Questions M1 (2067 questions)

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CAIE M1 Specimen Q2
6 marks Moderate -0.8
A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of 30° to the horizontal.
  1. Find the time taken for the particle to reach a speed of 2.5 m s\(^{-1}\). [3]
  2. Find the distance that the particle travels along the ground before it comes to rest. [3]
When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N.
CAIE M1 Specimen Q3
6 marks Standard +0.3
A lorry of mass 24 000 kg is travelling up a hill which is inclined at 3° to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N.
  1. When the speed of the lorry is 25 m s\(^{-1}\), its acceleration is 0.2 m s\(^{-2}\). Find the power developed by the lorry's engine. [4]
  2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N. [2]
CAIE M1 Specimen Q4
6 marks Standard +0.3
\includegraphics{figure_4} Blocks \(P\) and \(Q\), of mass \(m\) kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at 35° to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2. Find the set of values of \(m\) for which the two blocks remain at rest. [6]
CAIE M1 Specimen Q5
8 marks Standard +0.3
\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.
  1. Find the values of \(F\) and \(R\). [5]
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]
CAIE M1 Specimen Q6
10 marks Standard +0.3
A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$
  1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
  2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]
CAIE M1 Specimen Q7
10 marks Standard +0.3
A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration 0.5 m s\(^{-2}\) for a distance of 36 m. The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(AB\) is 210 m.
  1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). [5]
  2. Find the time that it takes from when the cyclist starts until the car overtakes her. [5]
24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration 4 m s\(^{-2}\) towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
Edexcel M1 2015 January Q1
7 marks Moderate -0.3
A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3u\) and the speed of truck \(B\) is \(2u\). In the collision the trucks join together. Modelling the trucks as particles, find
  1. the speed of \(A\) immediately after the collision, [3]
  2. the direction of motion of \(A\) immediately after the collision, [1]
  3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision. [3]
Edexcel M1 2015 January Q2
8 marks Standard +0.3
\includegraphics{figure_1} A block of mass 50 kg lies on a rough plane which is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{7}{24}\). The block is held at rest by a vertical rope, as shown in Figure 1, and is on the point of sliding down the plane. The block is modelled as a particle and the rope is modelled as a light inextensible string. Given that the friction force acting on the block has magnitude 65.8 N, find
  1. the tension in the rope, [4]
  2. the coefficient of friction between the block and the plane. [4]
Edexcel M1 2015 January Q3
7 marks Moderate -0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively.] A particle \(P\) is moving with constant velocity \((-6\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 0\), \(P\) passes through the point with position vector \((21\mathbf{i} + 5\mathbf{j})\) m, relative to a fixed origin \(O\).
  1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. [3]
  2. Write down the position vector of \(P\) at time \(t\) seconds. [1]
  3. Find the time at which \(P\) is north-west of \(O\). [3]
Edexcel M1 2015 January Q4
7 marks Standard +0.3
The points \(P\) and \(Q\) are at the same height \(h\) metres above horizontal ground. A small stone is dropped from rest from \(P\). Half a second later a second small stone is thrown vertically downwards from \(Q\) with speed 7.35 m s\(^{-1}\). Given that the stones hit the ground at the same time, find the value of \(h\). [7]
Edexcel M1 2015 January Q5
10 marks Standard +0.8
\includegraphics{figure_2} A particle \(P\) of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in the vertical plane which contains \(P\) and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is 0.5 Given that the acceleration of \(P\) is 1.45 m s\(^{-2}\), find the value of \(X\). [10]
Edexcel M1 2015 January Q6
10 marks Standard +0.3
A uniform rod \(AC\), of weight \(W\) and length \(3l\), rests horizontally on two supports, one at \(A\) and one at \(B\), where \(AB = 2l\). A particle of weight \(2W\) is placed on the rod at a distance \(x\) from \(A\). The rod remains horizontal and in equilibrium.
  1. Find the greatest possible value of \(x\). [5]
The magnitude of the reaction of the support at \(A\) is \(R\). Due to a weakness in the support at \(A\), the greatest possible value of \(R\) is \(4W\).
  1. find the least possible value of \(x\). [5]
Edexcel M1 2015 January Q7
10 marks Moderate -0.8
A train travels along a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration until it reaches its maximum speed of 108 km h\(^{-1}\). The train then travels at this speed before it moves with constant deceleration coming to rest at \(B\). The journey from \(A\) to \(B\) takes 8 minutes.
  1. Change 108 km h\(^{-1}\) into m s\(^{-1}\). [2]
  2. Sketch a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). [2]
Given that the distance between the two stations is 12 km and that the time spent decelerating is three times the time spent accelerating,
  1. find the acceleration, in m s\(^{-2}\), of the train. [6]
Edexcel M1 2015 January Q8
16 marks Standard +0.3
\includegraphics{figure_3} A particle \(A\) of mass \(3m\) is held at rest on a rough horizontal table. The particle is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass \(2m\), which hangs freely, vertically below \(P\). The system is released from rest, with the string taut, when \(A\) is 1.3 m from \(P\) and \(B\) is 1 m above the horizontal floor, as shown in Figure 3. Given that \(B\) hits the floor 2 s after release and does not rebound,
  1. find the acceleration of \(A\) during the first two seconds, [2]
  2. find the coefficient of friction between \(A\) and the table, [8]
  3. determine whether \(A\) reaches the pulley. [6]
Edexcel M1 2016 January Q1
7 marks Moderate -0.3
A truck of mass 2400 kg is pulling a trailer of mass \(M\) kg along a straight horizontal road. The tow bar, connecting the truck to the trailer, is horizontal and parallel to the direction of motion. The tow bar is modelled as being light and inextensible. The resistance forces acting on the truck and the trailer are constant and of magnitude 400 N and 200 N respectively. The acceleration of the truck is 0.5 m s\(^{-2}\) and the tension in the tow bar is 600 N.
