Questions — Edexcel M1 (663 questions)

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Edexcel M1 2022 October Q6
9 marks Moderate -0.3
[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors.] A particle \(A\) of mass 0.5 kg is at rest on a smooth horizontal plane. At time \(t = 0\), two forces, \(\mathbf{F}_1 = (-3\mathbf{i} + 2\mathbf{j})\) N and \(\mathbf{F}_2 = (p\mathbf{i} + q\mathbf{j})\) N, where \(p\) and \(q\) are constants, are applied to \(A\). Given that \(A\) moves in the direction of the vector \((\mathbf{i} - 2\mathbf{j})\),
  1. show that \(2p + q - 4 = 0\) [4] Given that \(p = 5\)
  2. Find the speed of \(A\) at time \(t = 4\) seconds. [5]
Edexcel M1 2022 October Q7
13 marks Standard +0.3
\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. Another particle \(Q\), also of mass \(m\), is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, \(P\) and \(Q\) all lie in the same vertical plane. The coefficient of friction between \(Q\) and the table is \(\mu\), where \(\mu < 1\) Particle \(Q\) is released from rest. The tension in the string before \(Q\) hits the pulley is \(kmg\), where \(k\) is a constant.
  1. Find \(k\) in terms of \(\mu\). [7] Given that \(Q\) is initially a distance \(d\) from the pulley,
  2. find, in terms of \(d\), \(g\) and \(\mu\), the time taken by \(Q\), after release, to reach the pulley. [4]
  3. Describe what would happen if \(\mu \geqslant 1\), giving a reason for your answer. [2]
Edexcel M1 2022 October Q8
16 marks Moderate -0.3
[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(A\) and \(B\), are moving with constant velocities. The velocity of \(A\) is \((3\mathbf{i} + 12\mathbf{j})\text{ kmh}^{-1}\) and the velocity of \(B\) is \((p\mathbf{i} + q\mathbf{j})\text{ kmh}^{-1}\)
  1. Find the speed of \(A\). [2] The ships are modelled as particles. At 12 noon, \(A\) is at the point with position vector \((-9\mathbf{i} + 6\mathbf{j})\) km and \(B\) is at the point with position vector \((16\mathbf{i} + 6\mathbf{j})\) km. At time \(t\) hours after 12 noon, $$\overrightarrow{AB} = [(25 - 12t)\mathbf{i} - 9t\mathbf{j}] \text{ km}$$
  2. Find the value of \(p\) and the value of \(q\). [7]
  3. Find the bearing of \(A\) from \(B\) when the ships are 15 km apart, giving your answer to the nearest degree. [7]
Edexcel M1 Specimen Q1
5 marks Moderate -0.8
A particle \(P\) is moving with constant velocity \((-3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 6\) s \(P\) is at the point with position vector \((-4\mathbf{i} - 7\mathbf{j})\) m. Find the distance of \(P\) from the origin at time \(t = 2\) s. [5]
Edexcel M1 Specimen Q2
7 marks Moderate -0.3
Particle \(P\) has mass \(m\) kg and particle \(Q\) has mass \(3m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4u\) m s\(^{-1}\) and \(Q\) has speed \(ku\) m s\(^{-1}\), where \(k\) is a constant. 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 \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\). [3]
Edexcel M1 Specimen Q3
7 marks Standard +0.3
\includegraphics{figure_1} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac{1}{2}\). The box is pushed by a force of magnitude 100 N which acts at an angle of 30° with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box. [7]
Edexcel M1 Specimen Q4
7 marks Standard +0.3
A beam \(AB\) has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 1\) m. Two children, Sophie and Tom, each of weight 500 N, stand on the beam with Sophie standing twice as far from the end \(B\) as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at \(D\) is three times the magnitude of the reaction at \(C\). By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end \(B\). [7]
Edexcel M1 Specimen Q5
12 marks Standard +0.8
Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = 7\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).
  1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
  2. Find the value of \(T\). [8]
Edexcel M1 Specimen Q6
10 marks Moderate -0.8
A ball is projected vertically upwards with a speed of 14.7 m s\(^{-1}\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
  1. the greatest height, above the ground, reached by the ball, [4]
  2. the speed with which the ball first strikes the ground, [3]
  3. the total time from when the ball is projected to when it first strikes the ground. [3]
Edexcel M1 Specimen Q7
10 marks Standard +0.3
\includegraphics{figure_2} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{3}\). Given that the particle is on the point of sliding up the plane, find
  1. the magnitude of the normal reaction between the particle and the plane, [5]
  2. the value of \(P\). [5]
Edexcel M1 Specimen Q8
17 marks Standard +0.3
\includegraphics{figure_3} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.
