Questions — Edexcel (10514 questions)

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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]
Edexcel M1 Q3
8 marks Moderate -0.8
A racing car is travelling on a straight horizontal road. Its initial speed is \(25\) m s\(^{-1}\) and it accelerates for \(4\) s to reach a speed of \(V\) m s\(^{-1}\). It then travels at a constant speed of \(V\) m s\(^{-1}\) for a further \(8\) s. The total distance travelled by the car during this \(12\) s period is \(600\) m.
  1. Sketch a speed-time graph to illustrate the motion of the car during this \(12\) s period. [2]
  2. Find the value of \(V\). [4]
  3. Find the acceleration of the car during the initial \(4\) s period. [2]
Edexcel M1 Q4
11 marks Standard +0.3
\includegraphics{figure_2} A plank \(AB\) has length \(4\) m. It lies on a horizontal platform, with the end \(A\) lying on the platform and the end \(B\) projecting over the edge, as shown in Fig. 2. The edge of the platform is at the point \(C\). Jack and Jill are experimenting with the plank. Jack has mass \(40\) kg and Jill has mass \(25\) kg. They discover that, if Jack stands at \(B\) and Jill stands at \(A\) and \(BC = 1.6\) m, the plank is in equilibrium and on the point of tilting about \(C\). By modelling the plank as a uniform rod, and Jack and Jill as particles,
  1. find the mass of the plank. [3]
They now alter the position of the plank in relation to the platform so that, when Jill stands at \(B\) and Jack stands at \(A\), the plank is again in equilibrium and on the point of tilting about \(C\).
  1. Find the distance \(BC\) in this position. [5]
  2. State how you have used the modelling assumptions that
    1. the plank is uniform,
    2. the plank is a rod,
    3. Jack and Jill are particles.
    [3]
Edexcel M1 Q5
13 marks Moderate -0.8
A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of \(1.6\) m above the top of the post.
  1. Show that the speed of the pile-driver just before it hits the post is \(5.6\) m s\(^{-1}\). [2]
The post has mass \(6\) kg and the pile-driver has mass \(78\) kg. When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.
  1. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. [3]
The post is brought to rest by the action of a resistive force from the ground acting for \(0.06\) s. By modelling this force as constant throughout this time,
  1. find the magnitude of the resistive force, [4]
  2. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest. [4]
Edexcel M1 Q6
13 marks Moderate -0.3
[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \((-5\mathbf{i} + 10\mathbf{j})\) km relative to \(O\) and has velocity \((2\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). Ship \(B\) is at the point with position vector \((3\mathbf{i} + 4\mathbf{j})\) km and has velocity \((-2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).
  1. Given that the two ships maintain these velocities, show that they collide. [6]
The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \((\mathbf{i} + \mathbf{j})\) km h\(^{-1}\). Given that \(A\) obeys this order and maintains this new constant velocity,
  1. find an expression for the vector \(\overrightarrow{AB}\) at time \(t\) hours after noon. [2]
  2. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours, [3]
  3. Find the time at which \(B\) will be due north of \(A\). [2]
Edexcel M1 Q7
15 marks Standard +0.3
\includegraphics{figure_3} A small parcel of mass \(2\) kg moves on a rough plane inclined at an angle of \(30°\) to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which it attached to it. The rope makes an angle of \(30°\) with the plane, as shown in Fig. 3. The coefficient of friction between the parcel and the plane is \(0.4\). Given that the tension in the rope is \(24\) N,
  1. find, to 2 significant figures, the acceleration of the parcel. [8]
The rope now breaks. The parcel slows down and comes to rest.
  1. Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again. [4]
  2. Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest. [3]
Edexcel M1 2003 November Q1
6 marks Easy -1.2
A small ball is projected vertically upwards from a point A. The greatest height reached by the ball is 40 m above A. Calculate
  1. the speed of projection. [3]
  2. the time between the instant that the ball is projected and the instant it returns to A. [3]
Edexcel M1 2003 November Q2
8 marks Moderate -0.3
A railway truck \(S\) of mass 2000 kg is travelling due east along a straight horizontal track with constant speed 12 m s\(^{-1}\). The truck \(S\) collides with a truck \(T\) which is travelling due west along the same track as \(S\) with constant speed 6 m s\(^{-1}\). The magnitude of the impulse of \(T\) on \(S\) is 28800 Ns.
  1. Calculate the speed of \(S\) immediately after the collision. [3]
  2. State the direction of motion of \(S\) immediately after the collision. [1]
Given that, immediately after the collision, the speed of \(T\) is 3.6 m s\(^{-1}\), and that \(T\) and \(S\) are moving in opposite directions,
  1. calculate the mass of \(T\). [4]
Edexcel M1 2003 November Q3
9 marks Standard +0.3
\includegraphics{figure_1} A heavy suitcase \(S\) of mass 50 kg is moving along a horizontal floor under the action of a force of magnitude \(P\) newtons. The force acts at 30° to the floor, as shown in Fig. 1, and \(S\) moves in a straight line at constant speed. The suitcase is modelled as a particle and the floor as a rough horizontal plane. The coefficient of friction between \(S\) and the floor is \(\frac{3}{4}\). Calculate the value of \(P\). [9]
Edexcel M1 2003 November Q4
12 marks Moderate -0.3
A car starts from rest at a point \(S\) on a straight racetrack. The car moves with constant acceleration for 20 s, reaching a speed of 25 m s\(^{-1}\). The car then travels at a constant speed of 25 m s\(^{-1}\) for 120 s. Finally it moves with constant deceleration, coming to rest at a point \(F\).
  1. In the space below, sketch a speed-time graph to illustrate the motion of the car. [2]
The distance between \(S\) and \(F\) is 4 km.
