Questions — CAIE (7646 questions)

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CAIE M1 2021 June Q1
3 marks Moderate -0.5
A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane inclined at 60° to the horizontal. The winch pulls the load a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N. Find the work done by the winch. [3]
CAIE M1 2021 June Q2
6 marks Standard +0.3
\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(m\) kg and 0.1 kg respectively, where \(m > 0.1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.9 m above horizontal ground (see diagram). The system is released from rest, and while both particles are in motion the tension in the string is 1.5 N. Particle \(B\) does not reach the pulley.
  1. Find \(m\). [4]
  2. Find the speed at which \(A\) reaches the ground. [2]
CAIE M1 2021 June Q3
6 marks Standard +0.3
Three particles \(P\), \(Q\) and \(R\), of masses 0.1 kg, 0.2 kg and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle \(P\) is projected towards \(Q\) at a speed of \(5 \text{ m s}^{-1}\). After \(P\) and \(Q\) collide, \(P\) rebounds with speed \(1 \text{ m s}^{-1}\).
  1. Find the speed of \(Q\) immediately after the collision with \(P\). [3]
\(Q\) now collides with \(R\). Immediately after the collision with \(Q\), \(R\) begins to move with speed \(V \text{ m s}^{-1}\).
  1. Given that there is no subsequent collision between \(P\) and \(Q\), find the greatest possible value of \(V\). [3]
CAIE M1 2021 June Q4
7 marks Moderate -0.3
Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point \(A\). Isabella accelerates for 5 s at a constant rate \(a \text{ m s}^{-2}\). She then travels at the constant speed she has reached for 10 s, before decelerating to rest at a constant rate over a period of 5 s. Maria accelerates at a constant rate, reaching a speed of \(5 \text{ m s}^{-1}\) in a distance of 27.5 m. She then maintains this speed for a period of 10 s, before decelerating to rest at a constant rate over a period of 5 s.
  1. Given that \(a = 1.1\), find which cyclist travels further. [5]
  2. Find the value of \(a\) for which the two cyclists travel the same distance. [2]
CAIE M1 2021 June Q5
8 marks Standard +0.3
A particle moving in a straight line starts from rest at a point \(A\) and comes instantaneously to rest at a point \(B\). The acceleration of the particle at time \(t\) s after leaving \(A\) is \(a \text{ m s}^{-2}\), where $$a = 6t^{\frac{1}{2}} - 2t.$$
  1. Find the value of \(t\) at point \(B\). [3]
  2. Find the distance travelled from \(A\) to the point at which the acceleration of the particle is again zero. [5]
CAIE M1 2021 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} Three coplanar forces of magnitudes 10 N, 25 N and 20 N act at a point \(O\) in the directions shown in the diagram.
  1. Given that the component of the resultant force in the \(x\)-direction is zero, find \(\alpha\), and hence find the magnitude of the resultant force. [4]
  2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces. [5]
CAIE M1 2021 June Q7
11 marks Standard +0.3
\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.
  1. It is given that the sloping section of the slide is smooth.
    1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
    2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
  2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]
CAIE M1 2022 June Q1
6 marks Moderate -0.8
A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m, reaching a speed of 25 m s\(^{-1}\). The car then travels at this speed for 400 m, before decelerating uniformly to rest over a period of 5 s.
  1. Find the time for which the car is accelerating. [2]
  2. Sketch the velocity–time graph for the motion of the car, showing the key points. [2]
  3. Find the average speed of the car during its motion. [2]
CAIE M1 2022 June Q2
5 marks Moderate -0.8
Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them. [5]
CAIE M1 2022 June Q3
5 marks Standard +0.3
A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]
CAIE M1 2022 June Q4
6 marks Standard +0.3
\includegraphics{figure_4} Three coplanar forces of magnitudes 20 N, 100 N and \(F\) N act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find \(F\) and \(\alpha\). [6]
CAIE M1 2022 June Q5
9 marks Standard +0.3
Two racing cars \(A\) and \(B\) are at rest alongside each other at a point \(O\) on a straight horizontal test track. The mass of \(A\) is 1200 kg. The engine of \(A\) produces a constant driving force of 4500 N. When \(A\) arrives at a point \(P\) its speed is 25 m s\(^{-1}\). The distance \(OP\) is \(d\) m. The work done against the resistance force experienced by \(A\) between \(O\) and \(P\) is 75 000 J.
