Questions — Edexcel (10514 questions)

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Edexcel M1 2002 January Q7
12 marks Standard +0.3
\includegraphics{figure_2} A ring of mass 0.3 kg is threaded on a fixed, rough horizontal curtain pole. A light inextensible string is attached to the ring. The string and the pole lie in the same vertical plane. The ring is pulled downwards by the string which makes an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{3}{4}\) as shown in Fig. 2. The tension in the string is 2.5 N. Given that, in this position, the ring is in limiting equilibrium,
  1. find the coefficient of friction between the ring and the pole. [8]
\includegraphics{figure_3} The direction of the string is now altered so that the ring is pulled upwards. The string lies in the same vertical plane as before and again makes an angle \(\alpha\) with the horizontal, as shown in Fig. 3. The tension in the string is again 2.5 N.
  1. Find the normal reaction exerted by the pole on the ring. [2]
  2. State whether the ring is in equilibrium in the position shown in Fig. 3, giving a brief justification for your answer. You need make no further detailed calculation of the forces acting. [2]
Edexcel M1 2002 January Q8
16 marks Standard +0.3
\includegraphics{figure_4} Two particles \(P\) and \(Q\) have masses \(3m\) and \(5m\) respectively. They are connected by a light inextensible string which passes over a small smooth light pulley fixed at the edge of a rough horizontal table. Particle \(P\) lies on the table and particle \(Q\) hangs freely below the pulley, as shown in Fig. 4. The coefficient of friction between \(P\) and the table is 0.6. The system is released from rest with the string taut. For the period before \(Q\) hits the floor or \(P\) reaches the pulley,
  1. write down an equation of motion for each particle separately, [4]
  2. find, in terms of \(g\), the acceleration of \(Q\), [4]
  3. find, in terms of \(m\) and \(g\), the tension in the string. [2]
When \(Q\) has moved a distance \(h\), it hits the floor and the string becomes slack. Given that \(P\) remains on the table during the subsequent motion and does not reach the pulley,
  1. find, in terms of \(h\), the distance moved by \(P\) after the string becomes slack until \(P\) comes to rest. [6]
Edexcel M1 2003 January Q1
5 marks Moderate -0.8
A railway truck \(P\) of mass 2000 kg is moving along a straight horizontal track with speed 10 m s\(^{-1}\). The truck \(P\) collides with a truck \(Q\) of mass 3000 kg, which is at rest on the same track. Immediately after the collision \(Q\) moves with speed 5 m s\(^{-1}\). Calculate
  1. the speed of \(P\) immediately after the collision, [3]
  2. the magnitude of the impulse exerted by \(P\) on \(Q\) during the collision. [2]
Edexcel M1 2003 January Q2
6 marks Moderate -0.3
\includegraphics{figure_1} In Fig. 1, \(\angle AOC = 90°\) and \(\angle BOC = \theta°\). A particle at \(O\) is in equilibrium under the action of three coplanar forces. The three forces have magnitude 8 N, 12 N and \(X\) N and act along \(OA\), \(OB\) and \(OC\) respectively. Calculate
  1. the value, to one decimal place, of \(\theta\), [3]
  2. the value, to 2 decimal places, of \(X\). [3]
Edexcel M1 2003 January Q3
6 marks Moderate -0.8
A particle \(P\) of mass 0.4 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. Initially the velocity of \(P\) is \((6\mathbf{i} - 2\mathbf{j})\) m s\(^{-1}\) and 4 s later the velocity of \(P\) is \((-14\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\).
  1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\). [3]
  2. Calculate the magnitude of \(\mathbf{F}\). [3]
Edexcel M1 2003 January Q4
8 marks Moderate -0.8
Two ships \(P\) and \(Q\) are moving along straight lines with constant velocities. Initially \(P\) is at a point \(O\) and the position vector of \(Q\) relative to \(O\) is \((6\mathbf{i} + 12\mathbf{j})\) km, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. The ship \(P\) is moving with velocity \(10\mathbf{j}\) km h\(^{-1}\) and \(Q\) is moving with velocity \((-8\mathbf{i} + 6\mathbf{j})\) km h\(^{-1}\). At time \(t\) hours the position vectors of \(P\) and \(Q\) relative to \(O\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively.
  1. Find \(\mathbf{p}\) and \(\mathbf{q}\) in terms of \(t\). [3]
  2. Calculate the distance of \(Q\) from \(P\) when \(t = 3\). [3]
  3. Calculate the value of \(t\) when \(Q\) is due north of \(P\). [2]
Edexcel M1 2003 January Q5
10 marks Standard +0.3
\includegraphics{figure_1} A box of mass 1.5 kg is placed on a plane which is inclined at an angle of 30° to the horizontal. The coefficient of friction between the box and plane is \(\frac{1}{4}\). The box is kept in equilibrium by a light string which lies in a vertical plane containing a line of greatest slope of the plane. The string makes an angle of 20° with the plane, as shown in Fig. 2. The box is in limiting equilibrium and is about to move up the plane. The tension in the string is \(T\) newtons. The box is modelled as a particle. Find the value of \(T\). [10]
Edexcel M1 2003 January Q6
10 marks Standard +0.3
\includegraphics{figure_3} A uniform rod \(AB\) has length 3 m and weight 120 N. The rod rests in equilibrium in a horizontal position, smoothly supported at points \(C\) and \(D\), where \(AC = 0.5\) m and \(AD = 2\) m, as shown in Fig. 3. A particle of weight \(W\) newtons is attached to the rod at a point \(E\) where \(AE = x\) metres. The rod remains in equilibrium and the magnitude of the reaction at \(C\) is now twice the magnitude of the reaction at \(D\).
  1. Show that \(W = \frac{60}{1-x}\). [8]
  2. Hence deduce the range of possible values of \(x\). [2]
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]