Questions — CAIE M1 (786 questions)

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CAIE M1 2024 November Q5
10 marks Standard +0.8
A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \text{ ms}^{-1}\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \text{ ms}^{-1}\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).
  1. Show that \(T = 3.5\). [4]
  2. Find the height above \(O\) at which the particles collide. [1]
  3. Find the time from \(A\) being projected until \(C\) returns to \(O\). [5]
CAIE M1 2024 November Q6
6 marks Standard +0.3
\includegraphics{figure_6} A particle of mass \(1.2\) kg is placed on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The particle is kept in equilibrium by a horizontal force of magnitude \(P\) N acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is \(0.15\). Find the least possible value of \(P\). [6]
CAIE M1 2024 November Q7
8 marks Standard +0.3
A car has mass \(1200\) kg. When the car is travelling at a speed of \(v \text{ ms}^{-1}\), there is a resistive force of magnitude \(kv\) N. The maximum power of the car's engine is \(92.16\) kW.
  1. The car travels along a straight level road.
    1. The car has a greatest possible constant speed of \(48 \text{ ms}^{-1}\). Show that \(k = 40\). [1]
    2. At an instant when its speed is \(45 \text{ ms}^{-1}\), find the greatest possible acceleration of the car. [3]
  2. The car now travels at a constant speed up a hill inclined at an angle of \(\sin^{-1} 0.15\) to the horizontal. Find the greatest possible speed of the car going up the hill. [4]
CAIE M1 2024 November Q8
7 marks Standard +0.8
A particle \(P\) moves in a straight line, passing through a point \(O\) with velocity \(4.2 \text{ ms}^{-1}\). At time \(t\) s after \(P\) passes \(O\), the acceleration, \(a \text{ ms}^{-2}\), of \(P\) is given by \(a = 0.6t - 2.7\). Find the distance \(P\) travels between the times at which it is at instantaneous rest. [7]
CAIE M1 2024 November Q1
4 marks Moderate -0.3
An athlete has mass \(m\) kg. The athlete runs along a horizontal road against a constant resistance force of magnitude 24 N. The total work done by the athlete in increasing his speed from 5 ms\(^{-1}\) to 6 ms\(^{-1}\) while running a distance of 50 metres is 1541 J. Find the value of \(m\). [4]
CAIE M1 2024 November Q2
6 marks Moderate -0.3
\includegraphics{figure_2} Coplanar forces of magnitudes 16 N, 12 N, 24 N and 8 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [6]
CAIE M1 2024 November Q3
6 marks Standard +0.3
A car of mass 1600 kg travels up a slope inclined at an angle of \(\sin^{-1}\) 0.08 to the horizontal. There is a constant resistance of magnitude 240 N acting on the car.
  1. It is given that the car travels at a constant speed of 32 ms\(^{-1}\). Find the power of the engine of the car. [3]
  2. Find the acceleration of the car when its speed is 24 ms\(^{-1}\) and the engine is working at 95\% of the power found in (a). [3]
CAIE M1 2024 November Q4
6 marks Standard +0.8
Two particles, \(A\) and \(B\), of masses 3 kg and 6 kg respectively, lie on a smooth horizontal plane. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with speed 8 ms\(^{-1}\). After \(A\) and \(B\) collide, \(A\) moves with speed 2 ms\(^{-1}\). Find the greater of the two possible total losses of kinetic energy due to the collision. [6]
CAIE M1 2024 November Q5
8 marks Standard +0.3
\includegraphics{figure_5} A particle of mass 12 kg is going to be pulled across a rough horizontal plane by a light inextensible string. The string is at an angle of 30° above the plane and has tension \(T\) N (see diagram). The coefficient of friction between the particle and the plane is 0.5.
  1. Given that the particle is on the point of moving, find the value of \(T\). [5]
  2. Given instead that the particle is accelerating at 0.2 ms\(^{-2}\), find the value of \(T\). [3]
CAIE M1 2024 November Q6
10 marks Moderate -0.8
A particle moves in a straight line. It starts from rest, at time \(t = 0\), and accelerates at 0.6 t ms\(^{-2}\) for 4 s, reaching a speed of \(V\) ms\(^{-1}\). The particle then travels at \(V\) ms\(^{-1}\) for 11 s, and finally slows down, with constant deceleration, stopping after a further 5 s.
