Questions — Edexcel (10514 questions)

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Edexcel M1 2004 November Q6
11 marks Moderate -0.8
Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. At time \(t = 0\), they are side by side, passing a point \(O\) on the road. Car \(A\) travels at a constant speed of \(30 \text{ m s}^{-1}\). Car \(B\) passes \(O\) with a speed of \(20 \text{ m s}^{-1}\), and has constant acceleration of \(4 \text{ m s}^{-2}\). Find
  1. the speed of \(B\) when it has travelled 78 m from \(O\), [2]
  2. the distance from \(O\) of \(A\) when \(B\) is 78 m from \(O\), [4]
  3. the time when \(B\) overtakes \(A\). [5]
Edexcel M1 2004 November Q7
12 marks Moderate -0.3
\includegraphics{figure_3} A sledge has mass 30 kg. The sledge is pulled in a straight line along horizontal ground by means of a rope. The rope makes an angle \(20°\) with the horizontal, as shown in Figure 3. The coefficient of friction between the sledge and the ground is 0.2. The sledge is modelled as a particle and the rope as a light inextensible string. The tension in the rope is 150 N. Find, to 3 significant figures,
  1. the normal reaction of the ground on the sledge, [3]
  2. the acceleration of the sledge. [3]
When the sledge is moving at \(12 \text{ m s}^{-1}\), the rope is released from the sledge.
  1. Find, to 3 significant figures, the distance travelled by the sledge from the moment when the rope is released to the moment when the sledge comes to rest. [6]
Edexcel M1 2004 November Q8
14 marks Moderate -0.3
\includegraphics{figure_4} A heavy package is held in equilibrium on a slope by a rope. The package is attached to one end of the rope, the other end being held by a man standing at the top of the slope. The package is modelled as a particle of mass 20 kg. The slope is modelled as a rough plane inclined at \(60°\) to the horizontal and the rope as a light inextensible string. The string is assumed to be parallel to a line of greatest slope of the plane, as shown in Figure 4. At the contact between the package and the slope, the coefficient of friction is 0.4.
  1. Find the minimum tension in the rope for the package to stay in equilibrium on the slope. [8]
The man now pulls the package up the slope. Given that the package moves at constant speed,
  1. find the tension in the rope. [4]
  2. State how you have used, in your answer to part (b), the fact that the package moves
    1. up the slope,
    2. at constant speed.
    [2]
Edexcel M1 Specimen Q1
7 marks Moderate -0.8
\includegraphics{figure_1} A tennis ball \(P\) is attached to one end of a light inextensible string, the other end of the string being attached to a the top of a fixed vertical pole. A girl applies a horizontal force of magnitude 50 N to \(P\), and \(P\) is in equilibrium under gravity with the string making an angle of \(40°\) with the pole, as shown in Fig. 1. By modelling the ball as a particle find, to 3 significant figures,
  1. the tension in the string, [3]
  2. the weight of \(P\). [4]
Edexcel M1 Specimen Q2
7 marks Moderate -0.8
A car starts from rest at a point \(O\) and moves in a straight line. The car moves with constant acceleration \(4 \text{ m s}^{-2}\) until it passes the point \(A\) when it is moving with speed \(10 \text{ m s}^{-1}\). It then moves with constant acceleration \(3 \text{ m s}^{-2}\) for 6 s until it reaches the point \(B\). Find
  1. the speed of the car at \(B\), [2]
  2. the distance \(OB\). [5]
Edexcel M1 Specimen Q3
9 marks Moderate -0.3
\includegraphics{figure_2} A non-uniform plank of wood \(AB\) has length 6 m and mass 90 kg. The plank is smoothly supported at its two ends \(A\) and \(B\), with \(A\) and \(B\) at the same horizontal level. A woman of mass 60 kg stands on the plank at the point \(C\), where \(AC = 2\) m, as shown in Fig. 2. The plank is in equilibrium and the magnitudes of the reactions on the plank at \(A\) and \(B\) are equal. The plank is modelled as a non-uniform rod and the woman as a particle. Find
  1. the magnitude of the reaction on the plank at \(B\), [2]
  2. the distance of the centre of mass of the plank from \(A\). [5]
  3. State briefly how you have used the modelling assumption that
    1. the plank is a rod,
    2. the woman is a particle.
    [2]
Edexcel M1 Specimen Q4
12 marks Moderate -0.8
A train \(T_1\) moves from rest at Station \(A\) with constant acceleration \(2 \text{ m s}^{-2}\) until it reaches a speed of \(36 \text{ m s}^{-1}\). In maintains this constant speed for 90 s before the brakes are applied, which produce constant retardation \(3 \text{ m s}^{-2}\). The train \(T_1\) comes to rest at station \(B\).
