Questions M1 (2067 questions)

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Edexcel M1 2005 January Q2
7 marks Moderate -0.3
\includegraphics{figure_1} A plank \(AB\) has mass 40 kg and length 3 m. A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(AB\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate
  1. the tension in the rope at \(C\), [2]
  2. the distance \(CB\). [5]
Edexcel M1 2005 January Q3
9 marks Moderate -0.8
\includegraphics{figure_2} A sprinter runs a race of 200 m. Her total time for running the race is 25 s. Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of 9 m s\(^{-1}\) in 4 s. The speed of 9 m s\(^{-1}\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u\) m s\(^{-1}\) at the end of the race. Calculate
  1. the distance covered by the sprinter in the first 20 s of the race, [2]
  2. the value of \(u\), [4]
  3. the deceleration of the sprinter in the last 5 s of the race. [3]
Edexcel M1 2005 January Q4
10 marks Moderate -0.8
\includegraphics{figure_3} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at 20° to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4. The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate
  1. the normal reaction of the plane on \(P\), [2]
  2. the value of \(X\). [4]
The force of magnitude \(X\) newtons is now removed.
  1. Show that \(P\) remains in equilibrium on the plane. [4]
Edexcel M1 2005 January Q5
13 marks Standard +0.3
\includegraphics{figure_4} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s. Modelling \(A\) and \(B\) as particles, calculate
  1. the acceleration of \(B\), [3]
  2. the tension in the string, [4]
  3. the value of \(\mu\). [5]
  4. State how in your calculations you have used the information that the string is inextensible. [1]
Edexcel M1 2005 January Q6
13 marks Moderate -0.3
A stone \(S\) is sliding on ice. The stone is moving along a straight horizontal line \(ABC\), where \(AB = 24\) m and \(AC = 30\) m. The stone is subject to a constant resistance to motion of magnitude 0.3 N. At \(A\) the speed of \(S\) is 20 m s\(^{-1}\), and at \(B\) the speed of \(S\) is 16 m s\(^{-1}\). Calculate
  1. the deceleration of \(S\), [2]
  2. the speed of \(S\) at \(C\). [3]
  3. Show that the mass of \(S\) is 0.1 kg. [2]
At \(C\), the stone \(S\) hits a vertical wall, rebounds from the wall and then slides back along the line \(CA\). The magnitude of the impulse of the wall on \(S\) is 2.4 Ns and the stone continues to move against a constant resistance of 0.3 N.
  1. Calculate the time between the instant that \(S\) rebounds from the wall and the instant that \(S\) comes to rest. [6]
Edexcel M1 2005 January Q7
16 marks Moderate -0.3
Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \((20\mathbf{i} + 10\mathbf{j})\) km relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \((14\mathbf{i} - 6\mathbf{j})\) km. Three hours later, \(P\) is at the point with position vector \((29\mathbf{i} + 34\mathbf{j})\) km. The ship \(Q\) travels with velocity \(12\mathbf{j}\) km h\(^{-1}\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively. Find
  1. the velocity of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
  2. expressions for \(\mathbf{p}\) and \(\mathbf{q}\), in terms of \(t\), \(\mathbf{i}\) and \(\mathbf{j}\). [4]
At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d\) km.
