Questions M1 (2067 questions)

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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]
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]