Questions M1 (2067 questions)

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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]
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]