Questions — CAIE M1 (786 questions)

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CAIE M1 2010 June Q1
4 marks Standard +0.3
A car of mass 1150 kg travels up a straight hill inclined at 1.2° to the horizontal. The resistance to motion of the car is 975 N. Find the acceleration of the car at an instant when it is moving with speed 16 m s\(^{-1}\) and the engine is working at a power of 35 kW. [4]
CAIE M1 2010 June Q2
5 marks Easy -1.2
\includegraphics{figure_2} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes 3 s to return to its starting position. Find
  1. the acceleration of the tool during the first 2 s of the motion, [1]
  2. the distance the tool moves forward while cutting, [2]
  3. the greatest speed of the tool during the return to its starting position. [2]
CAIE M1 2010 June Q3
5 marks Moderate -0.8
\includegraphics{figure_3} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45° to the horizontal (see diagram).
  1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N, correct to 3 significant figures. [2]
  2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]
CAIE M1 2010 June Q4
7 marks Standard +0.3
\includegraphics{figure_4} Coplanar forces of magnitudes 250 N, 160 N and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]
CAIE M1 2010 June Q5
7 marks Moderate -0.3
\(P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(PQ\).
  1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\). [3]
  2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J. Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\). [4]
CAIE M1 2010 June Q6
11 marks Standard +0.8
\includegraphics{figure_6} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.
  1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is 3.95 m s\(^{-1}\), correct to 3 significant figures. [7]
  2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]
CAIE M1 2010 June Q7
11 marks Standard +0.3
A vehicle is moving in a straight line. The velocity \(v\) m s\(^{-1}\) at time \(t\) s after the vehicle starts is given by $$v = A(t - 0.05t^2) \quad \text{for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t^2} \quad \text{for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is 225 m.
  1. Find the value of \(A\) and show that \(B = 3375\). [5]
  2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
  3. Find the speed of the vehicle when it has travelled a total distance of 315 m. [3]
CAIE M1 2010 June Q1
4 marks Moderate -0.3
A car of mass \(1150 \text{ kg}\) travels up a straight hill inclined at \(1.2°\) to the horizontal. The resistance to motion of the car is \(975 \text{ N}\). Find the acceleration of the car at an instant when it is moving with speed \(16 \text{ m s}^{-1}\) and the engine is working at a power of \(35 \text{ kW}\). [4]
CAIE M1 2010 June Q2
5 marks Easy -1.2
\includegraphics{figure_2} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for \(8 \text{ s}\) while cutting and then takes \(3 \text{ s}\) to return to its starting position. Find
  1. the acceleration of the tool during the first \(2 \text{ s}\) of the motion, [1]
  2. the distance the tool moves forward while cutting, [2]
  3. the greatest speed of the tool during the return to its starting position. [2]
CAIE M1 2010 June Q3
5 marks Moderate -0.8
\includegraphics{figure_3} A small ring of mass \(0.8 \text{ kg}\) is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude \(7 \text{ N}\) pulling upwards at \(45°\) to the horizontal (see diagram).
  1. Show that the normal component of the contact force acting on the ring has magnitude \(3.05 \text{ N}\), correct to 3 significant figures. [2]
  2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]
CAIE M1 2010 June Q4
7 marks Moderate -0.3
\includegraphics{figure_4} Coplanar forces of magnitudes \(250 \text{ N}\), \(160 \text{ N}\) and \(370 \text{ N}\) act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]
CAIE M1 2010 June Q5
7 marks Moderate -0.3
\(P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of \(0.45 \text{ m}\) above the level of \(P\). A particle of mass \(0.3 \text{ kg}\) moves upwards along the line \(PQ\).
  1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\). [3]
  2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is \(0.39 \text{ J}\). Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\). [4]
CAIE M1 2010 June Q6
11 marks Standard +0.3
\includegraphics{figure_6} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.45 \text{ kg}\) respectively, are connected by a light inextensible string of length \(2.8 \text{ m}\). The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is \(2 \text{ m}\) above the floor. Particle \(A\) is held in contact with the surface at a distance of \(2.1 \text{ m}\) from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is \(0.3\). Particle \(A\) is released and the system begins to move.
