Questions — Edexcel M1 (663 questions)

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Edexcel M1 2003 January Q7
14 marks Moderate -0.3
A ball is projected vertically upwards with a speed \(u\) m s\(^{-1}\) from a point \(A\) which is 1.5 m above the ground. The ball moves freely under gravity until it reaches the ground. The greatest height attained by the ball is 25.6 m above \(A\).
  1. Show that \(u = 22.4\). [3]
The ball reaches the ground 7 seconds after it has been projected from \(A\).
  1. Find, to 2 decimal places, the value of \(T\). [4]
The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The mass of the ball is 0.6 kg. The ground is assumed to exert a constant resistive force of magnitude \(F\) newtons.
  1. Find, to 3 significant figures, the value of \(F\). [6]
  2. State one physical factor which could be taken into account to make the model used in this question more realistic. [1]
Edexcel M1 2003 January Q8
16 marks Standard +0.3
\includegraphics{figure_4} A particle \(A\) of mass 0.8 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass 1.2 kg which hangs freely below the pulley, as shown in Fig. 4. The system is released from rest with the string taut and with \(B\) at a height of 0.6 m above the ground. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find
  1. the tension in the string before \(B\) reaches the ground, [5]
  2. the time taken by \(B\) to reach the ground. [3]
In a refinement of the model, it is assumed that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac{1}{4}\). Using this refined model,
  1. find the time taken by \(B\) to reach the ground. [8]
Edexcel M1 2004 January Q1
7 marks Moderate -0.8
Two trucks \(A\) and \(B\), moving in opposite directions on the same horizontal railway track, collide. The mass of \(A\) is 600 kg. The mass of \(B\) is \(m\) kg. Immediately before the collision, the speed of \(A\) is 4 m s\(^{-1}\) and the speed of \(B\) is 2 m s\(^{-1}\). Immediately after the collision, the trucks are joined together and move with the same speed 0.5 m s\(^{-1}\). The direction of motion of \(A\) is unchanged by the collision. Find
  1. the value of \(m\), [4]
  2. the magnitude of the impulse exerted on \(A\) in the collision. [3]
Edexcel M1 2004 January Q2
9 marks Moderate -0.8
\includegraphics{figure_1} A lever consists of a uniform steel rod \(AB\), of weight 100 N and length 2 m, which rests on a small smooth pivot at a point \(C\) of the rod. A load of weight 2200 N is suspended from the end \(B\) of the rod by a rope. The lever is held in equilibrium in a horizontal position by a vertical force applied at the end \(A\), as shown in Fig. 1. The rope is modelled as a light string. Given that \(BC = 0.2\) m,
  1. find the magnitude of the force applied at \(A\). [4]
The position of the pivot is changed so that the rod remains in equilibrium when the force at \(A\) has magnitude 1200 N.
  1. Find, to the nearest cm, the new distance of the pivot from \(B\). [5]
Edexcel M1 2004 January Q3
10 marks Moderate -0.3
The tile on a roof becomes loose and slides from rest down the roof. The roof is modelled as a plane surface inclined at 30° to the horizontal. The coefficient of friction between the tile and the roof is 0.4. The tile is modelled as a particle of mass \(m\) kg.
  1. Find the acceleration of the tile as it slides down the roof. [7]
The tile moves a distance 3 m before reaching the edge of the roof.
  1. Find the speed of the tile as it reaches the edge of the roof. [2]
  2. Write down the answer to part (a) if the tile had mass \(2m\) kg. [1]
Edexcel M1 2004 January Q4
10 marks Standard +0.3
\includegraphics{figure_2} Two small rings, \(A\) and \(B\), each of mass \(2m\), are threaded on a rough horizontal pole. The coefficient of friction between each ring and the pole is \(\mu\). The rings are attached to the ends of a light inextensible string. A smooth ring \(C\), of mass \(3m\), is threaded on the string and hangs in equilibrium below the pole. The rings \(A\) and \(B\) are in limiting equilibrium on the pole, with \(\angle BAC = \angle ABC = \theta\), where \(\tan \theta = \frac{3}{4}\), as shown in Fig. 2.