  1. Sketch a speed-time graph to illustrate the journey of \(T_1\) from \(A\) to \(B\). [3]
  2. Show that the distance between \(A\) and \(B\) is 3780 m. [5]
\includegraphics{figure_3} A second train \(T_2\) takes 150 s to move form rest at \(A\) to rest at \(B\). Figure 3 shows the speed-time graph illustrating this journey.
  1. Explain briefly one way in which \(T_1\)'s journey differs from \(T_2\)'s journey. [1]
  2. Find the greatest speed, in m s\(^{-1}\), attained by \(T_2\) during its journey. [3]
Edexcel M1 Specimen Q5
12 marks Moderate -0.3
A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \text{ m s}^{-1}\), the speed of \(B\) is \(4 \text{ m s}^{-1}\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.
  1. Find the speed and direction of \(C\) after the collision. [4]
  2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision. [3]
  3. State a modelling assumption which you have made about the trucks in your solution [1]
Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.
  1. Find the value of \(d\). [4]
Edexcel M1 Specimen Q6
13 marks Standard +0.3
\includegraphics{figure_4} A particle of mass \(m\) rests on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a small light smooth pulley \(P\) fixed at the top of the plane. The other end of the string is attached to a particle \(B\) of mass \(3m\), and \(B\) hangs freely below \(P\), as shown in Fig. 4. The particles are released from rest with the string taut. The particle \(B\) moves down with acceleration of magnitude \(\frac{1}{3}g\). Find
  1. the tension in the string, [4]
  2. the coefficient of friction between \(A\) and the plane. [9]
Edexcel M1 Specimen Q7
15 marks Moderate -0.3
Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \text{ m s}^{-1}\) due east, and the velocity of \(B\) is \((10\mathbf{i} + 10\mathbf{j}) \text{ m s}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300\mathbf{j}\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively.
  1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3]
  2. Hence write down an expression for \(\overrightarrow{AB}\) in terms of \(t\). [1]
  3. Find the time when the bearing of \(B\) from \(A\) is \(045°\). [5]
  4. Find the time when the cars are again 300 m apart. [6]
Edexcel M2 2014 January Q1
8 marks Moderate -0.3
A particle \(P\) of mass 2 kg is moving with velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \((2\mathbf{i} - 3\mathbf{j})\) m s\(^{-1}\).
  1. Find the magnitude of the impulse. [5]
  2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied. [3]
Edexcel M2 2014 January Q2
11 marks Standard +0.3
A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the direction of \(x\) increasing, where $$v = (t - 2)(3t - 10), \quad t \geq 0$$ When \(t = 0\), \(P\) is at the origin \(O\).
  1. Find the acceleration of \(P\) when \(t = 3\) [3]
  2. Find the total distance travelled by \(P\) in the first 3 seconds of its motion. [6]
  3. Show that \(P\) never returns to \(O\). [2]
Edexcel M2 2014 January Q3
12 marks Standard +0.3
A car has mass 550 kg. When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of 30 m s\(^{-1}\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of 25 m s\(^{-1}\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.
    1. Find the value of \(R\).
    2. Find the value of \(P\). [8]
  2. Find the acceleration of the car when it travels along the straight horizontal road at 20 m s\(^{-1}\) with the engine working at 50 kW. [4]
Edexcel M2 2014 January Q4
11 marks Standard +0.3
\includegraphics{figure_1} A uniform lamina \(ABCD\) is formed by removing the isosceles triangle \(ADC\) of height \(h\) metres, where \(h < 2\sqrt{3}\), from a uniform lamina \(ABC\) in the shape of an equilateral triangle of side 4 m, as shown in Figure 1. The centre of mass of \(ABCD\) is at \(D\).
  1. Show that \(h = \sqrt{3}\) [7]
The weight of the lamina \(ABCD\) is \(W\) newtons. The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied at \(B\) so that the lamina is in equilibrium with \(AB\) vertical. The horizontal force acts in the vertical plane containing the lamina.
  1. Find \(F\) in terms of \(W\). [4]
Edexcel M2 2014 January Q5
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a uniform rod \(AB\), of mass \(m\) and length \(2a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(AC = \frac{2}{3}a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(BD = 2a\).
  1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac{1}{2}mg \sin \theta\) [3]
  2. Find the magnitude of the normal reaction on the rod at \(B\). [5]
The rod is in limiting equilibrium when \(\tan \theta = \frac{4}{3}\).
  1. Find the coefficient of friction between the rod and the ground. [3]
Edexcel M2 2014 January Q6
11 marks Standard +0.8
[In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((3\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\) where \(v > 3\). The ball moves freely under gravity and passes through the point \(A\) before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac{15}{49}\) s after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac{1}{2}E\) joules.
  1. Find the value of \(v\). [8]
At another point \(B\) on the path of the ball the kinetic energy is also \(\frac{1}{2}E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.