  1. Show that \(d = 100\). [3]
Car \(B\) starts off at the same instant as car \(A\). The two cars arrive at \(P\) simultaneously and with the same speed. The engine of \(B\) produces a driving force of 3200 N and the car experiences a constant resistance to motion of 1200 N.
  1. Find the mass of \(B\). [3]
  2. Find the steady speed which \(B\) can maintain when its engine is working at the same rate as it is at \(P\). [3]
CAIE M1 2022 June Q6
10 marks Standard +0.3
A particle starts from a point \(O\) and moves in a straight line. The velocity \(v\) m s\(^{-1}\) of the particle at time \(t\) s after leaving \(O\) is given by $$v = k(3t^2 - 2t^3),$$ where \(k\) is a constant.
  1. Verify that the particle returns to \(O\) when \(t = 2\). [4]
  2. It is given that the acceleration of the particle is \(-13.5\) m s\(^{-2}\) for the positive value of \(t\) at which \(v = 0\). Find \(k\) and hence find the total distance travelled in the first two seconds of motion. [6]
CAIE M1 2022 June Q7
9 marks Standard +0.3
Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are 3 m s\(^{-1}\) and 2 m s\(^{-1}\) respectively. In the collision between the particles, the speed of \(A\) is reduced to 2.5 m s\(^{-1}\).
  1. Find the speed of \(B\) immediately after the collision. [2]
After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.
  1. Show that the speed of \(B\) immediately after it hits the barrier is 0.5 m s\(^{-1}\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\). [7]
CAIE M1 2022 June Q1
5 marks Moderate -0.8
Small smooth spheres \(A\) and \(B\), of equal radii and of masses \(5\text{kg}\) and \(3\text{kg}\) respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8.5\text{ms}^{-1}\). The spheres collide and after the collision \(A\) continues to move in the same direction but with a quarter of the speed of \(B\).
  1. Find the speed of \(B\) after the collision. [3]
  2. Find the loss of kinetic energy of the system due to the collision. [2]
CAIE M1 2022 June Q2
6 marks Moderate -0.3
\includegraphics{figure_2} Coplanar forces of magnitudes \(60\text{N}\), \(20\text{N}\), \(16\text{N}\) and \(14\text{N}\) act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force. [6]
CAIE M1 2022 June Q3
7 marks Standard +0.3
Two particles \(A\) and \(B\), of masses \(2.4\text{kg}\) and \(1.2\text{kg}\) respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at a distance of \(2.1\text{m}\) above a horizontal plane and \(B\) is \(1.5\text{m}\) above the plane. The particles hang vertically and are released from rest. In the subsequent motion \(A\) reaches the plane and does not rebound and \(B\) does not reach the pulley.
  1. Show that the tension in the string before \(A\) reaches the plane is \(16\text{N}\) and find the magnitude of the acceleration of the particles before \(A\) reaches the plane. [4]
  2. Find the greatest height of \(B\) above the plane. [3]
CAIE M1 2022 June Q4
9 marks Standard +0.3
A particle \(A\), moving along a straight horizontal track with constant speed \(8\text{ms}^{-1}\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20\text{ms}^{-1}\) when it passes \(O\) and has a constant deceleration of \(2\text{ms}^{-2}\). \(B\) comes to rest when it returns to \(O\).
