Questions — CAIE M1 (732 questions)

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CAIE M1 2022 March Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{19a41291-2692-48f4-86af-bb4930353959-08_645_611_258_767} Four coplanar forces act at a point. The magnitudes of the forces are \(10 \mathrm {~N} , F \mathrm {~N} , G \mathrm {~N}\) and \(2 F \mathrm {~N}\). The directions of the forces are as shown in the diagram.
  1. Given that the forces are in equilibrium, find the values of \(F\) and \(G\).
  2. Given instead that \(F = 3\), find the value of \(G\) for which the resultant of the forces is perpendicular to the 10 N force.
CAIE M1 2022 March Q6
6 A cyclist starts from rest at a fixed point \(O\) and moves in a straight line, before coming to rest \(k\) seconds later. The acceleration of the cyclist at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 2 t ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 5 } t ^ { \frac { 1 } { 2 } }\) for \(0 < t \leqslant k\).
  1. Find the value of \(k\).
  2. Find the maximum speed of the cyclist.
  3. Find an expression for the displacement from \(O\) in terms of \(t\). Hence find the total distance travelled by the cyclist from the time at which she reaches her maximum speed until she comes to rest.
CAIE M1 2022 March Q7
7 A bead, \(A\), of mass 0.1 kg is threaded on a long straight rigid wire which is inclined at \(\sin ^ { - 1 } \left( \frac { 7 } { 25 } \right)\) to the horizontal. \(A\) is released from rest and moves down the wire. The coefficient of friction between \(A\) and the wire is \(\mu\). When \(A\) has travelled 0.45 m down the wire, its speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(\mu = 0.25\).
    Another bead, \(B\), of mass 0.5 kg is also threaded on the wire. At the point where \(A\) has travelled 0.45 m down the wire, it hits \(B\) which is instantaneously at rest on the wire. \(A\) is brought to instantaneous rest in the collision. The coefficient of friction between \(B\) and the wire is 0.275 .
  2. Find the time from when the collision occurs until \(A\) collides with \(B\) 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 2023 March Q1
1 A crate of mass 200 kg is being pulled at constant speed along horizontal ground by a horizontal rope attached to a winch. The winch is working at a constant rate of 4.5 kW and there is a constant resistance to the motion of the crate of magnitude 600 N .
  1. Find the time that it takes for the crate to move a distance of 15 m .
    The rope breaks after the crate has moved 15 m .
  2. Find the time taken, after the rope breaks, for the crate to come to rest.
CAIE M1 2023 March Q2
2 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Find the speed of \(P\) when it is 10 m above the ground.
    At the same instant that \(P\) is projected, a second particle \(Q\) is dropped from a height of 18 m above the ground in the same vertical line as \(P\).
  2. Find the height above the ground at which the two particles collide.
CAIE M1 2023 March Q3
3 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 \(a = 4 t ^ { \frac { 1 } { 2 } }\).
  1. Find the speed of the particle when \(t = 9\).
  2. Find the time after leaving \(O\) at which the speed (in metres per second) and the distance travelled (in metres) are numerically equal.
CAIE M1 2023 March Q4
4 A toy railway locomotive of mass 0.8 kg is towing a truck of mass 0.4 kg on a straight horizontal track at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant resistance force of magnitude 0.2 N on the locomotive, but no resistance force on the truck. There is a light rigid horizontal coupling connecting the locomotive and the truck.
  1. State the tension in the coupling.
  2. Find the power produced by the locomotive's engine.
    The power produced by the locomotive's engine is now changed to 1.2 W .
  3. Find the magnitude of the tension in the coupling at the instant that the locomotive begins to accelerate.
    \includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-06_726_803_264_671} The diagram shows a block \(D\) of mass 100 kg supported by two sloping struts \(A D\) and \(B D\), each attached at an angle of \(45 ^ { \circ }\) to fixed points \(A\) and \(B\) respectively on a horizontal floor. The block is also held in place by a vertical rope \(C D\) attached to a fixed point \(C\) on a horizontal ceiling. The tension in the rope \(C D\) is 500 N and the block rests in equilibrium.
CAIE M1 2023 March Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-10_289_1191_269_475} The diagram shows a smooth track which lies in a vertical plane. The section \(A B\) is a quarter circle of radius 1.8 m with centre \(O\). The section \(B C\) is a horizontal straight line of length 7.0 m and \(O B\) is perpendicular to \(B C\). The section \(C F E\) is a straight line inclined at an angle of \(\theta ^ { \circ }\) above the horizontal. A particle \(P\) of mass 0.5 kg is released from rest at \(A\). Particle \(P\) collides with a particle \(Q\) of mass 0.1 kg which is at rest at \(B\). Immediately after the collision, the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(B C\). You should assume that \(P\) is moving horizontally when it collides with \(Q\).