  1. Show that the tension in the string is \(\frac{5}{2}mg\). [3]
  2. Find the value of \(\mu\). [7]
Edexcel M1 2004 January Q5
11 marks Standard +0.3
\includegraphics{figure_3} A particle \(A\) of mass 4 kg moves on the inclined face of a smooth wedge. This face is inclined at 30° to the horizontal. The wedge is fixed on horizontal ground. Particle \(A\) is connected to a particle \(B\), of mass 3 kg, by a light inextensible string. The string passes over a small light smooth pulley which is fixed at the top of the plane. The section of the string from \(A\) to the pulley lies in a line of greatest slope of the wedge. The particle \(B\) hangs freely below the pulley, as shown in Fig. 3. The system is released from rest with the string taut. For the motion before \(A\) reaches the pulley and before \(B\) hits the ground, find
  1. the tension in the string, [6]
  2. the magnitude of the resultant force exerted by the string on the pulley. [3]
  1. The string in this question is described as being 'light'.
    1. Write down what you understand by this description.
    2. State how you have used the fact that the string is light in your answer to part (a). [2]
Edexcel M1 2004 January Q6
14 marks Moderate -0.8
A train starts from rest at a station \(A\) and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s\(^{-2}\). For the next 24 s it moves at a constant acceleration 0.75 m s\(^{-2}\). It then moves with constant speed for \(T\) seconds. Finally it slows down with constant deceleration 3 m s\(^{-2}\) until it comes to a rest at station \(B\).
  1. Show that, 34 s after leaving \(A\), the speed of the train is 30 m s\(^{-1}\). [3]
  2. Sketch a speed-time graph to illustrate the motion of the train as it moves from \(A\) to \(B\). [3]
  3. Find the distance moved by the train during the first 34 s of its journey from \(A\). [4]
The distance from \(A\) to \(B\) is 3 km.
  1. Find the value of \(T\). [4]
Edexcel M1 2004 January Q7
14 marks Moderate -0.3
[In this question the vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors in the direction due east and due north respectively.] Two boats \(A\) and \(B\) are moving with constant velocities. Boat \(A\) moves with velocity \(9\mathbf{j}\) km h\(^{-1}\). Boat \(B\) moves with velocity \((3\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).
  1. Find the bearing on which \(B\) is moving. [2]
At noon, \(A\) is at point \(O\), and \(B\) is 10 km due west of \(O\). At time \(t\) hours after noon, the position vectors of \(A\) and \(B\) relative to \(O\) are \(\mathbf{a}\) km and \(\mathbf{b}\) km respectively.
  1. Find expressions for \(\mathbf{a}\) and \(\mathbf{b}\) in terms of \(t\), giving your answer in the form \(p\mathbf{i} + q\mathbf{j}\). [3]
  2. Find the time when \(B\) is due south of \(A\). [2]
At time \(t\) hours after noon, the distance between \(A\) and \(B\) is \(d\) km. By finding an expression for \(\overrightarrow{AB}\),
  1. show that \(d^2 = 25t^2 - 60t + 100\). [4]
At noon, the boats are 10 km apart.
  1. Find the time after noon at which the boats are again 10 km apart. [3]
Edexcel M1 2005 January Q1
7 marks Moderate -0.8
A particle \(P\) of mass 1.5 kg is moving along a straight horizontal line with speed 3 m s\(^{-1}\). Another particle \(Q\) of mass 2.5 kg is moving, in the opposite direction, along the same straight line with speed 4 m s\(^{-1}\). The particles collide. Immediately after the collision the direction of motion of \(P\) is reversed and its speed is 2.5 m s\(^{-1}\).
  1. Calculate the speed of \(Q\) immediately after the impact. [3]
  2. State whether or not the direction of motion of \(Q\) is changed by the collision. [1]
  3. Calculate the magnitude of the impulse exerted by \(Q\) on \(P\), giving the units of your answer. [3]
Edexcel M1 2005 January Q2
7 marks Moderate -0.3
\includegraphics{figure_1} A plank \(AB\) has mass 40 kg and length 3 m. A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(AB\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate
  1. the tension in the rope at \(C\), [2]
  2. the distance \(CB\). [5]
Edexcel M1 2005 January Q3
9 marks Moderate -0.8
\includegraphics{figure_2} A sprinter runs a race of 200 m. Her total time for running the race is 25 s. Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of 9 m s\(^{-1}\) in 4 s. The speed of 9 m s\(^{-1}\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u\) m s\(^{-1}\) at the end of the race. Calculate
  1. the distance covered by the sprinter in the first 20 s of the race, [2]
  2. the value of \(u\), [4]
  3. the deceleration of the sprinter in the last 5 s of the race. [3]
Edexcel M1 2005 January Q4
10 marks Moderate -0.8
\includegraphics{figure_3} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at 20° to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4. The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate
  1. the normal reaction of the plane on \(P\), [2]
  2. the value of \(X\). [4]
The force of magnitude \(X\) newtons is now removed.
