Questions M1 (2067 questions)

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Edexcel M1 2010 January Q6
14 marks Standard +0.3
\includegraphics{figure_4} Two particles \(A\) and \(B\) have masses \(5m\) and \(km\) respectively, where \(k < 5\). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut, the hanging parts of the string vertical and with \(A\) and \(B\) at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, \(A\) descends with acceleration \(\frac{1}{4}g\).
  1. Show that the tension in the string as \(A\) descends is \(\frac{15}{4}mg\). [3]
  2. Find the value of \(k\). [3]
  3. State how you have used the information that the pulley is smooth. [1]
After descending for 1.2 s, the particle \(A\) reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between \(B\) and the pulley is such that, in the subsequent motion, \(B\) does not reach the pulley.
  1. Find the greatest height reached by \(B\) above the plane. [7]
Edexcel M1 2010 January Q7
14 marks Moderate -0.3
[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving along a straight line with constant velocity. At time \(t\) hours the position vector of \(S\) is \(\mathbf{s}\) km. When \(t = 0\), \(\mathbf{s} = 9\mathbf{i} - 6\mathbf{j}\). When \(t = 4\), \(\mathbf{s} = 21\mathbf{i} + 10\mathbf{j}\). Find
  1. the speed of \(S\), [4]
  2. the direction in which \(S\) is moving, giving your answer as a bearing. [2]
  3. Show that \(\mathbf{s} = (3t + 9)\mathbf{i} + (4t - 6)\mathbf{j}\). [2]
A lighthouse \(L\) is located at the point with position vector \((18\mathbf{i} + 6\mathbf{j})\) km. When \(t = T\), the ship \(S\) is 10 km from \(L\).
  1. Find the possible values of \(T\). [6]
Edexcel M1 2013 January Q1
7 marks Moderate -0.8
Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2u\) and \(5u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac{1}{2}u\) and its direction of motion is reversed.
  1. Find the speed and direction of motion of \(Q\) after the collision. [4]
  2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision. [3]
Edexcel M1 2013 January Q2
9 marks Moderate -0.3
A steel girder \(AB\), of mass 200 kg and length 12 m, rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(AC = 2\) m and \(DB = 2\) m. A man of mass 80 kg stands on the girder at the point \(P\), where \(AP = 4\) m, as shown in Figure 1.
[diagram]
The man is modelled as a particle and the girder is modelled as a uniform rod.
  1. Find the magnitude of the reaction on the girder at the support at \(C\). [3]
The support at \(D\) is now moved to the point \(X\) on the girder, where \(XB = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2.
[diagram]
Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find
  1. the magnitude of the reaction at the support at \(X\), [2]
  2. the value of \(x\). [4]
Edexcel M1 2013 January Q3
8 marks Moderate -0.3
A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(OP\) at \(30°\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of \(30°\) to the horizontal, as shown in Figure 3. \includegraphics{figure_3} Find
  1. the value of \(F\),
  2. the tension in the string. [8]
Edexcel M1 2013 January Q4
9 marks Standard +0.3
A lifeboat slides down a straight ramp inclined at an angle of \(15°\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s\(^{-1}\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp. [9]
Edexcel M1 2013 January Q5
15 marks Moderate -0.8
\includegraphics{figure_4} The velocity-time graph in Figure 4 represents the journey of a train \(P\) travelling along a straight horizontal track between two stations which are 1.5 km apart. The train \(P\) leaves the first station, accelerating uniformly from rest for 300 m until it reaches a speed of 30 m s\(^{-1}\). The train then maintains this speed for 7 seconds before decelerating uniformly at 1.25 m s\(^{-2}\), coming to rest at the next station.
  1. Find the acceleration of \(P\) during the first 300 m of its journey. [2]
  2. Find the value of \(T\). [5]
A second train \(Q\) completes the same journey in the same total time. The train leaves the first station, accelerating uniformly from rest until it reaches a speed of \(V\) m s\(^{-1}\) and then immediately decelerates uniformly until it comes to rest at the next station.
