Questions — Edexcel M1 (663 questions)

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Edexcel M1 2011 June Q1
8 marks Moderate -0.8
At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.
  1. Show that the speed of projection is 28 m s\(^{-1}\). [3]
  2. Find the times, in seconds, when the ball is 33.6 m above \(O\). [5]
Edexcel M1 2011 June Q2
8 marks Moderate -0.3
Particle \(P\) has mass 3 kg and particle \(Q\) has mass 2 kg. The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, \(P\) has speed 3 m s\(^{-1}\) and \(Q\) has speed 2 m s\(^{-1}\). Immediately after the collision, both particles move in the same direction and the difference in their speeds is 1 m s\(^{-1}\).
  1. Find the speed of each particle after the collision. [5]
  2. Find the magnitude of the impulse exerted on \(P\) by \(Q\). [3]
Edexcel M1 2011 June Q3
9 marks Standard +0.3
\includegraphics{figure_1} A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 1. The coefficient of friction between the particle and the plane is \(\frac{1}{2}\). Given that the particle is on the point of sliding down the plane,
  1. show that the magnitude of the normal reaction between the particle and the plane is 20 N,
  2. find the value of \(W\). [9]
Edexcel M1 2011 June Q4
12 marks Moderate -0.8
A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s\(^{-1}\). She maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the finishing line with a speed of \(V\) m s\(^{-1}\).
  1. Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race. [2]
  2. Find the distance run by the girl in the first 64 s of the race. [3]
  3. Find the value of \(V\). [5]
  4. Find the deceleration of the girl in the final 20 s of her race. [2]
Edexcel M1 2011 June Q5
11 marks Moderate -0.8
A plank \(PQR\), of length 8 m and mass 20 kg, is in equilibrium in a horizontal position on two supports at \(P\) and \(Q\), where \(PQ = 6\) m. A child of mass 40 kg stands on the plank at a distance of 2 m from \(P\) and a block of mass \(M\) kg is placed on the plank at the end \(R\). The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at \(P\) is equal to the force exerted on the plank by the support at \(Q\). By modelling the plank as a uniform rod, and the child and the block as particles,
    1. find the magnitude of the force exerted on the plank by the support at \(P\),
    2. find the value of \(M\). [10]
  2. State how, in your calculations, you have used the fact that the child and the block can be modelled as particles. [1]
Edexcel M1 2011 June Q6
16 marks Standard +0.8
\includegraphics{figure_2} Two particles \(P\) and \(Q\) have masses 0.3 kg and \(m\) 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 at the top of a fixed rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\frac{1}{2}\). The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle \(P\) is held at rest on the inclined plane and the particle \(Q\) hangs freely below the pulley with the string taut, as shown in Figure 2. The system is released from rest and \(Q\) accelerates vertically downwards at 1.4 m s\(^{-2}\). Find
  1. the magnitude of the normal reaction of the inclined plane on \(P\), [2]
  2. the value of \(m\). [8]
When the particles have been moving for 0.5 s, the string breaks. Assuming that \(P\) does not reach the pulley,
  1. find the further time that elapses until \(P\) comes to instantaneous rest. [6]
Edexcel M1 2011 June Q7
11 marks Moderate -0.3
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).] Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \((2\mathbf{i} - 3\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) moves with velocity \((3\mathbf{i} + 4\mathbf{j})\) km h\(^{-1}\).
  1. Find, to the nearest degree, the bearing on which \(Q\) is moving. [2]
At 2 pm, ship \(P\) is at the point with position vector \((\mathbf{i} + \mathbf{j})\) km and ship \(Q\) is at the point with position vector \((-2\mathbf{j})\) km. At time \(t\) hours after 2 pm, the position vector of \(P\) is \(\mathbf{p}\) km and the position vector of \(Q\) is \(\mathbf{q}\) km.
  1. Write down expressions, in terms of \(t\), for
    1. \(\mathbf{p}\),
    2. \(\mathbf{q}\),
    3. \(\overrightarrow{PQ}\). [5]
  2. Find the time when
    1. \(Q\) is due north of \(P\),
    2. \(Q\) is north-west of \(P\). [4]
Edexcel M1 2013 June Q1
6 marks Moderate -0.8
Two particles \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately before the collision the speed of \(A\) is 5 m s\(^{-1}\) and the speed of \(B\) is 6 m s\(^{-1}\). The magnitude of the impulse exerted on \(B\) by \(A\) is 14 N s. Find
  1. the speed of \(A\) immediately after the collision, [3]
  2. the speed of \(B\) immediately after the collision. [3]
Edexcel M1 2013 June Q2
8 marks Moderate -0.3
\includegraphics{figure_1} A particle of weight 8 N is attached at \(C\) to the ends of two light inextensible strings \(AC\) and \(BC\). The other ends, \(A\) and \(B\), are attached to a fixed horizontal ceiling. The particle hangs at rest in equilibrium, with the strings in a vertical plane. The string \(AC\) is inclined at 35° to the horizontal and the string \(BC\) is inclined at 25° to the horizontal, as shown in Figure 1. Find
  1. the tension in the string \(AC\),
  2. the tension in the string \(BC\).