  1. Find the magnitude of the driving force of the truck. [3]
  2. Find the value of \(M\). [3]
  3. Explain how you have used the fact that the tow bar is inextensible in your calculations. [1]
Edexcel M1 2016 January Q2
8 marks Moderate -0.3
Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) is moving due east and particle \(Q\) is moving due west. Particle \(P\) has mass \(2m\) and particle \(Q\) has mass \(3m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(4u\) and the speed of \(Q\) is \(u\). The magnitude of the impulse in the collision is \(\frac{33}{5}mu\).
  1. Find the speed and direction of motion of \(P\) immediately after the collision. [4]
  2. Find the speed and direction of motion of \(Q\) immediately after the collision. [4]
Edexcel M1 2016 January Q3
8 marks Standard +0.3
\includegraphics{figure_1} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at 30° to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac{1}{5}\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope. [8]
Edexcel M1 2016 January Q4
13 marks Moderate -0.3
A small stone is projected vertically upwards from the point \(O\) and moves freely under gravity. The point \(A\) is 3.6 m vertically above \(O\). When the stone first reaches \(A\), the stone is moving upwards with speed 11.2 m s\(^{-1}\). The stone is modelled as a particle.
  1. Find the maximum height above \(O\) reached by the stone. [4]
  2. Find the total time between the instant when the stone was projected from \(O\) and the instant when it returns to \(O\). [5]
  3. Sketch a velocity-time graph to represent the motion of the stone from the instant when it passes through \(A\) moving upwards to the instant when it returns to \(O\). Show, on the axes, the coordinates of the points where your graph meets the axes. [4]
Edexcel M1 2016 January Q5
10 marks Moderate -0.3
\includegraphics{figure_2} A non-uniform rod \(AB\) has length 4 m and weight 120 N. The centre of mass of the rod is at the point \(G\) where \(AG = 2.2\) m. The rod is suspended in a horizontal position by two vertical light inextensible strings, one at each end, as shown in Figure 2. A particle of weight 40 N is placed on the rod at the point \(P\), where \(AP = x\) metres. The rod remains horizontal and in equilibrium.
  1. Find, in terms of \(x\),
    1. the tension in the string at \(A\), [6]
    2. the tension in the string at \(B\).
    Either string will break if the tension in it exceeds 84 N.
  2. Find the range of possible values of \(x\). [4]
Edexcel M1 2016 January Q6
13 marks Moderate -0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm, the position vector of ship \(P\) is \((5\mathbf{i} - 3\mathbf{j})\) km and the position vector of ship \(Q\) is \((7\mathbf{i} + 5\mathbf{j})\) km.
  1. Find the distance between \(P\) and \(Q\) at 2 pm. [3]
Ship \(P\) is moving with constant velocity \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) is moving with constant velocity \((-3\mathbf{i} - 15\mathbf{j})\) km h\(^{-1}\).
  1. Find the position vector of \(P\) at time \(t\) hours after 2 pm. [2]
  2. Find the position vector of \(Q\) at time \(t\) hours after 2 pm. [1]
  3. Show that \(Q\) will meet \(P\) and find the time at which they meet. [5]
  4. Find the position vector of the point at which they meet. [2]
Edexcel M1 2016 January Q7
16 marks Standard +0.3
\includegraphics{figure_3} A particle \(P\) of mass 2 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) is at rest on the table, a distance \(d\) metres from the pulley. Particle \(Q\) is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between \(P\) and the table is \(\frac{1}{4}\). The system is released from rest and \(P\) slides along the table towards the pulley. Assuming that \(P\) has not reached the pulley and that \(Q\) remains on the inclined plane,
  1. write down an equation of motion for \(P\), [2]
  2. write down an equation of motion for \(Q\), [2]
    1. find the acceleration of \(P\),
    2. find the tension in the string. [5]
When \(P\) has moved a distance 0.5 m from its initial position, the string breaks. Given that \(P\) comes to rest just as it reaches the pulley,
  1. find the value of \(d\). [7]
Edexcel M1 2016 June Q1
7 marks Moderate -0.3
A car is moving along a straight horizontal road with constant acceleration \(a\) m s\(^{-2}\) (\(a > 0\)). At time \(t = 0\) the car passes the point \(P\) moving with speed \(u\) m s\(^{-1}\). In the next 4 s, the car travels 76 m and then in the following 6 s it travels a further 219 m. Find
  1. the value of \(u\),
  2. the value of \(a\).
[7]
Edexcel M1 2016 June Q2
6 marks Moderate -0.3
Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(km\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2u\). As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.
  1. Find the value of \(k\). [4]
  2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision. [2]
Edexcel M1 2016 June Q3
10 marks Standard +0.3
A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.
  1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
  2. find the value of \(R\). [8]
Edexcel M1 2016 June Q4
10 marks Moderate -0.3
\includegraphics{figure_1} A diving board \(AB\) consists of a wooden plank of length 4 m and mass 30 kg. The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(AC = 0.6\) m, as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.
  1. Find
    1. the magnitude of the force acting on the plank at \(A\),
    2. the magnitude of the force acting on the plank at \(C\).
    [6] The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N.
  2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\). [3]
  3. Explain how you have used the fact that the diver is modelled as a particle. [1]