Questions — Edexcel M2 (623 questions)

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Edexcel M2 2006 January Q3
9 marks Standard +0.3
A car of mass 1000 kg is moving along a straight horizontal road. The resistance to motion is modelled as a constant force of magnitude \(R\) newtons. The engine of the car is working at a rate of 12 kW. When the car is moving with speed 15 m s\(^{-1}\), the acceleration of the car is 0.2 m s\(^{-2}\).
  1. Show that \(R = 600\). [4]
The car now moves with constant speed \(U\) m s\(^{-1}\) downhill on a straight road inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{30}\). The engine of the car is now working at a rate of 7 kW. The resistance to motion from non-gravitational forces remains of magnitude \(R\) newtons.
  1. Calculate the value of \(U\). [5]
Edexcel M2 2006 January Q4
13 marks Standard +0.3
A particle \(A\) of mass \(2m\) is moving with speed \(3u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\frac{8}{3}u\) and the direction of motion of \(B\) is reversed.
  1. Calculate the coefficient of restitution between \(A\) and \(B\). [6]
  2. Show that the kinetic energy lost in the collision is \(7mu^2\). [3]
After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\frac{14}{3}mu\).
  1. Calculate the coefficient of restitution between \(B\) and the wall. [4]
Edexcel M2 2006 January Q5
12 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a triangular lamina \(ABC\). The coordinates of \(A\), \(B\) and \(C\) are \((0, 4)\), \((9, 0)\) and \((0, -4)\) respectively. Particles of mass \(4m\), \(6m\) and \(2m\) are attached at \(A\), \(B\) and \(C\) respectively.
  1. Calculate the coordinates of the centre of mass of the three particles, without the lamina. [4]
The lamina \(ABC\) is uniform and of mass \(km\). The centre of mass of the combined system consisting of the three particles and the lamina has coordinates \((4, \lambda)\).
  1. Show that \(k = 6\). [3]
  2. Calculate the value of \(\lambda\). [2]
The combined system is freely suspended from \(O\) and hangs at rest.
  1. Calculate, in degrees to one decimal place, the angle between \(AC\) and the vertical. [3]
Edexcel M2 2006 January Q6
13 marks Standard +0.8
\includegraphics{figure_2} A ladder \(AB\), of weight \(W\) and length \(4a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\mu\). The other end \(B\) rests against a smooth vertical wall. The ladder makes an angle \(\theta\) with the horizontal, where \(\tan \theta = 2\). A load of weight \(4W\) is placed at the point \(C\) on the ladder, where \(AC = 3a\), as shown in Figure 2. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The load is modelled as a particle. Given that the system is in limiting equilibrium,
  1. show that \(\mu = 0.35\). [6]
A second load of weight \(kW\) is now placed on the ladder at \(A\). The load of weight \(4W\) is removed from \(C\) and placed on the ladder at \(B\). The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The loads are modelled as particles. Given that the ladder and the loads are in equilibrium,
  1. Find the range of possible values of \(k\). [7]
Edexcel M2 2006 January Q7
14 marks Standard +0.3
\includegraphics{figure_3} The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m. The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of 11 m s\(^{-1}\) at an angle of elevation of \(30°\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.
  1. Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\). [3]
  2. Show that \(C\) is approximately 0.63 m below \(T\). [4]
The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V\) m s\(^{-1}\), the angle of elevation remaining \(30°\). This time \(B\) hits \(T\).
  1. Calculate the value of \(V\). [6]
  2. Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target. [1]
Edexcel M2 2007 January Q1
6 marks Moderate -0.8
A particle of mass 0.8 kg is moving in a straight line on a rough horizontal plane. The speed of the particle is reduced from 15 m s\(^{-1}\) to 10 m s\(^{-1}\) as the particle moves 20 m. Assuming that the only resistance to motion is the friction between the particle and the plane, find
  1. the work done by friction in reducing the speed of the particle from 15 m s\(^{-1}\) to 10 m s\(^{-1}\), [2]
  2. the coefficient of friction between the particle and the plane. [4]
Edexcel M2 2007 January Q2
8 marks Standard +0.3
A car of mass 800 kg is moving at a constant speed of 15 m s\(^{-1}\) down a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{3}{4}\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 900 N.
  1. Find, in kW, the rate of working of the engine of the car. [4]
When the car is travelling down the road at 15 m s\(^{-1}\), the engine is switched off. The car comes to rest in time \(T\) seconds after the engine is switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 900 N.
  1. Find the value of \(T\). [4]
Edexcel M2 2007 January Q3
10 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm, from a uniform circular lamina, of centre \(O\) and radius 24 cm. The point \(X\) lies on the diameter \(AOB\) of the lamina and \(AX = 16\) cm. The centre of mass of \(T\) is at the point \(G\).