[8]
Edexcel M1 2013 June Q3
9 marks Standard +0.3
\includegraphics{figure_2} A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley \(P\) is fixed at the top of the plane. Two particles \(A\) and \(B\), of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley \(P\). The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs freely below \(P\), as shown in Figure 2. The coefficient of friction between \(A\) and the plane is \(\frac{1}{\sqrt{3}}\). Initially \(A\) is held at rest on the plane. The particles are released from rest with the string taut and \(A\) moves up the plane. Find the tension in the string immediately after the particles are released. [9]
Edexcel M1 2013 June Q4
7 marks Moderate -0.3
At time \(t = 0\), two balls \(A\) and \(B\) are projected vertically upwards. The ball \(A\) is projected vertically upwards with speed 2 m s\(^{-1}\) from a point 50 m above the horizontal ground. The ball \(B\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). At time \(t = T\) seconds, the two balls are at the same vertical height, \(h\) metres, above the ground. The balls are modelled as particles moving freely under gravity. Find
  1. the value of \(T\), [5]
  2. the value of \(h\). [2]
Edexcel M1 2013 June Q5
10 marks Standard +0.3
\includegraphics{figure_3} A particle \(P\) of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points \(A\) and \(B\), where \(AB = 10\) m, as shown in Figure 3. The speed of \(P\) at \(A\) is 2 m s\(^{-1}\). The particle \(P\) takes 3.5 s to move from \(A\) to \(B\). Find
  1. the speed of \(P\) at \(B\), [3]
  2. the acceleration of \(P\), [2]
  3. the coefficient of friction between \(P\) and the plane. [5]
Edexcel M1 2013 June Q6
11 marks Moderate -0.3
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin \(O\).] A ship \(S\) is moving with constant velocity \((3\mathbf{i} + 3\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(S\) is \((-4\mathbf{i} + 2\mathbf{j})\) km.
  1. Find the position vector of \(S\) at time \(t\) hours. [2]
A ship \(T\) is moving with constant velocity \((-2\mathbf{i} + n\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(T\) is \((6\mathbf{i} + \mathbf{j})\) km. The two ships meet at the point \(P\).
  1. Find the value of \(n\). [5]
  2. Find the distance \(OP\). [4]
Edexcel M1 2013 June Q7
11 marks Standard +0.3
\includegraphics{figure_4} A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle \(\theta\) to the road, as shown in Figure 4. The vehicles are travelling at 20 m s\(^{-1}\) as they enter a zone where the speed limit is 14 m s\(^{-1}\). The truck's brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s\(^{-1}\) is 100 m.
  1. Find the deceleration of the truck and the car. [3]
The constant braking force on the truck has magnitude \(R\) newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos \(\theta = 0.9\), find
  1. the force in the towbar, [4]
  2. the value of \(R\). [4]
Edexcel M1 2013 June Q8
13 marks Standard +0.3
\includegraphics{figure_5} A uniform rod \(AB\) has length 2 m and mass 50 kg. The rod is in equilibrium in a horizontal position, resting on two smooth supports at \(C\) and \(D\), where \(AC = 0.2\) metres and \(DB = x\) metres, as shown in Figure 5. Given that the magnitude of the reaction on the rod at \(D\) is twice the magnitude of the reaction on the rod at \(C\),
  1. find the value of \(x\). [6]
The support at \(D\) is now moved to the point \(E\) on the rod, where \(EB = 0.4\) metres. A particle of mass \(m\) kg is placed on the rod at \(B\), and the rod remains in equilibrium in a horizontal position. Given that the magnitude of the reaction on the rod at \(E\) is four times the magnitude of the reaction on the rod at \(C\),
  1. find the value of \(m\). [7]
Edexcel M1 2013 June Q1
6 marks Moderate -0.8
Particle \(P\) has mass 3 kg and particle \(Q\) has mass \(m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision, the speed of \(P\) is \(4 \text{ m s}^{-1}\) and the speed of \(Q\) is \(3 \text{ m s}^{-1}\). In the collision the direction of motion of \(P\) is unchanged and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(1 \text{ m s}^{-1}\) and the speed of \(Q\) is \(1.5 \text{ m s}^{-1}\).