  1. Sketch on the diagram above, a velocity-time graph which represents the journey of train \(Q\). [2]
  2. Find the value of \(V\). [6]
Edexcel M1 2013 January Q6
11 marks Standard +0.3
[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \((4\mathbf{i} - 8\mathbf{j})\) km. At 9.30 am the ship is at the point with position vector \((\mathbf{i} - 4\mathbf{j})\) km.
  1. Find the speed of the ship in km h\(^{-1}\). [4]
  2. Show that the position vector \(\mathbf{r}\) km of the ship, \(t\) hours after 9 am, is given by \(\mathbf{r} = (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}\). [2]
At 10 am, a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am, the passenger observes that \(L\) is now south-west of the ship.
  1. Find the position vector of \(L\). [5]
Edexcel M1 2013 January Q7
16 marks Standard +0.8
\includegraphics{figure_5} Figure 5 shows two particles \(A\) and \(B\), of mass \(2m\) and \(4m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,
  1. give a reason why the magnitudes of the accelerations of the two particles are the same, [1]
  2. write down an equation of motion for each particle, [4]
  3. find the acceleration of each particle. [5]
Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),
  1. find the distance \(XY\) in terms of \(h\). [6]
Edexcel M1 2002 June Q1
6 marks Moderate -0.8
A car moves with constant acceleration along a straight horizontal road. The car passes the point \(A\) with speed \(5 \text{ m s}^{-1}\) and \(4 \text{ s}\) later it passes the point \(B\), where \(AB = 50\text{m}\).
  1. Find the acceleration of the car. [3]
When the car passes the point \(C\), it has speed \(30 \text{ m s}^{-1}\).
  1. Find the distance \(AC\). [3]
Edexcel M1 2002 June Q2
7 marks Standard +0.3
The masses of two particles \(A\) and \(B\) are \(0.5 \text{ kg}\) and \(m \text{ kg}\) respectively. The particles are moving on a smooth horizontal table in opposite directions and collide directly. Immediately before the collision the speed of \(A\) is \(5 \text{ m s}^{-1}\) and the speed of \(B\) is \(3 \text{ m s}^{-1}\). In the collision, the magnitude of the impulse exerted by \(B\) on \(A\) is \(3.6 \text{ Ns}\). As a result of the collision the direction of motion of \(A\) is reversed.
  1. Find the speed of \(A\) immediately after the collision. [3]
The speed of \(B\) immediately after the collision is \(1 \text{ m s}^{-1}\).
  1. Find the two possible values of \(m\). [4]
Edexcel M1 2002 June Q3
8 marks Moderate -0.8
\includegraphics{figure_1} A uniform rod \(AB\) has length \(100 \text{ cm}\). Two light pans are suspended, one from each end of the rod, by two strings which are assumed to be light and inextensible. The system forms a balance with the rod resting horizontally on a smooth pivot, as shown in Fig. 1. A particle of weight \(16 \text{ N}\) is placed in the pan at \(A\) and a particle of weight \(5 \text{ N}\) is placed in the pan at \(B\). The rod rests horizontally in equilibrium when the pivot is at the point \(C\) on the rod, where \(AC = 30 \text{ cm}\).
  1. Find the weight of the rod. [3]
The particle in the pan at \(A\) is replaced by a particle of weight \(3.5 \text{ N}\). The particle of weight \(5 \text{ N}\) remains in the pan at \(B\). The rod now rests horizontally in equilibrium when the pivot is moved to the point \(D\).
  1. Find the distance \(AD\). [4]
  2. Explain briefly where the assumption that the strings are light has been used in your answer to part (a). [1]
Edexcel M1 2002 June Q4
12 marks Standard +0.3
\includegraphics{figure_2} A box of mass \(6 \text{ kg}\) lies on a rough plane inclined at an angle of \(30°\) to the horizontal. The box is held in equilibrium by means of a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. The line of action of the force is in the same vertical plane as a line of greatest slope of the plane. The coefficient of friction between the box and the plane is \(0.4\). The box is modelled as a particle. Given that the box is in limiting equilibrium and on the point of moving up the plane, find,
  1. the normal reaction exerted on the box by the plane, [4]
  2. the value of \(P\). [3]
The horizontal force is removed.