  1. By finding an expression for \(\overrightarrow{PQ}\), show that $$d^2 = 25t^2 - 92t + 292.$$ [5]
Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
  1. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer. [5]
Edexcel M1 2006 January Q1
6 marks Moderate -0.8
A stone is thrown vertically upwards with speed \(16 \text{ m s}^{-1}\) from a point \(h\) metres above the ground. The stone hits the ground \(4\) s later. Find
  1. the value of \(h\), [3]
  2. the speed of the stone as it hits the ground. [3]
Edexcel M1 2006 January Q2
8 marks Moderate -0.8
  1. Two particles \(A\) and \(B\), of mass \(3\) kg and \(2\) kg respectively, are moving in the same direction on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4 \text{ m s}^{-1}\) and the speed of \(B\) is \(1.5 \text{ m s}^{-1}\). In the collision, the particles join to form a single particle \(C\). Find the speed of \(C\) immediately after the collision. [3]
  2. Two particles \(P\) and \(Q\) have mass \(3\) kg and \(m\) kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. Each particle has speed \(4 \text{ m s}^{-1}\), when they collide directly. In this collision, the direction of motion of each particle is reversed. The speed of \(P\) immediately after the collision is \(2 \text{ m s}^{-1}\) and the speed of \(Q\) is \(1 \text{ m s}^{-1}\). Find
    1. the value of \(m\), [3]
    2. the magnitude of the impulse exerted on \(Q\) in the collision. [2]
Edexcel M1 2006 January Q3
8 marks Moderate -0.8
\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,
  1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
  2. State what is implied by the modelling assumption that the beam is uniform. [1]
David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,
  1. find the distance of the centre of mass of the beam from \(C\). [4]
Edexcel M1 2006 January Q4
9 marks Moderate -0.3
Two forces \(\mathbf{P}\) and \(\mathbf{Q}\) act on a particle. The force \(\mathbf{P}\) has magnitude \(7\) N and acts due north. The resultant of \(\mathbf{P}\) and \(\mathbf{Q}\) is a force of magnitude \(10\) N acting in a direction with bearing \(120°\). Find
  1. the magnitude of \(\mathbf{Q}\),
  2. the direction of \(\mathbf{Q}\), giving your answer as a bearing.
[9]
Edexcel M1 2006 January Q5
14 marks Standard +0.3
\includegraphics{figure_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find
  1. the value of \(P\), [4]
  2. the value of \(\mu\). [5]
The horizontal force is removed.
  1. Determine whether or not the parcel moves. [5]
Edexcel M1 2006 January Q6
16 marks Moderate -0.8
[In this question the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are due east and due north respectively.] A model boat \(A\) moves on a lake with constant velocity \((-\mathbf{i} + 6\mathbf{j}) \text{ m s}^{-1}\). At time \(t = 0\), \(A\) is at the point with position vector \((2\mathbf{i} - 10\mathbf{j})\) m. Find
  1. the speed of \(A\), [2]
  2. the direction in which \(A\) is moving, giving your answer as a bearing. [3]
At time \(t = 0\), a second boat \(B\) is at the point with position vector \((-26\mathbf{i} + 4\mathbf{j})\) m. Given that the velocity of \(B\) is \((3\mathbf{i} + 4\mathbf{j}) \text{ m s}^{-1}\),
  1. show that \(A\) and \(B\) will collide at a point \(P\) and find the position vector of \(P\). [5]
Given instead that \(B\) has speed \(8 \text{ m s}^{-1}\) and moves in the direction of the vector \((3\mathbf{i} + 4\mathbf{j})\),
  1. find the distance of \(B\) from \(P\) when \(t = 7\) s. [6]
Edexcel M1 2006 January Q7
14 marks Standard +0.3
\includegraphics{figure_3} A fixed wedge has two plane faces, each inclined at \(30°\) to the horizontal. Two particles \(A\) and \(B\), of mass \(3m\) and \(m\) respectively, are attached to the ends of a light inextensible string. Each particle moves on one of the plane faces of the wedge. The string passes over a small smooth light pulley fixed at the top of the wedge. The face on which \(A\) moves is smooth. The face on which \(B\) moves is rough. The coefficient of friction between \(B\) and this face is \(\mu\). Particle \(A\) is held at rest with the string taut. The string lies in the same vertical plane as lines of greatest slope on each plane face of the wedge, as shown in Figure 3. The particles are released from rest and start to move. Particle \(A\) moves downwards and \(B\) moves upwards. The accelerations of \(A\) and \(B\) each have magnitude \(\frac{1}{10}g\).