  1. Find \(AG\). [6]
The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(OB\). A small stud of mass \(\frac{1}{4}m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(AB\) horizontal. Modelling the stud as a particle,
  1. find the mass of \(T\) in terms of \(m\). [4]
Edexcel M2 2007 January Q4
12 marks Standard +0.3
A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal table. Another particle \(Q\) of mass \(km\) is at rest on the table. The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. After the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(3v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{2}\).
  1. Find, in terms of \(v\) only, the speed of \(P\) before the collision. [3]
  2. Find the value of \(k\). [3]
After being struck by \(P\), the particle \(Q\) collides directly with a particle \(R\) of mass \(11m\) which is at rest on the table. After this second collision, \(Q\) and \(R\) have the same speed and are moving in opposite directions. Show that
  1. the coefficient of restitution between \(Q\) and \(R\) is \(\frac{1}{4}\), [4]
  2. there will be a further collision between \(P\) and \(Q\). [2]
Edexcel M2 2007 January Q5
12 marks Standard +0.3
\includegraphics{figure_2} A horizontal uniform rod \(AB\) has mass \(m\) and length \(4a\). The end \(A\) rests against a rough vertical wall. A particle of mass \(2m\) is attached to the rod at the point \(C\), where \(AC = 3a\). One end of a light inextensible string \(BD\) is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\), where \(D\) is vertically above \(A\). The rod is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{3}{4}\), as shown in Figure 2.
  1. Find the tension in the string. [5]
  2. Show that the horizontal component of the force exerted by the wall on the rod has magnitude \(\frac{5}{8}mg\). [3]
The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
  1. find the value of \(\mu\). [4]
Edexcel M2 2007 January Q6
13 marks Standard +0.3
A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds, \(\mathbf{F} = (1.5t^2 - 3)\mathbf{i} + 2t\mathbf{j}\). When \(t = 2\), the velocity of \(P\) is \((-4\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).
  1. Find the acceleration of \(P\) at time \(t\) seconds. [2]
  2. Show that, when \(t = 3\), the velocity of \(P\) is \((9\mathbf{i} + 15\mathbf{j})\) m s\(^{-1}\). [5]
When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf{Q}\) N s. Immediately after the impulse the velocity of \(P\) is \((-3\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\). Find
  1. the magnitude of \(\mathbf{Q}\), [3]
  2. the angle between \(\mathbf{Q}\) and \(\mathbf{i}\). [3]
Edexcel M2 2007 January Q7
14 marks Standard +0.3
\includegraphics{figure_3} A particle \(P\) is projected from a point \(A\) with speed \(u\) m s\(^{-1}\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac{4}{5}\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(AB = 45\) m. After projection, \(P\) moves freely under gravity passing through a point \(C\), 30 m above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed 24.5 m s\(^{-1}\),
  1. using conservation of energy, or otherwise, show that \(u = 17.5\), [4]
  2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\), [3]
  3. find the distance \(BD\). [7]
Edexcel M2 2008 January Q1
5 marks Moderate -0.8
A parcel of mass 2.5 kg is moving in a straight line on a smooth horizontal floor. Initially the parcel is moving with speed 8 m s\(^{-1}\). The parcel is brought to rest in a distance of 20 m by a constant horizontal force of magnitude \(R\) newtons. Modelling the parcel as a particle, find
  1. the kinetic energy lost by the parcel in coming to rest, [2]
  2. the value of \(R\). [3]
Edexcel M2 2008 January Q2
9 marks Moderate -0.3
At time \(t\) seconds \((t \geq 0)\), a particle \(P\) has position vector \(\mathbf{p}\) metres, with respect to a fixed origin \(O\), where $$\mathbf{p} = (3t^2 - 6t + 4)\mathbf{i} + (3t^3 - 4t)\mathbf{j}.$$ Find
  1. the velocity of \(P\) at time \(t\) seconds, [2]
  2. the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf{i}\). [3]
When \(t = 1\), the particle \(P\) receives an impulse of \((2\mathbf{i} - 6\mathbf{j})\) N s. Given that the mass of \(P\) is 0.5 kg,
  1. find the velocity of \(P\) immediately after the impulse. [4]
Edexcel M2 2008 January Q3
9 marks Standard +0.3
A car of mass 1000 kg is moving at a constant speed of 16 m s\(^{-1}\) up a straight road inclined at an angle \(\theta\) to the horizontal. The rate of working of the engine of the car is 20 kW and the resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 550 N.
  1. Show that \(\sin \theta = \frac{1}{14}\). [5]
When the car is travelling up the road at 16 m s\(^{-1}\), the engine is switched off. The car comes to rest, without braking, having moved a distance \(y\) metres from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 550 N.
