Questions — CAIE M1 (786 questions)

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CAIE M1 2020 November Q7
12 marks Standard +0.3
\includegraphics{figure_7} Three points \(A\), \(B\) and \(C\) lie on a line of greatest slope of a plane inclined at an angle of \(30°\) to the horizontal, with \(AB = 1\) m and \(BC = 1\) 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.
  1. Given that \(\mu = \frac{1}{2}\sqrt{3}\), find the speed of the particle at \(C\). [8]
  2. Given instead that the particle comes to rest at \(C\), find the exact value of \(\mu\). [4]
CAIE M1 2022 November Q1
5 marks Moderate -0.5
\includegraphics{figure_1} Coplanar forces of magnitudes \(P\) N, \(Q\) N, 16 N and 22 N act at a point in the directions shown in the diagram. The forces are in equilibrium. Find the values of \(P\) and \(Q\). [5]
CAIE M1 2022 November Q2
5 marks Standard +0.3
Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed 5 m s\(^{-1}\) and \(B\) is moving towards \(A\) with speed 3 m s\(^{-1}\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is 2 m s\(^{-1}\). Find the loss of kinetic energy of the system due to the collision. [5]
CAIE M1 2022 November Q3
9 marks Moderate -0.3
A constant resistance of magnitude 1400 N acts on a car of mass 1250 kg.
  1. The car is moving along a straight level road at a constant speed of 28 m s\(^{-1}\). Find, in kW, the rate at which the engine of the car is working. [2]
  2. The car now travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.12\), with the engine working at 43.5 kW. Find this speed. [3]
  3. On another occasion, the car pulls a trailer of mass 600 kg up the same hill. The system of the car and the trailer is modelled as particles connected by a light inextensible cable. The car's engine produces a driving force of 5000 N and the resistance to the motion of the trailer is 300 N. The resistance to the motion of the car remains 1400 N. Find the acceleration of the system and the tension in the cable. [4]
CAIE M1 2022 November Q4
9 marks Standard +0.3
\includegraphics{figure_4} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of 18° to the horizontal. The block is pulled up the plane by a light string that makes an angle of 26° above a line of greatest slope. The tension in the string is \(T\) N (see diagram). The coefficient of friction between the block and plane is 0.65.
  1. The acceleration of the block is 0.2 m s\(^{-2}\). Find \(T\). [7]
  2. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion. [2]
CAIE M1 2022 November Q5
10 marks Standard +0.3
A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(-20\) m s\(^{-1}\). The acceleration \(a\) m s\(^{-2}\) at time \(t\) s after leaving \(O\) is given by \(a = 12 - 2t\).
  1. Sketch a velocity-time graph for \(0 \leq t \leq 12\), indicating the times when \(P\) is at rest. [5]
  2. Find the total distance travelled by \(P\) in the interval \(0 \leq t \leq 12\). [5]
CAIE M1 2022 November Q6
12 marks Standard +0.3
\includegraphics{figure_6_1} **Fig. 6.1** Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of 30° to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.
  1. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane. [6]
  2. \includegraphics{figure_6_2} **Fig. 6.2** It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level. [6]
CAIE M1 2022 November Q1
3 marks Moderate -0.5
A cyclist is riding a bicycle along a straight horizontal road \(AB\) of length 50 m. The cyclist starts from rest at \(A\) and reaches a speed of \(6 \text{ m s}^{-1}\) at \(B\). The cyclist produces a constant driving force of magnitude 100 N. There is a resistance force, and the work done against the resistance force from \(A\) to \(B\) is 3560 J. Find the total mass of the cyclist and bicycle. [3]
CAIE M1 2022 November Q2
7 marks Moderate -0.3
A particle \(P\) of mass 0.4 kg is in limiting equilibrium on a plane inclined at \(30°\) to the horizontal.
  1. Show that the coefficient of friction between the particle and the plane is \(\frac{1}{3}\sqrt{3}\). [3]
A force of magnitude 7.2 N is now applied to \(P\) directly up a line of greatest slope of the plane.