  1. Calculate the total time the car takes to travel from \(S\) to \(F\). [3]
A motorcycle starts at \(S\), 10 s after the car has left \(S\). The motorcycle moves with constant acceleration from rest and passes the car at a point \(P\) which is 1.5 km from \(S\). When the motorcycle passes the car, the motorcycle is still accelerating and the car is moving at a constant speed. Calculate
  1. the time the motorcycle takes to travel from \(S\) to \(P\), [5]
  2. the speed of the motorcycle at \(P\). [2]
Edexcel M1 2003 November Q5
12 marks Moderate -0.3
A particle \(P\) of mass 3 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. At \(t = 0\), \(P\) has velocity \((3\mathbf{i} - 5\mathbf{j})\) m s\(^{-1}\). At \(t = 4\) s, the velocity of \(P\) is \((-5\mathbf{i} + 11\mathbf{j})\) 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]
At \(t = 6\) s, \(P\) is at the point \(A\) with position vector \((6\mathbf{i} - 29\mathbf{j})\) m relative to a fixed origin \(O\). At this instant the force \(\mathbf{F}\) newtons is removed and \(P\) then moves with constant velocity. Three seconds after the force has been removed, \(P\) is at the point \(B\).
  1. Calculate the distance of \(B\) from \(O\). [6]
Edexcel M1 2003 November Q6
12 marks Standard +0.3
\includegraphics{figure_2} A non-uniform rod \(AB\) has length 5 m and weight 200 N. The rod rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1.5\) m and \(DB = 1\) m, as shown in Fig. 2. The centre of mass of \(AB\) is \(x\) metres from \(A\). A particle of weight \(W\) newtons is placed on the rod at \(A\). The rod remains in equilibrium and the magnitude of the reaction of \(C\) on the rod is 160 N.
  1. Show that \(50x - W = 100\). [5]
The particle is now removed from \(A\) and placed on the rod at \(B\). The rod remains in equilibrium and the reaction of \(C\) on the rod now has magnitude 50 N.
  1. Obtain another equation connecting \(W\) and \(x\). [3]
  2. Calculate the value of \(x\) and the value of \(W\). [4]
Edexcel M1 2003 November Q7
16 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\), of mass \(m\) kg and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac{1}{8}g\).
  1. Write down an equation of motion for \(B\). [2]
  2. Find the tension in the string. [2]
  3. Prove that \(m = \frac{16}{35}\). [4]
  4. State where in the calculations you have used the information that \(P\) is a light smooth pulley. [1]
On release, \(B\) is at a height of one metre above the ground and \(AP = 1.4\) m. The particle \(B\) strikes the ground and does not rebound.
  1. Calculate the speed of \(B\) as it reaches the ground. [2]
  2. Show that \(A\) comes to rest as it reaches \(P\). [5]
END
Edexcel M1 2004 November Q1
5 marks Moderate -0.8
A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P\), 300 m from \(S\), its speed is \(30 \text{ m s}^{-1}\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).
  1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\). [2]
  2. Find the time taken by the car to travel from \(P\) to \(S\). [3]
Edexcel M1 2004 November Q2
7 marks Moderate -0.3
\includegraphics{figure_1} The particles have mass 3 kg and \(m\) kg, where \(m < 3\). They are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are held in position with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The particles are then released from rest. The initial acceleration of each particle has magnitude \(\frac{1}{2}g\). Find
  1. the tension in the string immediately after the particles are released, [3]
  2. the value of \(m\). [4]
Edexcel M1 2004 November Q3
8 marks Moderate -0.3
\includegraphics{figure_2} A plank of wood \(AB\) has mass 10 kg and length 4 m. It rests in a horizontal position on two smooth supports. One support is at the end \(A\). The other is at the point \(C\), 0.4 m from \(B\), as shown in Figure 2. A girl of mass 30 kg stands at \(B\) with the plank in equilibrium. By modelling the plank as a uniform rod and the girl as a particle,
  1. find the reaction on the plank at \(A\). [4]
The girl gets off the plank. A boulder of mass \(m\) kg is placed on the plank at \(A\) and a man of mass 80 kg stands on the plank at \(B\). The plank remains in equilibrium and is on the point of tilting about \(C\). By modelling the plank again as a uniform rod, and the man and the boulder as particles,
  1. find the value of \(m\). [4]
Edexcel M1 2004 November Q4
8 marks Moderate -0.8
A tent peg is driven into soft ground by a blow from a hammer. The tent peg has mass 0.2 kg and the hammer has mass 3 kg. The hammer strikes the peg vertically. Immediately before the impact, the speed of the hammer is \(16 \text{ m s}^{-1}\). It is assumed that, immediately after the impact, the hammer and the peg move together vertically downwards.
  1. Find the common speed of the peg and the hammer immediately after the impact. [3]
Until the peg and hammer come to rest, the resistance exerted by the ground is assumed to be constant and of magnitude \(R\) newtons. The hammer and peg are brought to rest 0.05 s after the impact.
  1. Find, to 3 significant figures, the value of \(R\). [5]
Edexcel M1 2004 November Q5
10 marks Moderate -0.8
A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \((-\mathbf{i} + 2\mathbf{j}) \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \((2\mathbf{i} - 3\mathbf{j}) \text{ m s}^{-1}\).
  1. Find, to the nearest degree, the angle between the vector \(\mathbf{j}\) and the direction of motion of \(P\) when \(t = 0\). [3]
At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). Find
  1. an expression for \(\mathbf{v}\) in terms of \(t\), in the form \(a\mathbf{i} + b\mathbf{j}\), [2]
  2. the speed of \(P\) when \(t = 3\), [3]
  3. the time when \(P\) is moving parallel to \(\mathbf{i}\). [2]