  1. Show that the box will now start to move down the plane. [5]
Edexcel M1 2002 June Q5
13 marks Moderate -0.3
A particle \(P\) of mass \(2 \text{ kg}\) moves in a plane under the action of a single constant force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). When \(t = 0\), \(\mathbf{v} = (-5\mathbf{i} + 7\mathbf{j})\) and when \(t = 3\), \(\mathbf{v} = (\mathbf{i} - 2\mathbf{j})\).
  1. Find in degrees the angle between the direction of motion of \(P\) when \(t = 3\) and the vector \(\mathbf{j}\). [3]
  2. Find the acceleration of \(P\). [2]
  3. Find the magnitude of \(\mathbf{F}\). [3]
  4. Find in terms of \(t\) the velocity of \(P\). [2]
  5. Find the time at which \(P\) is moving parallel to the vector \(\mathbf{i} + \mathbf{j}\). [3]
Edexcel M1 2002 June Q6
14 marks Moderate -0.3
A man travels in a lift to the top of a tall office block. The lift starts from rest on the ground floor and moves vertically. It comes to rest again at the top floor, having moved a vertical distance of \(27 \text{ m}\). The lift initially accelerates with a constant acceleration of \(2 \text{ m s}^{-1}\) until it reaches a speed of \(3 \text{ m s}^{-1}\). It then moves with a constant speed of \(3 \text{ m s}^{-1}\) for \(T\) seconds. Finally it decelerates with a constant deceleration for \(2.5 \text{ s}\) before coming to rest at the top floor.
  1. Sketch a speed-time graph for the motion of the lift. [2]
  2. Hence, or otherwise, find the value of \(T\). [3]
  3. Sketch an acceleration-time graph for the motion of the lift. [3]
The mass of the man is \(80 \text{ kg}\) and the mass of the lift is \(120 \text{ kg}\). The lift is pulled up by means of a vertical cable attached to the top of the lift. By modelling the cable as light and inextensible, find
  1. the tension in the cable when the lift is accelerating, [3]
  2. the magnitude of the force exerted by the lift on the man during the last \(2.5 \text{ s}\) of the motion. [3]
Edexcel M1 2002 June Q7
15 marks Standard +0.3
\includegraphics{figure_3} Particles \(A\) and \(B\), of mass \(2m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle \(A\) is held on the table, while \(B\) rests on a smooth plane inclined at \(30°\) to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between \(A\) and the table is \(\mu\). The particle \(A\) is released from rest and begins to move. By writing down an equation of motion for each particle,
  1. show that, while both particles move with the string taut. Each particle has an acceleration of magnitude \(\frac{1}{5}(1 - 4\mu)g\). [7]
When each particle has moved a distance \(h\), the string breaks. The particle \(A\) comes to rest before reaching the pulley. Given that \(\mu = 0.2\),
  1. find, in terms of \(h\), the total distance moved by \(A\). [6]
For the model described above,
  1. state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic. [2]
Edexcel M1 2004 June Q1
6 marks Moderate -0.8
\includegraphics{figure_1} A particle of weight \(W\) newtons is attached at \(C\) to the ends of two light inextensible strings \(AC\) and \(BC\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(AC\) and \(BC\) inclined to the horizontal at \(30°\) and \(60°\) respectively, as shown in Fig. 1. Given the tension in \(AC\) is 50 N, calculate
  1. the tension in \(BC\), to 3 significant figures, [3]
  2. the value of \(W\). [3]
Edexcel M1 2004 June Q2
7 marks Moderate -0.8
A particle \(P\) is moving with constant acceleration along a straight horizontal line \(ABC\), where \(AC = 24\) m. Initially \(P\) is at \(A\) and is moving with speed \(5\) m s\(^{-1}\) in the direction \(AB\). After \(1.5\) s, the direction of motion of \(P\) is unchanged and \(P\) is at \(B\) with speed \(9.5\) m s\(^{-1}\).
  1. Show that the speed of \(P\) at \(C\) is \(13\) m s\(^{-1}\). [4]
The mass of \(P\) is \(2\) kg. When \(P\) reaches \(C\), an impulse of magnitude \(30\) Ns is applied to \(P\) in the direction \(CB\).