  1. Given that \(P\) starts from rest, find the time that it takes for \(P\) to move 1 m up the plane. [4]
CAIE M1 2022 November Q3
6 marks Standard +0.3
\includegraphics{figure_3} A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of \(60°\) to a horizontal ceiling. The other string is attached at an angle \(α°\) to a vertical wall (see diagram). The tension in the string attached to the ceiling is 4 N. Find the tension in the string which is attached to the wall and find the value of \(α\). [6]
CAIE M1 2022 November Q4
6 marks Standard +0.3
A car of mass 1200 kg is travelling along a straight horizontal road \(AB\). There is a constant resistance force of magnitude 500 N. When the car passes point \(A\), it has a speed of \(15 \text{ m s}^{-1}\) and an acceleration of \(0.8 \text{ m s}^{-2}\).
  1. Find the power of the car's engine at the point \(A\). [3]
The car continues to work with this power as it travels from \(A\) to \(B\). The car takes 53 seconds to travel from \(A\) to \(B\) and the speed of the car at \(B\) is \(32 \text{ m s}^{-1}\).
  1. Show that the distance \(AB\) is 1362.6 m. [3]
CAIE M1 2022 November Q5
7 marks Standard +0.3
\includegraphics{figure_5} A block \(A\) of mass 80 kg is connected by a light, inextensible rope to a block \(B\) of mass 40 kg. The rope joining the two blocks is taut and is parallel to a line of greatest slope of a plane which is inclined at an angle of \(20°\) to the horizontal. A force of magnitude 500 N inclined at an angle of \(15°\) above the same line of greatest slope acts on \(A\) (see diagram). The blocks move up the plane and there is a resistance force of 50 N on \(B\), but no resistance force on \(A\).
  1. Find the acceleration of the blocks and the tension in the rope. [5]
  1. Find the time that it takes for the blocks to reach a speed of \(1.2 \text{ m s}^{-1}\) from rest. [2]
CAIE M1 2022 November Q6
9 marks Moderate -0.3
Three particles \(A\), \(B\) and \(C\) of masses 0.3 kg, 0.4 kg and \(m\) kg respectively lie at rest in a straight line on a smooth horizontal plane. The distance between \(B\) and \(C\) is 2.1 m. \(A\) is projected directly towards \(B\) with speed \(2 \text{ m s}^{-1}\). After \(A\) collides with \(B\) the speed of \(A\) is reduced to \(0.6 \text{ m s}^{-1}\), still moving in the same direction.
  1. Show that the speed of \(B\) after the collision is \(1.05 \text{ m s}^{-1}\). [2]
After the collision between \(A\) and \(B\), \(B\) moves directly towards \(C\). Particle \(B\) now collides with \(C\). After this collision, the two particles coalesce and have a combined speed of \(0.5 \text{ m s}^{-1}\).
  1. Find \(m\). [2]
  1. Find the time that it takes, from the instant when \(B\) and \(C\) collide, until \(A\) collides with the combined particle. [5]
CAIE M1 2022 November Q7
12 marks Standard +0.3
A particle \(P\) travels in a straight line, starting at rest from a point \(O\). The acceleration of \(P\) at time \(t\) s after leaving \(O\) is denoted by \(a \text{ m s}^{-2}\), where $$a = 0.3t^{\frac{1}{2}} \quad \text{for } 0 \leqslant t \leqslant 4,$$ $$a = -kt^{-\frac{1}{2}} \quad \text{for } 4 < t \leqslant T,$$ where \(k\) and \(T\) are constants.
  1. Find the velocity of \(P\) at \(t = 4\). [2]
  1. It is given that there is no change in the velocity of \(P\) at \(t = 4\) and that the velocity of \(P\) at \(t = 16\) is \(0.3 \text{ m s}^{-1}\). Show that \(k = 2.6\) and find an expression, in terms of \(t\), for the velocity of \(P\) for \(4 \leqslant t \leqslant T\). [4]
  1. Given that \(P\) comes to instantaneous rest at \(t = T\), find the exact value of \(T\). [2]
  1. Find the total distance travelled between \(t = 0\) and \(t = T\). [4]
CAIE M1 2023 November Q1
4 marks Moderate -0.5
A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the top of the plane is 2 m s\(^{-1}\) and the speed of the block at the bottom of the plane is 4 m s\(^{-1}\). Find the work done against the resistance to motion of the block. [4]
CAIE M1 2023 November Q2
5 marks Standard +0.3
\includegraphics{figure_2} The diagram shows a smooth ring \(R\), of mass \(m\) kg, threaded on a light inextensible string. A horizontal force of magnitude 2 N acts on \(R\). The ends of the string are attached to fixed points \(A\) and \(B\) on a vertical wall. The part \(AR\) of the string makes an angle of 30° with the vertical, the part \(BR\) makes an angle of 40° with the vertical and the string is taut. The ring is in equilibrium. Find the tension in the string and find the value of \(m\). [5]
CAIE M1 2023 November Q3
5 marks Standard +0.3
\includegraphics{figure_3} A block of mass 10 kg is at rest on a rough plane inclined at an angle of 30° to the horizontal. A force of 120 N is applied to the block at an angle of 20° above a line of greatest slope (see diagram). There is a force resisting the motion of the block and 200 J of work is done against this force when the block has moved a distance of 5 m up the plane from rest. Find the speed of the block when it has moved a distance of 5 m up the plane from rest. [5]
CAIE M1 2023 November Q4
7 marks Moderate -0.8
A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).