  1. By considering the motion of \(A\), find, in terms of \(m\) and \(g\), the tension in the string. [3]
  2. By considering the motion of \(B\), find the value of \(\mu\). [8]
  3. Find the resultant force exerted by the string on the pulley, giving its magnitude and direction. [3]
Edexcel M1 2007 January Q1
6 marks Moderate -0.8
\includegraphics{figure_1} A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of 30° to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find
  1. the value of \(P\), [3]
  2. the value of \(Q\). [3]
Edexcel M1 2007 January Q2
10 marks Moderate -0.3
\includegraphics{figure_2} A uniform plank \(AB\) has weight 120 N and length 3 m. The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1\) m and \(CD = x\) m, as shown in Figure 2. The reaction of the support on the plank at \(D\) has magnitude 80 N. Modelling the plank as a rod,
  1. show that \(x = 0.75\) [3]
A rock is now placed at \(B\) and the plank is on the point of tilting about \(D\). Modelling the rock as a particle, find
  1. the weight of the rock, [4]
  2. the magnitude of the reaction of the support on the plank at \(D\). [2]
  3. State how you have used the model of the rock as a particle. [1]
Edexcel M1 2007 January Q3
9 marks Moderate -0.8
A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) 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]
  3. the velocity of \(P\) at time \(t = 6\) s. [3]
Edexcel M1 2007 January Q4
10 marks Moderate -0.8
A particle \(P\) of mass 0.3 kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal table. The particle \(P\) collides directly with a particle \(Q\) of mass 0.6 kg, which is at rest on the table. Immediately after the particles collide, \(P\) has speed 2 m s\(^{-1}\) and \(Q\) has speed 5 m s\(^{-1}\). The direction of motion of \(P\) is reversed by the collision. Find
  1. the value of \(u\), [4]
  2. the magnitude of the impulse exerted by \(P\) on \(Q\). [2]
Immediately after the collision, a constant force of magnitude \(R\) newtons is applied to \(Q\) in the direction directly opposite to the direction of motion of \(Q\). As a result \(Q\) is brought to rest in 1.5 s.
  1. Find the value of \(R\). [4]
Edexcel M1 2007 January Q5
10 marks Moderate -0.8
A ball is projected vertically upwards with speed 21 m s\(^{-1}\) from a point \(A\), which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modelling the ball as a particle, find
  1. the greatest height above \(A\) reached by the ball, [3]
  2. the speed of the ball as it reaches the ground, [3]
  3. the time between the instant when the ball is projected from \(A\) and the instant when the ball reaches the ground. [4]
Edexcel M1 2007 January Q6
14 marks Moderate -0.3
\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.
  1. Find the value of \(P\). [8]
The tension in the rope is now increased to 150 N.
  1. Find the acceleration of the box. [6]
Edexcel M1 2007 January Q7
16 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.
  1. Write down an equation of motion for \(P\) and an equation of motion for \(Q\). [4]
  2. Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). [2]
  3. Find the tension in the string. [2]
  4. State where in your calculations you have used the information that the string is inextensible. [1]
On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find
  1. the speed of \(Q\) as it reaches the ground, [2]
  2. the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. [5]
Edexcel M1 2010 January Q1
6 marks Moderate -0.8
A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed 12 m s\(^{-1}\). Another particle \(B\) of mass \(m\) kg is moving along the same straight line, in the opposite direction to \(A\), with speed 8 m s\(^{-1}\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed 3 m s\(^{-1}\) and \(B\) is moving with speed 4 m s\(^{-1}\). Find
  1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision, [2]
  2. the value of \(m\). [4]
Edexcel M1 2010 January Q2
8 marks Moderate -0.8
An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s\(^{-1}\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.
  1. In the space below, sketch a speed-time graph to illustrate the motion of the athlete. [3]
  2. Calculate the value of \(T\). [5]
Edexcel M1 2010 January Q3
8 marks Moderate -0.3
\includegraphics{figure_1} A particle of mass \(m\) kg is attached at \(C\) to two light inextensible strings \(AC\) and \(BC\). The other ends of the strings are attached to 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 Figure 1. Given that the tension in \(AC\) is 20 N, find
  1. the tension in \(BC\), [4]
  2. the value of \(m\). [4]
Edexcel M1 2010 January Q4
10 marks Moderate -0.3
\includegraphics{figure_2} A pole \(AB\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(AC = 1.8\) m, as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.
  1. Show that the tension in the rope attached to the pole at \(C\) is \(\left(\frac{5}{6}W + \frac{100}{3}\right)\) N. [4]
  2. Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). [3]
Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),
  1. find the value of \(W\). [3]
Edexcel M1 2010 January Q5
15 marks Standard +0.3
A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find
  1. the acceleration of the particle, [3]
  2. the coefficient of friction between the particle and the plane. [5]
The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \includegraphics{figure_3}
  1. Find the value of \(X\). [7]