Questions — Edexcel M2 (623 questions)

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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]
Edexcel M2 Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a decoration which is made by cutting 2 circular discs from a sheet of uniform card. The discs are joined so that they touch at a point \(D\) on the circumference of both discs. The discs are coplanar and have centres \(A\) and \(B\) with radii 10 cm and 20 cm respectively.
  1. Find the distance of the centre of mass of the decoration from \(B\). [5]
The point \(C\) lies on the circumference of the smaller disc and \(\angle CAB\) is a right angle. The decoration is freely suspended from \(C\) and hangs in equilibrium.
  1. Find, in degrees to one decimal place, the angle between \(AB\) and the vertical. [4]
Edexcel M2 Q3
9 marks Challenging +1.2
A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(0.15\) and \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of \(30°\) with the wall. A man of mass \(5m\) stands on the ladder, which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(6a\). Find the value of \(k\). [9]
Edexcel M2 Q4
10 marks Moderate -0.3
The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) lie in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) vertical. A ball of mass \(0.1\) kg is hit by a bat which gives it an impulse of \((3.5\mathbf{i} + 3\mathbf{j})\) Ns. The velocity of the ball immediately after being hit is \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).
  1. Find the velocity of the ball immediately before it is hit. [3]
In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.
  1. Find the greatest height of the ball above the ground in the subsequent motion. [3]
The ball is caught when it is again 1 m above the ground.
  1. Find the distance from the point where the ball is hit to the point where it is caught. [4]
Edexcel M2 Q5
10 marks Standard +0.3
A child is playing with a small model of a fire-engine of mass \(0.5\) kg and a straight, rigid plank. The plank is inclined at an angle \(α\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(α = 20°\) the fire-engine is projected with an initial speed of \(5\) m s\(^{-1}\) and first comes to rest after travelling 2 m.
  1. Find, to 3 significant figures, the value of \(R\). [7]
When \(α = 40°\) the fire-engine is again projected with an initial speed of \(5\) m s\(^{-1}\).
  1. Find how far the fire-engine travels before first coming to rest. [3]
Edexcel M2 Q6
16 marks Standard +0.3
A particle \(A\) of mass \(2m\) is moving with speed \(2u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).
  1. Show that the speed of \(B\) after the collision is \(\frac{3}{2}u\). [6]
  1. Find the speed of \(A\) after the collision. [2]
Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,
  1. find the range of possible values for \(e\). [8]
Edexcel M2 Q7
16 marks Standard +0.3
\includegraphics{figure_2} At time \(t = 0\) a small package is projected from a point \(B\) which is \(2.4\) m above a point \(A\) on horizontal ground. The package is projected with speed \(23.75\) m s\(^{-1}\) at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{5}\). The package strikes the ground at point \(C\), as shown in Fig. 2. The package is modelled as a particle moving freely under gravity.
  1. Find the time taken for the package to reach \(C\). [5]
A lorry moves along the line \(AC\), approaching \(A\) with constant speed 18 m s\(^{-1}\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest 7 seconds later. The acceleration, \(a\) m s\(^{-2}\) of the lorry at time \(t\) seconds is given by $$a = -\frac{1}{4}t^2, \quad 0 \leq t \leq 7.$$
  1. Find the speed of the lorry at time \(t\) seconds. [3]
  1. Hence show that \(T = 6\). [3]
  1. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry. [5]
END
Edexcel M2 Q1
4 marks Moderate -0.8
A particle of mass 4 kg is moving in a straight horizontal line. There is a constant resistive force of magnitude \(R\) newtons. The speed of the particle is reduced from 25 m s\(^{-1}\) to rest over a distance of 200 m. Use the work-energy principle to calculate the value of \(R\). [4]
Edexcel M2 Q2
9 marks Standard +0.3
A van of mass 1500 kg is driving up a straight road inclined at an angle \(α\) to the horizontal, where \(\sin α = \frac{1}{16}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is operating at a rate of 60 kW,
  1. calculate the magnitude of the initial deceleration of the van. [4]
When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,
  1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]
  1. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]
Edexcel M2 Q3
10 marks Standard +0.3
A particle \(P\) of mass \(0.3\) kg is moving under the action of a single force \(F\) newtons. At time \(t\) seconds the velocity of \(P\), v m s\(^{-1}\), is given by $$\mathbf{v} = 3t^2\mathbf{i} + (6t - 4)\mathbf{j}.$$
  1. Calculate, to 3 significant figures, the magnitude of \(\mathbf{F}\) when \(t = 2\). [5]
When \(t = 0\), \(P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \((3\mathbf{i} - 4\mathbf{j})\) m. When \(t = 4\), \(P\) is at the point \(B\).
  1. Find the position vector of \(B\). [5]
Edexcel M2 Q4
11 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a template made by removing a square \(WXYZ\) from a uniform triangular lamina \(ABC\). The lamina is isosceles with \(CA = CB\) and \(AB = 12a\). The midpoint of \(AB\) is \(N\) and \(NC = 8a\). The centre \(O\) of the square lies on \(NC\) and \(ON = 2a\). The sides \(WX\) and \(ZY\) are parallel to \(AB\) and \(WZ = 2a\). The centre of mass of the template is at \(G\).
  1. Show that \(NG = \frac{8}{7}a\). [7]
The template has mass \(M\). A small metal stud of mass \(kM\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(YZ\). By modelling the stud as a particle,
  1. calculate the value of \(k\). [4]
Edexcel M2 Q5
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a horizontal uniform pole \(AB\), of weight \(W\) and length \(2a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2W\) is attached to the pole at \(C\), where \(BC = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(θ\) to the horizontal, where \(\tan θ = \frac{5}{3}\). The pole is modelled as a uniform rod.
  1. Show that the tension in \(BD\) is \(\frac{5(5a - 2x)}{6a}W\). [5]
The vertical component of the force exerted by the wall on the pole is \(\frac{1}{2}W\). Find
  1. x in terms of \(a\), [3]
  1. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]
Edexcel M2 Q6
14 marks Standard +0.3
A smooth sphere \(P\) of mass \(m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(2m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). After the collision the direction of motion of \(P\) is unchanged. The spheres have the same radii and the coefficient of restitution between \(P\) and \(Q\) is \(e\). By modelling the spheres as particles,
  1. show that the speed of \(Q\) immediately after the collision is \(\frac{1}{3}(1 + e)u\), [5]
  1. find the range of possible values of \(e\). [4]
Given that \(e = \frac{1}{4}\),
  1. find the loss of kinetic energy in the collision. [4]
  1. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]
TURN OVER FOR QUESTION 7