  1. Show that \(P\) remains in equilibrium on the plane. [4]
Edexcel M1 2005 January Q5
13 marks Standard +0.3
\includegraphics{figure_4} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s. Modelling \(A\) and \(B\) as particles, calculate
  1. the acceleration of \(B\), [3]
  2. the tension in the string, [4]
  3. the value of \(\mu\). [5]
  4. State how in your calculations you have used the information that the string is inextensible. [1]
Edexcel M1 2005 January Q6
13 marks Moderate -0.3
A stone \(S\) is sliding on ice. The stone is moving along a straight horizontal line \(ABC\), where \(AB = 24\) m and \(AC = 30\) m. The stone is subject to a constant resistance to motion of magnitude 0.3 N. At \(A\) the speed of \(S\) is 20 m s\(^{-1}\), and at \(B\) the speed of \(S\) is 16 m s\(^{-1}\). Calculate
  1. the deceleration of \(S\), [2]
  2. the speed of \(S\) at \(C\). [3]
  3. Show that the mass of \(S\) is 0.1 kg. [2]
At \(C\), the stone \(S\) hits a vertical wall, rebounds from the wall and then slides back along the line \(CA\). The magnitude of the impulse of the wall on \(S\) is 2.4 Ns and the stone continues to move against a constant resistance of 0.3 N.
  1. Calculate the time between the instant that \(S\) rebounds from the wall and the instant that \(S\) comes to rest. [6]
Edexcel M1 2005 January Q7
16 marks Moderate -0.3
Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \((20\mathbf{i} + 10\mathbf{j})\) km relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \((14\mathbf{i} - 6\mathbf{j})\) km. Three hours later, \(P\) is at the point with position vector \((29\mathbf{i} + 34\mathbf{j})\) km. The ship \(Q\) travels with velocity \(12\mathbf{j}\) km h\(^{-1}\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively. Find
  1. the velocity of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
  2. expressions for \(\mathbf{p}\) and \(\mathbf{q}\), in terms of \(t\), \(\mathbf{i}\) and \(\mathbf{j}\). [4]
At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d\) km.
  1. By finding an expression for \(\overrightarrow{PQ}\), show that $$d^2 = 25t^2 - 92t + 292.$$ [5]
Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
  1. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer. [5]
Edexcel M1 2006 January Q1
6 marks Moderate -0.8
A stone is thrown vertically upwards with speed \(16 \text{ m s}^{-1}\) from a point \(h\) metres above the ground. The stone hits the ground \(4\) s later. Find
  1. the value of \(h\), [3]
  2. the speed of the stone as it hits the ground. [3]
Edexcel M1 2006 January Q2
8 marks Moderate -0.8
  1. Two particles \(A\) and \(B\), of mass \(3\) kg and \(2\) kg respectively, are moving in the same direction on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4 \text{ m s}^{-1}\) and the speed of \(B\) is \(1.5 \text{ m s}^{-1}\). In the collision, the particles join to form a single particle \(C\). Find the speed of \(C\) immediately after the collision. [3]
  2. Two particles \(P\) and \(Q\) have mass \(3\) kg and \(m\) kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. Each particle has speed \(4 \text{ m s}^{-1}\), when they collide directly. In this collision, the direction of motion of each particle is reversed. The speed of \(P\) immediately after the collision is \(2 \text{ m s}^{-1}\) and the speed of \(Q\) is \(1 \text{ m s}^{-1}\). Find
    1. the value of \(m\), [3]
    2. the magnitude of the impulse exerted on \(Q\) in the collision. [2]
Edexcel M1 2006 January Q3
8 marks Moderate -0.8
\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,
  1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
  2. State what is implied by the modelling assumption that the beam is uniform. [1]
David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,
  1. find the distance of the centre of mass of the beam from \(C\). [4]
Edexcel M1 2006 January Q4
9 marks Moderate -0.3
Two forces \(\mathbf{P}\) and \(\mathbf{Q}\) act on a particle. The force \(\mathbf{P}\) has magnitude \(7\) N and acts due north. The resultant of \(\mathbf{P}\) and \(\mathbf{Q}\) is a force of magnitude \(10\) N acting in a direction with bearing \(120°\). Find
  1. the magnitude of \(\mathbf{Q}\),
  2. the direction of \(\mathbf{Q}\), giving your answer as a bearing.
