Questions M1 (2067 questions)

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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]
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]