  1. Show that \(V = 4.8\). [1]
  2. Sketch a velocity-time graph for the motion. [3]
  3. Find an expression, in terms of \(t\), for the velocity of the particle for \(15 \leqslant t \leqslant 20\). [2]
  4. Find the total distance travelled by the particle. [4]
CAIE M1 2024 November Q7
10 marks Standard +0.3
\includegraphics{figure_7} Two particles, \(A\) and \(B\), of masses 3 kg and 5 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held with the string taut and its straight parts vertical. Particle \(A\) is 1 m above a horizontal plane, and particle \(B\) is 2 m above the plane (see diagram). The particles are released from rest. In the subsequent motion, \(A\) does not reach the pulley, and after \(B\) reaches the plane it remains in contact with the plane.
  1. Find the tension in the string and the time taken for \(B\) to reach the plane. [6]
  2. Find the time for which \(A\) is at least 3.25 m above the plane. [4]
CAIE M1 2005 June Q1
3 marks Moderate -0.8
A small block is pulled along a rough horizontal floor at a constant speed of \(1.5 \text{ m s}^{-1}\) by a constant force of magnitude \(30 \text{ N}\) acting at an angle of \(\theta°\) upwards from the horizontal. Given that the work done by the force in \(20 \text{ s}\) is \(720 \text{ J}\), calculate the value of \(\theta\). [3]
CAIE M1 2005 June Q2
6 marks Moderate -0.3
\includegraphics{figure_2} Three coplanar forces act at a point. The magnitudes of the forces are \(5 \text{ N}\), \(6 \text{ N}\) and \(7 \text{ N}\), and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the three forces. [6]
CAIE M1 2005 June Q3
6 marks Standard +0.3
\(A\) and \(B\) are points on the same line of greatest slope of a rough plane inclined at \(30°\) to the horizontal. \(A\) is higher up the plane than \(B\) and the distance \(AB\) is \(2.25 \text{ m}\). A particle \(P\), of mass \(m \text{ kg}\), is released from rest at \(A\) and reaches \(B\) \(1.5 \text{ s}\) later. Find the coefficient of friction between \(P\) and the plane. [6]
CAIE M1 2005 June Q4
7 marks Standard +0.3
\includegraphics{figure_4} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.3 \text{ kg}\) respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. Particle \(A\) hangs freely and particle \(B\) is in contact with the table (see diagram).
  1. The system is in limiting equilibrium with the string taut and \(A\) about to move downwards. Find the coefficient of friction between \(B\) and the table. [4]
A force now acts on particle \(B\). This force has a vertical component of \(1.8 \text{ N}\) upwards and a horizontal component of \(X \text{ N}\) directed away from the pulley.
  1. The system is now in limiting equilibrium with the string taut and \(A\) about to move upwards. Find \(X\). [3]
CAIE M1 2005 June Q5
7 marks Moderate -0.8
A particle \(P\) moves along the \(x\)-axis in the positive direction. The velocity of \(P\) at time \(t \text{ s}\) is \(0.03t^2 \text{ m s}^{-1}\). When \(t = 5\) the displacement of \(P\) from the origin \(O\) is \(2.5 \text{ m}\).
  1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\). [4]
  2. Find the velocity of \(P\) when its displacement from \(O\) is \(11.25 \text{ m}\). [3]
CAIE M1 2005 June Q6
9 marks Moderate -0.8
\includegraphics{figure_6} The diagram shows the velocity-time graph for a lift moving between floors in a building. The graph consists of straight line segments. In the first stage the lift travels downwards from the ground floor for \(5 \text{ s}\), coming to rest at the basement after travelling \(10 \text{ m}\).
  1. Find the greatest speed reached during this stage. [2]
The second stage consists of a \(10 \text{ s}\) wait at the basement. In the third stage, the lift travels upwards until it comes to rest at a floor \(34.5 \text{ m}\) above the basement, arriving \(24.5 \text{ s}\) after the start of the first stage. The lift accelerates at \(2 \text{ m s}^{-2}\) for the first \(3 \text{ s}\) of the third stage, reaching a speed of \(V \text{ m s}^{-1}\). Find
  1. the value of \(V\), [2]
  2. the time during the third stage for which the lift is moving at constant speed, [3]
  3. the deceleration of the lift in the final part of the third stage. [2]
CAIE M1 2005 June Q7
12 marks Standard +0.3
A car of mass \(1200 \text{ kg}\) travels along a horizontal straight road. The power provided by the car's engine is constant and equal to \(20 \text{ kW}\). The resistance to the car's motion is constant and equal to \(500 \text{ N}\). The car passes through the points \(A\) and \(B\) with speeds \(10 \text{ m s}^{-1}\) and \(25 \text{ m s}^{-1}\) respectively. The car takes \(30.5 \text{ s}\) to travel from \(A\) to \(B\).