  1. Show that the speed of \(Q\) immediately after the collision is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    When \(Q\) reaches \(C\), it collides with a particle \(R\) of mass 0.4 kg which is at rest at \(C\). The two particles coalesce. The combined particle comes instantaneously to rest at \(F\). You should assume that there is no instantaneous change in speed as the combined particle leaves \(C\), nor when it passes through \(C\) again as it returns down the slope.
  2. Given that the distance \(C F\) is 0.4 m , find the value of \(\theta\).
  3. Find the distance from \(B\) at which \(P\) collides with the combined particle.
    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 2024 March Q2
2 A particle is projected vertically upwards from horizontal ground. The speed of the particle 2 seconds after it is projected is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it is travelling downwards.
  1. Find the speed of projection of the particle.
  2. Find the distance travelled by the particle between the two times at which its speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2024 March Q3
3 A crate of mass 600 kg is being pulled up a line of greatest slope of a rough plane at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) by a rope attached to a winch. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and the rope is parallel to the plane. The winch is working at a constant rate of 8 kW . Find the coefficient of friction between the crate and the plane.
\includegraphics[max width=\textwidth, alt={}, center]{1ca74dfc-9bef-475c-a7d1-77b95c487f4b-05_483_953_269_557} Four coplanar forces act at a point. The magnitudes of the forces are \(F N , 2 F N , 3 F N\) and \(30 N\). The directions of the forces are as shown in the diagram. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).
CAIE M1 2024 March Q5
5 A particle moves in a straight line starting from a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle \(t \mathrm {~s}\) after leaving \(O\) is given by $$\mathrm { v } = \mathrm { t } ^ { 3 } - \frac { 9 } { 2 } \mathrm { t } ^ { 2 } + 1 \text { for } 0 \leqslant t \leqslant 4$$ You may assume that the velocity of the particle is positive for \(t < \frac { 1 } { 2 }\), is zero at \(t = \frac { 1 } { 2 }\) and is negative for \(t > \frac { 1 } { 2 }\).
  1. Find the distance travelled between \(t = 0\) and \(t = \frac { 1 } { 2 }\).
  2. Find the positive value of \(t\) at which the acceleration is zero. Hence find the total distance travelled between \(t = 0\) and this instant.
CAIE M1 2024 March Q6
6 A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F \mathrm {~N}\) acting on the trailer. The driving force of the car's engine is 3000 N .
  1. It is given that \(F = 100\). Find the acceleration of the car and the tension in the tow-bar.
  2. It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J . The speed of the car at the start of the 50 m is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car at the end of the 50 m .
    \includegraphics[max width=\textwidth, alt={}, center]{1ca74dfc-9bef-475c-a7d1-77b95c487f4b-10_680_887_269_596} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(A B C\). Particles \(P\) and \(Q\) are each of mass \(m \mathrm {~kg}\). The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(A B\) is 0.75 m and the length of \(B C\) is 3.25 m . The section \(A B\) of the plane is smooth and the section \(B C\) is rough. The coefficient of friction between each particle and the section \(B C\) is 0.25 . Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).
  3. Verify that particle \(P\) reaches \(B 0.5 \mathrm {~s}\) after it is released, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Find the time that it takes from the instant the two particles are released until they collide.
    The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25 .
  5. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE M1 2020 November Q1
1 A particle \(B\) of mass 5 kg is at rest on a smooth horizontal table. A particle \(A\) of mass 2.5 kg moves on the table with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and collides directly with \(B\). In the collision the two particles coalesce.
  1. Find the speed of the combined particle after the collision.
  2. Find the loss of kinetic energy of the system due to the collision.
CAIE M1 2020 November Q2
2 A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude 350 N .
  1. Find, in kW , the rate at which the engine of the car is working when it is travelling at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine is working at 15 kW .
CAIE M1 2020 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-04_378_969_258_587} Coplanar forces of magnitudes \(8 \mathrm {~N} , 12 \mathrm {~N} , 10 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\).
CAIE M1 2020 November Q4
4 A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t \mathrm {~s}\) after leaving \(O\) it has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 6 t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest.
\includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-06_540_606_258_767} Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is released from rest with both particles 0.5 m above a horizontal floor (see diagram). In the subsequent motion the 0.2 kg particle does not reach the pulley.
  1. Show that the magnitude of the acceleration of the particles is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. When the 0.8 kg particle reaches the floor it comes to rest. Find the greatest height of the 0.2 kg particle above the floor.