  1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \text{ m s}^{-1}\), correct to 3 significant figures. [7]
  2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]
CAIE M1 2010 June Q7
11 marks Standard +0.3
A vehicle is moving in a straight line. The velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\) after the vehicle starts is given by $$v = A(t - 0.05t^2) \text{ for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t} \text{ for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is \(225 \text{ m}\).
  1. Find the value of \(A\) and show that \(B = 3375\). [5]
  2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
  3. Find the speed of the vehicle when it has travelled a total distance of \(315 \text{ m}\). [3]
CAIE M1 2014 June Q1
6 marks Moderate -0.8
A particle moves in a straight line. At time \(t\) seconds, its displacement from a fixed point is \(s\) metres, where $$s = t^3 - 6t^2 + 9t$$
  1. Find expressions for the velocity and acceleration of the particle at time \(t\). [4]
  2. Find the times when the particle is at rest. [2]
CAIE M1 2014 June Q2
5 marks Easy -1.2
A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).
  1. Calculate the maximum frictional force that can act on the block. [2]
  2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]
CAIE M1 2014 June Q3
6 marks Moderate -0.5
\includegraphics{figure_3} A particle is moving under the action of three forces as shown in the diagram. The particle is in equilibrium. Find the magnitudes of forces \(P\) and \(Q\). [6]
CAIE M1 2014 June Q4
5 marks Moderate -0.5
A particle of mass \(0.5\) kg moves in a straight line under the action of a variable force. At time \(t\) seconds, the force is \((3t - 2)\) N in the direction of motion. Given that the particle starts from rest, find the velocity of the particle when \(t = 4\). [5]
CAIE M1 2014 June Q5
6 marks Moderate -0.8
\includegraphics{figure_5} A uniform rod AB has length \(2\) m and weight \(20\) N. The rod rests horizontally in equilibrium on two supports at points C and D, where AC = \(0.4\) m and BD = \(0.6\) m.
  1. Find the reaction at each support. [4]
  2. State what happens if the support at D is removed. [2]
CAIE M1 2014 June Q6
8 marks Easy -1.2
\includegraphics{figure_6} A particle is projected vertically upward from ground level with speed \(u\) m s\(^{-1}\). The particle moves under gravity alone.
  1. Find an expression for the maximum height reached by the particle. [3]
\includegraphics{figure_6b} The diagram shows a velocity-time graph for the motion of the particle.
  1. Use the graph to find the value of \(u\). [2]
  2. Find the time taken for the particle to return to ground level. [3]
CAIE M1 2015 June Q1
4 marks Easy -1.2
A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and the block moves at a constant speed of \(2.75 \text{ m s}^{-1}\). Find the work done by the tension in 40 s and find the power applied by the tension. [4]
CAIE M1 2015 June Q2
5 marks Moderate -0.3
\includegraphics{figure_2} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h\) m above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \text{ m s}^{-1}\).
  1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \text{ m s}^{-1}\). [1]
  2. Find the value of \(h\). [4]
CAIE M1 2015 June Q3
6 marks Standard +0.3
A car of mass 860 kg travels along a straight horizontal road. The power provided by the car's engine is \(P\) W and the resistance to the car's motion is \(R\) N. The car passes through one point with speed \(4.5 \text{ m s}^{-1}\) and acceleration \(4 \text{ m s}^{-2}\). The car passes through another point with speed \(22.5 \text{ m s}^{-1}\) and acceleration \(0.3 \text{ m s}^{-2}\). Find the values of \(P\) and \(R\). [6]
CAIE M1 2015 June Q4
6 marks Standard +0.3
A lorry of mass 12 000 kg moves up a straight hill of length 500 m, starting at the bottom with a speed of \(24 \text{ m s}^{-1}\) and reaching the top with a speed of \(16 \text{ m s}^{-1}\). The top of the hill is 25 m above the level of the bottom of the hill. The resistance to motion of the lorry is 7500 N. Find the driving force of the lorry. [6]
CAIE M1 2015 June Q5
7 marks Moderate -0.8
\includegraphics{figure_1} Four coplanar forces of magnitudes 4 N, 8 N, 12 N and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.
  1. Find the magnitude and direction of the resultant of the four forces. [5]
\includegraphics{figure_2} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).
  1. State the magnitude and direction of the resultant of the four forces in Fig. 2. [2]