  1. Find the acceleration of the car at \(A\). [4]
  2. By considering work and energy, find the distance \(AB\). [8]
CAIE M1 2009 June Q1
3 marks Easy -1.2
\includegraphics{figure_1} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground. [3]
CAIE M1 2009 June Q2
3 marks Moderate -0.5
\includegraphics{figure_2} A crate \(C\) is pulled at constant speed up a straight inclined path by a constant force of magnitude \(F\) N, acting upwards at an angle of 15° to the path. \(C\) passes through points \(P\) and \(Q\) which are 100 m apart (see diagram). As \(C\) travels from \(P\) to \(Q\) the work done against the resistance to \(C\)'s motion is 900 J, and the gain in \(C\)'s potential energy is 2100 J. Write down the work done by the pulling force as \(C\) travels from \(P\) to \(Q\), and hence find the value of \(F\). [3]
CAIE M1 2009 June Q3
5 marks Moderate -0.8
\includegraphics{figure_3} Forces of magnitudes 7 N, 10 N and 15 N act on a particle in the directions shown in the diagram.
  1. Find the component of the resultant of the three forces
    1. in the \(x\)-direction,
    2. in the \(y\)-direction.
    [3]
  2. Hence find the direction of the resultant. [2]
CAIE M1 2009 June Q4
6 marks Moderate -0.3
\includegraphics{figure_4} A block of mass 8 kg is at rest on a plane inclined at 20° to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).
  1. Given that the tension in the string is 13 N, find the frictional and normal components of the force exerted on the block by the plane. [4]
The string is cut; the block remains at rest, but is on the point of slipping down the plane.
  1. Find the coefficient of friction between the block and the plane. [2]
CAIE M1 2009 June Q5
9 marks Standard +0.3
\includegraphics{figure_5} A cyclist and his machine have a total mass of 80 kg. The cyclist starts from rest at the top \(A\) of a straight path and freewheels (moves without pedalling or braking) down the path to \(B\). The path \(AB\) is inclined at 2.6° to the horizontal and is of length 250 m (see diagram).
  1. Given that the cyclist passes through \(B\) with speed 9 m s\(^{-1}\), find the gain in kinetic energy and the loss in potential energy of the cyclist and his machine. Hence find the work done against the resistance to motion of the cyclist and his machine. [3]
The cyclist continues to freewheel along a horizontal straight path \(BD\) until he reaches the point \(C\), where the distance \(BC\) is \(d\) m. His speed at \(C\) is 5 m s\(^{-1}\). The resistance to motion is constant, and is the same on \(BD\) as on \(AB\).
  1. Find the value of \(d\). [3]
The cyclist starts to pedal at \(C\), generating 425 W of power.
  1. Find the acceleration of the cyclist immediately after passing through \(C\). [3]
CAIE M1 2009 June Q6
10 marks Standard +0.3
\includegraphics{figure_6} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s.
  1. Find the acceleration of \(A\) as it falls. [2]
The mass of \(A\) is 0.45 kg. Find
  1. the tension in the string while \(A\) is falling, [2]
  2. the mass of \(B\), [3]
  3. the maximum height above the floor reached by \(B\). [3]
CAIE M1 2009 June Q7
14 marks Standard +0.3
A particle \(P\) travels in a straight line from \(A\) to \(D\), passing through the points \(B\) and \(C\). For the section \(AB\) the velocity of the particle is \((0.5t - 0.01t^2)\) m s\(^{-1}\), where \(t\) s is the time after leaving \(A\).
  1. Given that the acceleration of \(P\) at \(B\) is 0.1 m s\(^{-2}\), find the time taken for \(P\) to travel from \(A\) to \(B\). [3]
The acceleration of \(P\) from \(B\) to \(C\) is constant and equal to 0.1 m s\(^{-2}\).
  1. Given that \(P\) reaches \(C\) with speed 14 m s\(^{-1}\), find the time taken for \(P\) to travel from \(B\) to \(C\). [3]
\(P\) travels with constant deceleration 0.3 m s\(^{-2}\) from \(C\) to \(D\). Given that the distance \(CD\) is 300 m, find
  1. the speed with which \(P\) reaches \(D\), [2]
  2. the distance \(AD\). [6]