Questions — Edexcel M1 (663 questions)

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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]
Edexcel M1 2005 June Q3
7 marks Standard +0.3
\includegraphics{figure_1} A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(AC\). The bead \(B\) is vertically below \(C\) and \(\angle BAC = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac{3}{4}\), find
  1. the tension in the string, [3]
  2. the weight of the bead. [4]
Edexcel M1 2005 June Q4
8 marks Moderate -0.3
\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find
  1. the normal reaction of the plane on the box, [3]
  2. the acceleration of the box. [5]
Edexcel M1 2005 June Q5
10 marks Moderate -0.8
A train is travelling at \(10 \text{ m s}^{-1}\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s, reducing its speed to \(3 \text{ m s}^{-1}\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \text{ m s}^{-1}\) for a further 15 s. He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal.
  1. Sketch a speed-time graph to show the motion of the train, [3]
  2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \text{ m s}^{-1}\). [2]
  3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest. [5]
Edexcel M1 2005 June Q6
10 marks Standard +0.3
\includegraphics{figure_3} A uniform beam \(AB\) has mass 12 kg and length 3 m. The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at the end \(A\), the other at a point \(C\) on the beam, where \(BC = 1\) m, as shown in Figure 3. The beam is modelled as a uniform rod.
  1. Find the reaction on the beam at \(C\). [3]
A woman of mass 48 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The reactions on the beam at \(A\) and \(C\) are now equal.
  1. Find the distance \(AD\). [7]
Edexcel M1 2005 June Q7
13 marks Moderate -0.3
\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find
  1. the acceleration of the lorry and the car, [3]
  2. the tension in the towbar. [4]
When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,
  1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
  2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]
Edexcel M1 2005 June Q8
13 marks Moderate -0.8
[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal vectors due east and north respectively.] At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j})\) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \((5\mathbf{i} + 8\mathbf{j}) \text{ m s}^{-1}\). Find
  1. the speed of the ball, [2]
  2. the position vector of the ball after \(t\) seconds. [2]
The point \(B\) on the field has position vector \((10\mathbf{i} + 7\mathbf{j})\) m.
  1. Find the time when the ball is due north of \(B\). [2]
At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \text{ m s}^{-1}\). Given that he intercepts the ball,
  1. find the value of \(v\). [6]
  2. State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic. [1]
Edexcel M1 2009 June Q1
7 marks Moderate -0.3
Three posts \(P\), \(Q\) and \(R\) are fixed in that order at the side of a straight horizontal road. The distance from \(P\) to \(Q\) is 45 m and the distance from \(Q\) to \(R\) is 120 m. A car is moving along the road with constant acceleration \(a\) m s\(^{-2}\). The speed of the car, as it passes \(P\), is \(u\) m s\(^{-1}\). The car passes \(Q\) two seconds after passing \(P\), and the car passes \(R\) four seconds after passing \(Q\). Find
  1. the value of \(u\),
  2. the value of \(a\).
[7]
Edexcel M1 2009 June Q2
6 marks Moderate -0.8
A particle is acted upon by two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\), given by \(\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})\) N, \(\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})\) N, where \(p\) is a positive constant.
  1. Find the angle between \(\mathbf{F}_2\) and \(\mathbf{j}\). [2]
The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\). Given that \(\mathbf{R}\) is parallel to \(\mathbf{i}\),
  1. find the value of \(p\). [4]
Edexcel M1 2009 June Q3
6 marks Moderate -0.3
Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(2m\) and the mass of \(B\) is \(m\). The particles are moving along the same straight line but in opposite directions and they collide directly. Immediately before they collide the speed of \(A\) is \(2u\) and the speed of \(B\) is \(3u\). The magnitude of the impulse received by each particle in the collision is \(\frac{7mu}{2}\). Find
  1. the speed of \(A\) immediately after the collision, [3]
  2. the speed of \(B\) immediately after the collision. [3]
Edexcel M1 2009 June Q4
9 marks Standard +0.3
A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]
Edexcel M1 2009 June Q5
9 marks Moderate -0.3
\includegraphics{figure_1} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude \(P\) newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of \(P\), giving your answer to 2 significant figures. [9]
Edexcel M1 2009 June Q6
13 marks Moderate -0.3
A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find
  1. the acceleration of the car and trailer, [3]
  2. the magnitude of the tension in the towbar. [3]
The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,
  1. find the value of \(F\). [7]
Edexcel M1 2009 June Q7
12 marks Standard +0.3
\includegraphics{figure_2} A beam \(AB\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(AP = 0.3\) m and \(BQ = 0.3\) m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(PX = x\) m, \(0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position.
  1. Show that the tension in the rope attached to the beam at \(P\) is \((588 - 350x)\) N. [3]
  2. Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\). [3]
  3. Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. [3]
Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),
  1. find the value of \(x\). [3]
Edexcel M1 2009 June Q8
13 marks Moderate -0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \((1.2\mathbf{i} - 0.9\mathbf{j})\) m s\(^{-1}\).
  1. Find the speed of \(H\). [2]
\includegraphics{figure_3} A horizontal field \(OABC\) is rectangular with \(OA\) due east and \(OC\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100\mathbf{j}\) m, relative to the fixed origin \(O\).
  1. Write down the position vector of \(H\) at time \(t\) seconds after noon. [2]
At noon, another hiker \(K\) is at the point with position vector \((9\mathbf{i} + 46\mathbf{j})\) m. Hiker \(K\) is moving with constant velocity \((0.75\mathbf{i} + 1.8\mathbf{j})\) m s\(^{-1}\).
  1. Show that, at time \(t\) seconds after noon, $$\overrightarrow{HK} = [(9 - 0.45t)\mathbf{i} + (2.7t - 54)\mathbf{j}] \text{ metres.}$$ [4]
Hence,
  1. show that the two hikers meet and find the position vector of the point where they meet. [5]
Edexcel M1 2010 June Q5
12 marks Standard +0.3
Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = T\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).
  1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
  2. Find the value of \(T\). [8]