\),
    2. the distance \(A B\).
    3. 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 \(B\).
      1. 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 ,
      2. the work done against the force resisting the motion.
      3. 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.
        1. 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.
        2. Find the time that elapses until the string becomes taut again.
        3. 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
    1. in the direction of \(P Q\),
    2. in the direction perpendicular to \(P Q\).
    3. 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 \(B\).