[9]
Edexcel M1 2006 January Q5
14 marks Standard +0.3
\includegraphics{figure_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find
  1. the value of \(P\), [4]
  2. the value of \(\mu\). [5]
The horizontal force is removed.
  1. Determine whether or not the parcel moves. [5]
Edexcel M1 2006 January Q6
16 marks Moderate -0.8
[In this question the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are due east and due north respectively.] A model boat \(A\) moves on a lake with constant velocity \((-\mathbf{i} + 6\mathbf{j}) \text{ m s}^{-1}\). At time \(t = 0\), \(A\) is at the point with position vector \((2\mathbf{i} - 10\mathbf{j})\) m. Find
  1. the speed of \(A\), [2]
  2. the direction in which \(A\) is moving, giving your answer as a bearing. [3]
At time \(t = 0\), a second boat \(B\) is at the point with position vector \((-26\mathbf{i} + 4\mathbf{j})\) m. Given that the velocity of \(B\) is \((3\mathbf{i} + 4\mathbf{j}) \text{ m s}^{-1}\),
  1. show that \(A\) and \(B\) will collide at a point \(P\) and find the position vector of \(P\). [5]
Given instead that \(B\) has speed \(8 \text{ m s}^{-1}\) and moves in the direction of the vector \((3\mathbf{i} + 4\mathbf{j})\),
  1. find the distance of \(B\) from \(P\) when \(t = 7\) s. [6]
Edexcel M1 2006 January Q7
14 marks Standard +0.3
\includegraphics{figure_3} A fixed wedge has two plane faces, each inclined at \(30°\) to the horizontal. Two particles \(A\) and \(B\), of mass \(3m\) and \(m\) respectively, are attached to the ends of a light inextensible string. Each particle moves on one of the plane faces of the wedge. The string passes over a small smooth light pulley fixed at the top of the wedge. The face on which \(A\) moves is smooth. The face on which \(B\) moves is rough. The coefficient of friction between \(B\) and this face is \(\mu\). Particle \(A\) is held at rest with the string taut. The string lies in the same vertical plane as lines of greatest slope on each plane face of the wedge, as shown in Figure 3. The particles are released from rest and start to move. Particle \(A\) moves downwards and \(B\) moves upwards. The accelerations of \(A\) and \(B\) each have magnitude \(\frac{1}{10}g\).
  1. By considering the motion of \(A\), find, in terms of \(m\) and \(g\), the tension in the string. [3]
  2. By considering the motion of \(B\), find the value of \(\mu\). [8]
  3. Find the resultant force exerted by the string on the pulley, giving its magnitude and direction. [3]
Edexcel M1 2007 January Q1
6 marks Moderate -0.8
\includegraphics{figure_1} A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of 30° to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find
  1. the value of \(P\), [3]
  2. the value of \(Q\). [3]
Edexcel M1 2007 January Q2
10 marks Moderate -0.3
\includegraphics{figure_2} A uniform plank \(AB\) has weight 120 N and length 3 m. The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1\) m and \(CD = x\) m, as shown in Figure 2. The reaction of the support on the plank at \(D\) has magnitude 80 N. Modelling the plank as a rod,
  1. show that \(x = 0.75\) [3]
A rock is now placed at \(B\) and the plank is on the point of tilting about \(D\). Modelling the rock as a particle, find
  1. the weight of the rock, [4]
  2. the magnitude of the reaction of the support on the plank at \(D\). [2]
  3. State how you have used the model of the rock as a particle. [1]