CAIE M1 2020 November Q6
5 marks
6 A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of \(\sin ^ { - 1 } 0.08\) to the horizontal. The resistances to the motion of the car and trailer are 400 N and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer have speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill and \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
  1. Use an energy method to find the constant driving force as the car and trailer travel up the hill.
    [0pt] [5]
    ..................................................................................................................................
    After reaching the top of the hill the system consisting of the car and trailer travels along a straight level road. The driving force of the car's engine is 2400 N and the resistances to motion are unchanged.
  2. Find the acceleration of the system and the tension in the tow-bar.
    \includegraphics[max width=\textwidth, alt={}, center]{de5edcfa-595b-4a9b-b3b3-7803670759cf-10_440_738_262_699} Three points \(A , B\) and \(C\) lie on a line of greatest slope of a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal, with \(A B = 1 \mathrm {~m}\) and \(B C = 1 \mathrm {~m}\), as shown in the diagram. A particle of mass 0.2 kg is released from rest at \(A\) and slides down the plane. The part of the plane from \(A\) to \(B\) is smooth. The part of the plane from \(B\) to \(C\) is rough, with coefficient of friction \(\mu\) between the plane and the particle.
  3. Given that \(\mu = \frac { 1 } { 2 } \sqrt { 3 }\), find the speed of the particle at \(C\).
  4. Given instead that the particle comes to rest at \(C\), find the exact value of \(\mu\).
    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 2020 November Q1
1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the momentum of \(P\).
  2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
CAIE M1 2020 November Q2
2 A car of mass 1800 kg is travelling along a straight horizontal road. The power of the car's engine is constant. There is a constant resistance to motion of 650 N .
  1. Find the power of the car's engine, given that the car's acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the steady speed which the car can maintain with the engine working at this power.
CAIE M1 2020 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-05_446_851_260_646} A block of mass \(m \mathrm {~kg}\) is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). The other string is inclined at \(60 ^ { \circ }\) to the horizontal and the tension in this string is 20 N . Find \(T\) and \(m\).
CAIE M1 2020 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-06_602_1203_260_470} The diagram shows a velocity-time graph which models the motion of a car. The graph consists of four straight line segments. The car accelerates at a constant rate of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(T \mathrm {~s}\). It then decelerates at a constant rate for 5 seconds before travelling at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 27.5 s . The car then decelerates to rest at a constant rate over a period of 5 s .
  1. Find \(T\).
  2. Given that the distance travelled up to the point at which the car begins to move with constant speed is one third of the total distance travelled, find \(V\).
CAIE M1 2020 November Q5
5 A particle is projected vertically upwards with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) alongside a building of height \(h \mathrm {~m}\).
  1. Given that the particle is above the level of the top of the building for 4 s , find \(h\).
  2. One second after the first particle is projected, a second particle is projected vertically upwards from the top of the building with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Denoting the time after projection of the first particle by \(t \mathrm {~s}\), find the value of \(t\) for which the two particles are at the same height above the ground.
CAIE M1 2020 November Q6
6 A block of mass 5 kg is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is \(\mu\).
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-10_424_709_392_760} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} When a force of magnitude 40 N is applied to the block, acting up the plane parallel to a line of greatest slope, the block begins to slide up the plane (see Fig. 6.1). Show that \(\mu < \frac { 1 } { 5 } \sqrt { 3 }\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-11_422_727_264_749} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure} When a force of magnitude 40 N is applied horizontally, in a vertical plane containing a line of greatest slope, the block does not move (see Fig. 6.2). Show that, correct to 3 decimal places, the least possible value of \(\mu\) is 0.152 .
CAIE M1 2020 November Q7
3 marks
7 A particle \(P\) moves in a straight line, starting from a point \(O\) with velocity \(1.72 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle, \(t \mathrm {~s}\) after leaving \(O\), is given by \(a = 0.1 t ^ { \frac { 3 } { 2 } }\).
  1. Find the value of \(t\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the displacement of \(P\) from \(O\) when \(t = 2\), giving your answer correct to 2 decimal places. [3]
CAIE M1 2020 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-14_388_1216_264_461} Two particles \(A\) and \(B\), 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 which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with \(A\) on the horizontal plane and \(B\) on the inclined plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The string is taut and \(B\) can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to \(B\) acting down the plane (see diagram).
  1. Given that both planes are smooth, find the tension in the string and the acceleration of \(B\).
  2. It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J . Use an energy method to find the speed of \(B\) when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that \(A\) does not hit the pulley when it moves 0.6 m .]
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.