  1. Find the magnitude of the impulse exerted on \(P\) in the collision. [3]
  2. Find the value of \(m\). [3]
Edexcel M1 2013 June Q2
6 marks Moderate -0.3
A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is 950 kg. The lift is being raised vertically by a vertical cable which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of \(2 \text{ m s}^{-2}\). By modelling the cable as being light and inextensible, find
  1. the tension in the cable; [3]
  2. the magnitude of the force exerted on the woman by the floor of the lift. [3]
Edexcel M1 2013 June Q3
8 marks Standard +0.3
\includegraphics{figure_1} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), and the plane is at an angle of \(30°\) to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is \(\frac{1}{2}\) and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope. [8]
Edexcel M1 2013 June Q4
9 marks Moderate -0.3
A lorry is moving along a straight horizontal road with constant acceleration. The lorry passes a point \(A\) with speed \(u \text{ m s}^{-1}\), \((u < 34)\), and 10 seconds later passes a point \(B\) with speed \(34 \text{ m s}^{-1}\). Given that \(AB = 240\) m, find
  1. the value of \(u\), [3]
  2. the time taken for the lorry to move from \(A\) to the mid-point of \(AB\). [6]
Edexcel M1 2013 June Q5
11 marks Moderate -0.3
A car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is \(22 \text{ m s}^{-1}\). The car maintains this speed for \(T\) seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.
  1. Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights. [2]
  2. Find the value of \(T\). [3]
A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, \(a \text{ m s}^{-2}\), and passes the car at the point \(A\) which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed \(22 \text{ m s}^{-1}\).
  1. Find the time it takes for the motorcycle to move from the first set of traffic lights to the point \(A\). [4]
  2. Find the value of \(a\). [2]
Edexcel M1 2013 June Q6
14 marks Standard +0.3
A beam \(AB\) has length 15 m. The beam rests horizontally in equilibrium on two smooth supports at the points \(P\) and \(Q\), where \(AP = 2\) m and \(QB = 3\) m. When a child of mass 50 kg stands on the beam at \(A\), the beam remains in equilibrium and is on the point of tilting about \(P\). When the same child of mass 50 kg stands on the beam at \(B\), the beam remains in equilibrium and is on the point of tilting about \(Q\). The child is modelled as a particle and the beam is modelled as a non-uniform rod.
    1. Find the mass of the beam.
    2. Find the distance of the centre of mass of the beam from \(A\). [8]
When the child stands at the point \(X\) on the beam, it remains horizontal and in equilibrium. Given that the reactions at the two supports are equal in magnitude,
  1. find \(AX\). [6]
Edexcel M1 2013 June Q7
11 marks Moderate -0.3
[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due east and due north respectively.] The velocity, \(\mathbf{v} \text{ m s}^{-1}\), of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (3t - 3)\mathbf{j}$$
  1. Find the speed of \(P\) when \(t = 0\) [3]
  2. Find the bearing on which \(P\) is moving when \(t = 2\) [2]
  3. Find the value of \(t\) when \(P\) is moving
    1. parallel to \(\mathbf{j}\),
    2. parallel to \((-\mathbf{i} - 3\mathbf{j})\). [6]
Edexcel M1 2013 June Q8
10 marks Moderate -0.3
\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(2m\) and \(3m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle \(B\) hangs at rest vertically below the pulley with the string taut, as shown in Figure 2. Particle \(A\) is released from rest. Assuming that \(A\) has not reached the pulley, find
  1. the acceleration of \(B\), [5]
  2. the tension in the string, [1]
  3. the magnitude and direction of the force exerted on the pulley by the string. [4]
Edexcel M1 Q1
7 marks Moderate -0.8
An aircraft moves along a straight horizontal runway with constant acceleration. It passes a point \(A\) on the runway with speed \(16\) m s\(^{-1}\). It then passes the point \(B\) on the runway with speed \(34\) m s\(^{-1}\). The distance from \(A\) to \(B\) is \(150\) m.
  1. Find the acceleration of the aircraft. [3]
  2. Find the time taken by the aircraft in moving from \(A\) to \(B\). [2]
  3. Find, to 3 significant figures, the speed of the aircraft when it passes the point mid-way between \(A\) and \(B\). [2]
Edexcel M1 Q2
8 marks Standard +0.3
\includegraphics{figure_1} A particle has mass \(2\) kg. It is attached at \(B\) to the ends of two light inextensible strings \(AB\) and \(BC\). When the particle hangs in equilibrium, \(AB\) makes an angle of \(30°\) with the vertical, as shown in Fig. 1. The magnitude of the tension in \(BC\) is twice the magnitude of the tension in \(AB\).
  1. Find, in degrees to one decimal place, the size of the angle that \(BC\) makes with the vertical. [4]
  2. Hence find, to 3 significant figures, the magnitude of the tension in \(AB\). [4]