  1. Find the value of \(T\). [3]
Edexcel M2 2014 January Q7
11 marks Standard +0.3
Three particles \(A\), \(B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5u\) and \(4u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3u\). In the subsequent motion, \(A\), \(B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. [7]
Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),
  1. find the set of possible values of \(e\). [4]
Edexcel M2 2015 June Q1
6 marks Standard +0.3
A particle of mass 0.3 kg is moving with velocity \((5\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) when it receives an impulse \((-3\mathbf{i} + 3\mathbf{j})\) N s. Find the change in the kinetic energy of the particle due to the impulse. [6]
Edexcel M2 2015 June Q2
10 marks Standard +0.3
At time \(t\) seconds, \(t \geq 0\), a particle \(P\) has velocity \(\mathbf{v}\) m s\(^{-1}\), where $$\mathbf{v} = (27 - 3t^2)\mathbf{i} + (8 - t^3)\mathbf{j}$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf{r}\) m relative to a fixed origin \(O\), where \(\mathbf{r} = -5\mathbf{i} + 2\mathbf{j}\) Find
  1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf{i}\), [5]
  2. the position vector of \(P\) at the instant when \(t = 3\) [5]
Edexcel M2 2015 June Q3
10 marks Standard +0.8
A thin uniform wire of mass \(12m\) is bent to form a right-angled triangle \(ABC\). The lengths of the sides \(AB\), \(BC\) and \(AC\) are \(3a\), \(4a\) and \(5a\) respectively. A particle of mass \(2m\) is attached to the triangle at \(B\) and a particle of mass \(3m\) is attached to the triangle at \(C\). The bent wire and the two particles form the system \(S\). The system \(S\) is freely suspended from \(A\) and hangs in equilibrium. Find the size of the angle between \(AB\) and the downward vertical. [10]
Edexcel M2 2015 June Q4
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 A particle \(P\) of mass 6.5 kg is projected up a fixed rough plane with initial speed 6 m s\(^{-1}\) from a point \(X\) on the plane, as shown in Figure 1. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\), where \(XY = d\) metres. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between \(P\) and the plane is \(\frac{1}{3}\).
  1. Use the work-energy principle to show that, to 2 significant figures, \(d = 2.7\) [7]
After coming to rest at \(Y\), the particle \(P\) slides back down the plane.
  1. Find the speed of \(P\) as it passes through \(X\). [4]
Edexcel M2 2015 June Q5
13 marks Standard +0.3
Three particles \(A\), \(B\) and \(C\) lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The masses of \(A\), \(B\) and \(C\) are \(3m\), \(4m\), and \(5m\) respectively. Particle \(A\) is projected with speed \(u\) towards particle \(B\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\).
  1. Show that the impulse exerted by \(A\) on \(B\) in this collision has magnitude \(\frac{16}{7}mu\) [7]
After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\). After this collision between \(B\) and \(C\), the kinetic energy of \(C\) is \(\frac{72}{245}mu^2\)
  1. Find the coefficient of restitution between \(B\) and \(C\). [6]
Edexcel M2 2015 June Q6
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 A uniform rod \(AB\) has length \(4a\) and weight \(W\). A particle of weight \(kW\), \(k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(AC = 3a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{1}{3}\). The peg is perpendicular to the vertical plane containing \(AB\).
  1. Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod. [1]
  2. Show that the magnitude of the force acting on the rod at \(C\) is $$\frac{\sqrt{10}}{5}W(1 + 2k)$$ [4]
The coefficient of friction between the rod and the ground is \(\frac{3}{4}\).
  1. Show that for the rod to remain in equilibrium \(k \leq \frac{2}{11}\). [7]
Edexcel M2 2015 June Q7
13 marks Standard +0.3
[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.] At time \(t = 0\), a particle \(P\) is projected with velocity \((4\mathbf{i} + 9\mathbf{j})\) m s\(^{-1}\) from a fixed point \(O\) on horizontal ground. The particle moves freely under gravity. When \(P\) is at the point \(H\) on its path, \(P\) is at its greatest height above the ground.
  1. Find the time taken by \(P\) to reach \(H\). [2]
At the point \(A\) on its path, the position vector of \(P\) relative to \(O\) is \((k\mathbf{i} + k\mathbf{j})\) m, where \(k\) is a positive constant.
  1. Find the value of \(k\). [4]
  2. Find, in terms of \(k\), the position vector of the other point on the path of \(P\) which is at the same vertical height above the ground as the point \(A\). [3]
At time \(T\) seconds the particle is at the point \(B\) and is moving perpendicular to \((4\mathbf{i} + 9\mathbf{j})\)
  1. Find the value of \(T\). [4]
Edexcel M2 Q1
5 marks Moderate -0.8
At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$r = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]