  1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\). [3]
  2. Find the values of \(t\) when the particles are the same distance from \(O\). [3]
  3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from \(0\) to \(20\). [3] $$s \text{ (m)}$$ $$200$$ $$100$$ $$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$
CAIE M1 2022 June Q5
6 marks Standard +0.3
\includegraphics{figure_5} A block of mass \(12\text{kg}\) is placed on a plane which is inclined at an angle of \(24°\) to the horizontal. A light string, making an angle of \(36°\) above a line of greatest slope, is attached to the block. The tension in the string is \(65\text{N}\) (see diagram). The coefficient of friction between the block and plane is \(\mu\). The block is in limiting equilibrium and is on the point of sliding up the plane. Find \(\mu\). [6]
CAIE M1 2022 June Q6
8 marks Standard +0.3
A car of mass \(900\text{kg}\) is moving up a hill inclined at \(\sin^{-1} 0.12\) to the horizontal. The initial speed of the car is \(11\text{ms}^{-1}\). After \(12\text{s}\), the car has travelled \(150\text{m}\) up the hill and has speed \(16\text{ms}^{-1}\). The engine of the car is working at a constant rate of \(24\text{kW}\).
  1. Find the work done against the resistive forces during the \(12\text{s}\). [5]
  2. The car then travels along a straight horizontal road. There is a resistance to the motion of the car of \((1520 + 4v)\text{N}\) when the speed of the car is \(v\text{ms}^{-1}\). The car travels at a constant speed with the engine working at a constant rate of \(32\text{kW}\). Find this speed. [3]
CAIE M1 2022 June Q7
9 marks Standard +0.3
A particle \(P\) moves in a straight line. The velocity \(v\text{ms}^{-1}\) at time \(t\) seconds is given by $$v = 0.5t \quad \text{for } 0 \leqslant t \leqslant 10,$$ $$v = 0.25t^2 - 8t + 60 \quad \text{for } 10 < t \leqslant 20.$$
  1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\). [3]
  2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\). [6]
CAIE M1 2023 June Q1
4 marks Moderate -0.8
Two particles \(P\) and \(Q\), of masses \(m\) kg and \(0.3\) kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(5\) m s\(^{-1}\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(2\) m s\(^{-1}\) in the same direction as it was originally moving.
  1. Find, in terms of \(m\), the speed of \(Q\) after the collision. [2]
After this collision, \(Q\) moves directly towards a third particle \(R\), of mass \(0.6\) kg, which is at rest on the plane. \(Q\) is brought to rest in the collision with \(R\), and \(R\) begins to move with a speed of \(1.5\) m s\(^{-1}\).
  1. Find the value of \(m\). [2]
CAIE M1 2023 June Q2
6 marks Moderate -0.3
A particle \(P\) of mass \(0.4\) kg is projected vertically upwards from horizontal ground with speed \(10\) m s\(^{-1}\).
  1. Find the greatest height above the ground reached by \(P\). [2]
When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses \(7.2\) J of energy.
  1. Find the time between the first and second instants at which \(P\) hits the ground. [4]
CAIE M1 2023 June Q3
4 marks Moderate -0.3
A particle moves in a straight line starting from rest. The displacement \(s\) m of the particle from a fixed point \(O\) on the line at time \(t\) s is given by $$s = t^2 - \frac{15}{4}t^2 + 6.$$ Find the value of \(s\) when the particle is again at rest. [4]
CAIE M1 2023 June Q4
8 marks Moderate -0.3
\includegraphics{figure_4} The velocity of a particle at time \(t\) s after leaving a fixed point \(O\) is \(v\) m s\(^{-1}\). The diagram shows a velocity-time graph which models the motion of the particle. The graph consists of \(5\) straight line segments. The particle accelerates to a speed of \(0.9\) m s\(^{-1}\) in a period of \(3\) s, then travels at constant speed for \(6\) s, then comes instantaneously to rest \(1\) s later. The particle then moves back and returns to rest at \(O\) at time \(T\) s.
  1. Find the distance travelled by the particle in the first \(10\) s of its motion. [2]
  2. Given that \(T = 12\), find the minimum velocity of the particle. [2]
  3. Given instead that the greatest speed of the particle is \(3\) m s\(^{-1}\), find the value of \(T\) and hence find the average speed of the particle for the whole of the motion. [4]