  1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane. [3]
  2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3, find the distance travelled by \(P\) in the third second of its motion. [4]
CAIE M1 2023 November Q5
8 marks Standard +0.3
A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s\(^{-1}\).
  1. Find the speed of \(A\) when it reaches a height of 20 m above the ground. [2]
When \(A\) reaches a height of 20 m, it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed 32.5 m s\(^{-1}\). In the collision between the two particles, \(B\) is brought to instantaneous rest.
  1. Show that the velocity of \(A\) immediately after the collision is 4.5 m s\(^{-1}\) downwards. [2]
  2. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground. [4]
CAIE M1 2023 November Q6
9 marks Standard +0.3
A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.
  1. Find the acceleration of the engine and find the tension in the coupling. [5]
At an instant when the engine is travelling at 30 m s\(^{-1}\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.
  1. Assuming that the resistance forces remain unchanged, find the value of \(\beta\). [4]
CAIE M1 2023 November Q7
12 marks Standard +0.3
A particle \(X\) travels in a straight line. The velocity of \(X\) at time \(t\) s after leaving a fixed point \(O\) is denoted by \(v\) m s\(^{-1}\), where $$v = -0.1t^3 + 1.8t^2 - 6t + 5.6.$$ The acceleration of \(X\) is zero at \(t = p\) and \(t = q\), where \(p < q\).
  1. Find the value of \(p\) and the value of \(q\). [4]
It is given that the velocity of \(X\) is zero at \(t = 14\).
  1. Find the velocities of \(X\) at \(t = p\) and at \(t = q\), and hence sketch the velocity-time graph for the motion of \(X\) for \(0 \leq t \leq 15\). [3]
  2. Find the total distance travelled by \(X\) between \(t = 0\) and \(t = 15\). [5]
CAIE M1 2024 November Q1
4 marks Moderate -0.8
Two particles, of masses \(1.8\) kg and \(1.2\) kg, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest. Find the magnitude of the acceleration of the particles and find the tension in the string. [4]
CAIE M1 2024 November Q2
5 marks Moderate -0.8
\includegraphics{figure_2} A particle of mass \(7.5\) kg, starting from rest at \(A\), slides down an inclined plane \(AB\). The point \(B\) is \(12.5\) metres vertically below the level of \(A\), as shown in the diagram.
  1. Given that the plane is smooth, use an energy method to find the speed of the particle at \(B\). [2]
  2. It is given instead that the plane is rough and the particle reaches \(B\) with a speed of \(8 \text{ ms}^{-1}\). The plane is \(25\) m long and the constant frictional force has magnitude \(F\) N. Find the value of \(F\). [3]
CAIE M1 2024 November Q3
4 marks Standard +0.3
\includegraphics{figure_3} Coplanar forces of magnitudes \(52\) N, \(39\) N and \(P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\). [4]
CAIE M1 2024 November Q4
6 marks Moderate -0.3
A bus travels between two stops, \(A\) and \(B\). The bus starts from rest at \(A\) and accelerates at a constant rate of \(a \text{ ms}^{-2}\) until it reaches a speed of \(16 \text{ ms}^{-1}\). It then travels at this constant speed before decelerating at a constant rate of \(0.75 \text{ ms}^{-2}\), coming to rest at \(B\). The total time for the journey is \(240\) s.
  1. Sketch the velocity-time graph for the bus's journey from \(A\) to \(B\). [1]
  2. Find an expression, in terms of \(a\), for the length of time that the bus is travelling with constant speed. [2]
  3. Given that the distance from \(A\) to \(B\) is \(3000\) m, find the value of \(a\). [3]