Questions — Edexcel M1 (663 questions)

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Edexcel M1 2007 January Q3
9 marks Moderate -0.8
A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\). Find
  1. the acceleration of \(P\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
  2. the magnitude of \(\mathbf{F}\), [4]
  3. the velocity of \(P\) at time \(t = 6\) s. [3]
Edexcel M1 2007 January Q4
10 marks Moderate -0.8
A particle \(P\) of mass 0.3 kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal table. The particle \(P\) collides directly with a particle \(Q\) of mass 0.6 kg, which is at rest on the table. Immediately after the particles collide, \(P\) has speed 2 m s\(^{-1}\) and \(Q\) has speed 5 m s\(^{-1}\). The direction of motion of \(P\) is reversed by the collision. Find
  1. the value of \(u\), [4]
  2. the magnitude of the impulse exerted by \(P\) on \(Q\). [2]
Immediately after the collision, a constant force of magnitude \(R\) newtons is applied to \(Q\) in the direction directly opposite to the direction of motion of \(Q\). As a result \(Q\) is brought to rest in 1.5 s.
  1. Find the value of \(R\). [4]
Edexcel M1 2007 January Q5
10 marks Moderate -0.8
A ball is projected vertically upwards with speed 21 m s\(^{-1}\) from a point \(A\), which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modelling the ball as a particle, find
  1. the greatest height above \(A\) reached by the ball, [3]
  2. the speed of the ball as it reaches the ground, [3]
  3. the time between the instant when the ball is projected from \(A\) and the instant when the ball reaches the ground. [4]
Edexcel M1 2007 January Q6
14 marks Moderate -0.3
\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.
  1. Find the value of \(P\). [8]
The tension in the rope is now increased to 150 N.
  1. Find the acceleration of the box. [6]
Edexcel M1 2007 January Q7
16 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.
  1. Write down an equation of motion for \(P\) and an equation of motion for \(Q\). [4]
  2. Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). [2]
  3. Find the tension in the string. [2]
  4. State where in your calculations you have used the information that the string is inextensible. [1]
On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find
  1. the speed of \(Q\) as it reaches the ground, [2]
  2. the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. [5]
Edexcel M1 2010 January Q1
6 marks Moderate -0.8
A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed 12 m s\(^{-1}\). Another particle \(B\) of mass \(m\) kg is moving along the same straight line, in the opposite direction to \(A\), with speed 8 m s\(^{-1}\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed 3 m s\(^{-1}\) and \(B\) is moving with speed 4 m s\(^{-1}\). Find
  1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision, [2]
  2. the value of \(m\). [4]
Edexcel M1 2010 January Q2
8 marks Moderate -0.8
An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s\(^{-1}\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.
  1. In the space below, sketch a speed-time graph to illustrate the motion of the athlete. [3]
  2. Calculate the value of \(T\). [5]
Edexcel M1 2010 January Q3
8 marks Moderate -0.3
\includegraphics{figure_1} A particle of mass \(m\) kg is attached at \(C\) to two light inextensible strings \(AC\) and \(BC\). The other ends of the strings are attached to 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 Figure 1. Given that the tension in \(AC\) is 20 N, find
  1. the tension in \(BC\), [4]
  2. the value of \(m\). [4]
Edexcel M1 2010 January Q4
10 marks Moderate -0.3
\includegraphics{figure_2} A pole \(AB\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(AC = 1.8\) m, as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.
  1. Show that the tension in the rope attached to the pole at \(C\) is \(\left(\frac{5}{6}W + \frac{100}{3}\right)\) N. [4]
  2. Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). [3]
Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),
  1. find the value of \(W\). [3]
Edexcel M1 2010 January Q5
15 marks Standard +0.3
A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find
  1. the acceleration of the particle, [3]
  2. the coefficient of friction between the particle and the plane. [5]
The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \includegraphics{figure_3}
  1. Find the value of \(X\). [7]
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]