  1. Find the value of \(y\). [4]
Edexcel M2 2008 January Q4
12 marks Standard +0.3
\includegraphics{figure_1} A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(ABC\), with \(\angle ABC = 90°\), \(AB = 12\) cm and \(BC = 21\) cm. The point \(O\) is 5 cm from \(AB\) and 5 cm from \(BC\), as shown in Figure 1.
  1. Find the distance of the centre of mass of \(S\) from
    1. \(AB\),
    2. \(BC\). [9]
The set square is freely suspended from \(C\) and hangs in equilibrium.
  1. Find, to the nearest degree, the angle between \(CB\) and the vertical. [3]
Edexcel M2 2008 January Q5
10 marks Standard +0.3
\includegraphics{figure_2} A ladder \(AB\), of mass \(m\) and length \(4a\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. A load of mass \(3m\) is fixed on the ladder at the point \(C\), where \(AC = a\). The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of 30° with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground. [10]
Edexcel M2 2008 January Q6
13 marks Standard +0.3
\includegraphics{figure_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 vertical.] A particle \(P\) is projected from the point \(A\) which has position vector \(47.5\mathbf{j}\) metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \((2u\mathbf{i} + 5u\mathbf{j})\) m s\(^{-1}\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30\mathbf{i}\) metres, as shown in Figure 3.
  1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s. [6]
  2. Find the value of \(u\). [2]
  3. Find the speed of \(P\) at \(B\). [5]
Edexcel M2 2008 January Q7
17 marks Standard +0.8
A particle \(P\) of mass \(2m\) is moving with speed \(2u\) in a straight line on a smooth horizontal plane. A particle \(Q\) of mass \(3m\) is moving with speed \(u\) in the same direction as \(P\). The particles collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\).
  1. Show that the speed of \(Q\) immediately after the collision is \(\frac{3}{2}u\). [5]
  2. Find the total kinetic energy lost in the collision. [5]
After the collision between \(P\) and \(Q\), the particle \(Q\) collides directly with a particle \(R\) of mass \(m\) which is at rest on the plane. The coefficient of restitution between \(Q\) and \(R\) is \(e\).
  1. Calculate the range of values of \(e\) for which there will be a second collision between \(P\) and \(Q\). [7]
Edexcel M2 2010 January Q1
8 marks Standard +0.3
A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \text{ ms}^{-1}\) in the positive \(x\)-direction, where \(v = 3t^2 - 4t + 3\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity. [8]
Edexcel M2 2010 January Q2
7 marks Moderate -0.3
Two particles, \(P\), of mass \(2m\), and \(Q\), of mass \(m\), are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between the particles is \(e\), where \(e < 1\). Find, in terms of \(u\) and \(e\),
  1. the speed of \(P\) immediately after the collision,
  2. the speed of \(Q\) immediately after the collision.
[7]
Edexcel M2 2010 January Q3
6 marks Moderate -0.3
A particle of mass \(0.5\) kg is projected vertically upwards from ground level with a speed of \(20 \text{ ms}^{-1}\). It comes to instantaneous rest at a height of \(10\) m above the ground. As the particle moves it is subject to air resistance of constant magnitude \(R\) newtons. Using the work-energy principle, or otherwise, find the value of \(R\). [6]
Edexcel M2 2010 January Q4
8 marks Standard +0.3
\includegraphics{figure_1} The points \(A\), \(B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass \(0.25\) kg. Immediately before being struck, the ball is moving along the horizontal line \(AB\) with speed \(30 \text{ ms}^{-1}\). Immediately after being struck, the ball moves along the horizontal line \(BC\) with speed \(40 \text{ ms}^{-1}\). The line \(BC\) makes an angle of \(60°\) with the original direction of motion \(AB\), as shown in Figure 1. Find, to 3 significant figures,
  1. the magnitude of the impulse given to the ball,
  2. the size of the angle that the direction of this impulse makes with the original direction of motion \(AB\).
[8]
Edexcel M2 2010 January Q5
11 marks Standard +0.3
A cyclist and her bicycle have a total mass of \(70\) kg. She cycles along a straight horizontal road with constant speed \(3.5 \text{ ms}^{-1}\). She is working at a constant rate of \(490\) W.
  1. Find the magnitude of the resistance to motion. [4]
The cyclist now cycles down a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), at a constant speed \(U \text{ ms}^{-1}\). The magnitude of the non-gravitational resistance to motion is modelled as \(40U\) newtons. She is now working at a constant rate of \(24\) W.
  1. Find the value of \(U\). [7]
Edexcel M2 2010 January Q6
7 marks Standard +0.3
\includegraphics{figure_2} A uniform rod \(AB\), of mass \(20\) kg and length \(4\) m, rests with one end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\), by a force acting at \(B\), as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is \(0.5\). Find the magnitude of the normal reaction of the ground on the rod at \(A\). [7]