  1. Find the tension in the string immediately after the particles are released. [6]
  2. Find the acceleration of \(A\) immediately after the particles are released. [2]
When the particles have been moving for 0.5 s, the string breaks.
  1. Find the further time that elapses until \(B\) hits the floor. [9]
Edexcel M1 2002 January Q1
3 marks Easy -1.2
A ball of mass 0.3 kg is moving vertically downwards with speed 8 m s\(^{-1}\) when it hits the floor which is smooth and horizontal. It rebounds vertically from the floor with speed 6 m s\(^{-1}\). Find the magnitude of the impulse exerted by the floor on the ball. [3]
Edexcel M1 2002 January Q2
6 marks Moderate -0.8
A railway truck \(A\) of mass 1800 kg is moving along a straight horizontal track with speed 4 m s\(^{-1}\). It collides directly with a stationary truck \(B\) of mass 1200 kg on the same track. In the collision, \(A\) and \(B\) are coupled and move off together.
  1. Find the speed of the trucks immediately after the collision. [3]
After the collision, the trucks experience a constant resistive force of magnitude \(R\) newtons. They come to rest 8 s after the collision.
  1. Find \(R\). [3]
Edexcel M1 2002 January Q3
8 marks Easy -1.2
A racing car moves with constant acceleration along a straight horizontal road. It passes the point \(O\) with speed 12 m s\(^{-1}\). It passes the point \(A\) 4 s later with speed 60 m s\(^{-1}\).
  1. Show that the acceleration of the car is 12 m s\(^{-2}\). [2]
  2. Find the distance \(OA\). [3]
The point \(B\) is the mid-point of \(OA\).
  1. Find, to 3 significant figures, the speed of the car when it passes \(B\). [3]
Edexcel M1 2002 January Q4
9 marks Standard +0.3
A motor scooter and a van set off along a straight road. They both start from rest at the same time and level with each other. The scooter accelerates with constant acceleration until it reaches its top speed of 20 m s\(^{-1}\). It then maintains a constant speed of 20 m s\(^{-1}\). The van accelerates with constant acceleration for 10 s until it reaches its top speed \(V\) m s\(^{-1}\), \(V > 20\). It then maintains a constant speed of \(V\) m s\(^{-1}\). The van draws level with the scooter when the scooter has been travelling for 40 s at its top speed. The total distance travelled by each vehicle is then 850 m.
  1. Sketch on the same diagram the speed-time graphs of both vehicles to illustrate their motion from the time when they start to the time when the van overtakes the scooter. [3]
  2. Find the time for which the scooter is accelerating. [3]
  3. Find the top speed of the van. [3]
Edexcel M1 2002 January Q5
10 marks Moderate -0.3
\includegraphics{figure_1} A heavy uniform steel girder \(AB\) has length 10 m. A load of weight 150 N is attached to the girder at \(A\) and a load of weight 250 N is attached to the girder at \(B\). The loaded girder hangs in equilibrium in a horizontal position, held by two vertical steel cables attached to the girder at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 3\) m, as shown in Fig. 1. The girder is modelled as a uniform rod, the loads as particles and the cables as light inextensible strings. The tension in the cable at \(D\) is three times the tension in the cable at \(C\).
  1. Draw a diagram showing all the forces acting on the girder. [2]
Find
  1. the tension in the cable at \(C\), [5]
  2. the weight of the girder. [2]
  3. Explain how you have used the fact that the girder is uniform. [1]
Edexcel M1 2002 January Q6
11 marks Moderate -0.8
A particle \(P\), of mass 3 kg, moves under the action of two constant forces (6\(\mathbf{i}\) + 2\(\mathbf{j}\)) N and (3\(\mathbf{i}\) - 5\(\mathbf{j}\)) N.
  1. Find, in the form (\(a\mathbf{i}\) + \(b\mathbf{j}\)) N, the resultant force \(\mathbf{F}\) acting on \(P\). [1]
  2. Find, in degrees to one decimal place, the angle between \(\mathbf{F}\) and \(\mathbf{j}\). [3]
  3. Find the acceleration of \(P\), giving your answer as a vector. [2]
The initial velocity of \(P\) is (-2\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\).
  1. Find, to 3 significant figures, the speed of \(P\) after 2 s. [5]
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]