  1. Find the velocity of \(P\) immediately after the impulse has been applied, stating clearly the direction of motion of \(P\) at this instant. [3]
Edexcel M1 2004 June Q3
9 marks Moderate -0.3
A particle \(P\) of mass \(2\) kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4\) kg which is at rest on the same horizontal plane. Immediately after the collision, \(P\) and \(Q\) are moving in opposite directions and the speed of \(P\) is one-third the speed of \(Q\).
  1. Show that the speed of \(P\) immediately after the collision is \(\frac{1}{5}u\) m s\(^{-1}\). [4]
After the collision \(P\) continues to move in the same straight line and is brought to rest by a constant resistive force of magnitude \(10\) N. The distance between the point of collision and the point where \(P\) comes to rest is \(1.6\) m.
  1. Calculate the value of \(u\). [5]
Edexcel M1 2004 June Q4
11 marks Moderate -0.3
\includegraphics{figure_2} A plank \(AE\), of length \(6\) m and mass \(10\) kg, rests in a horizontal position on supports at \(B\) and \(D\), where \(AB = 1\) m and \(DE = 2\) m. A child of mass \(20\) kg stands at \(C\), the mid-point of \(BD\), as shown in Fig. 2. The child is modelled as a particle and the plank as a uniform rod. The child and the plank are in equilibrium. Calculate
  1. the magnitude of the force exerted by the support on the plank at \(B\), [4]
  2. the magnitude of the force exerted by the support on the plank at \(D\). [3]
The child now stands at a point \(F\) on the plank. The plank is in equilibrium and on the point of tilting about \(D\).
  1. Calculate the distance \(DF\). [4]
Edexcel M1 2004 June Q5
12 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a boat \(B\) of mass \(400\) kg held at rest on a slipway by a rope. The boat is modelled as a particle and the slipway as a rough plane inclined at \(15°\) to the horizontal. The coefficient of friction between \(B\) and the slipway is \(0.2\). The rope is modelled as a light, inextensible string, parallel to a line of greatest slope of the plane. The boat is in equilibrium and on the point of sliding down the slipway.
  1. Calculate the tension in the rope. [6]
The boat is \(50\) m from the bottom of the slipway. The rope is detached from the boat and the boat slides down the slipway.
  1. Calculate the time taken for the boat to slide to the bottom of the slipway. [6]
Edexcel M1 2004 June Q6
13 marks Moderate -0.3
A small boat \(S\), drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 0900, \(S\) is at a point with position vector \((4\mathbf{i} - 6\mathbf{j})\) km relative to a fixed origin \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. At 0945, \(S\) is at the point with position vector \((7\mathbf{i} - 7.5\mathbf{j})\) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf{s}\) km.
  1. Calculate the bearing on which \(S\) is drifting. [4]
  2. Find an expression for \(\mathbf{s}\) in terms of \(t\). [3]
At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity \((p\mathbf{i} + q\mathbf{j})\) km h\(^{-1}\). Given that \(M\) intercepts \(S\) at 1015,
  1. calculate the value of \(p\) and the value of \(q\). [6]
Edexcel M1 2004 June Q7
17 marks Standard +0.3
\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.
  1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
  2. Calculate the tension in the string when the system is moving. [3]
  3. State how you have used the information that the string is inextensible. [1]
After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.
  1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
  2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]
Edexcel M1 2005 June Q1
6 marks Moderate -0.8
In taking off, an aircraft moves on a straight runway \(AB\) of length 1.2 km. The aircraft moves from \(A\) with initial speed \(2 \text{ m s}^{-1}\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \text{ m s}^{-1}\). Find
  1. the acceleration of the aircraft, [2]
  2. the distance \(BC\). [4]
Edexcel M1 2005 June Q2
8 marks Moderate -0.3
Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \text{ m s}^{-1}\) and the speed of \(B\) is \(2 \text{ m s}^{-1}\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
  1. the speed of \(A\) immediately after the collision, [5]
  2. the magnitude of the impulse exerted on \(B\) in the collision. [3]