Questions — Edexcel (10514 questions)

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Edexcel M2 2002 January 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 \(BD\) is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). 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 \(\frac{7}{4}W\). Find
  1. \(x\) in terms of \(a\), [3]
  2. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]
Edexcel M2 2002 January 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]
  2. 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]
  2. Give one possible form of energy into which the lost kinetic energy has been transformed. [1]
Edexcel M2 2002 January Q7
15 marks Standard +0.3
\includegraphics{figure_3} A rocket \(R\) of mass 100 kg is projected from a point \(A\) with speed 80 m s\(^{-1}\) at an angle of elevation of 60°, as shown in Fig. 3. The point \(A\) is 20 m vertically above a point \(O\) which is on horizontal ground. The rocket \(R\) moves freely under gravity. At \(B\) the velocity of \(R\) is horizontal. By modelling \(R\) as a particle, find
  1. the height in m of \(B\) above the ground, [4]
  2. the time taken for \(R\) to reach \(B\) from \(A\). [2]
When \(R\) is at \(B\), there is an internal explosion and \(R\) breaks into two parts \(P\) and \(Q\) of masses 60 kg and 40 kg respectively. Immediately after the explosion the velocity of \(P\) is 80 m s\(^{-1}\) horizontally away from \(A\). After the explosion the paths of \(P\) and \(Q\) remain in the plane \(OAB\). Part \(Q\) strikes the ground at \(C\). By modelling \(P\) and \(Q\) as particles,
  1. show that the speed of \(Q\) immediately after the explosion is 20 m s\(^{-1}\), [3]
  2. find the distance \(OC\). [6]
Edexcel M2 2003 January Q1
7 marks Moderate -0.8
Three particles of mass \(3m\), \(5m\) and \(\lambda m\) are placed at points with coordinates \((4, 0)\), \((0, -3)\) and \((4, 2)\) respectively. The centre of mass of the system of three particles is at \((2, k)\).
  1. Show that \(\lambda = 2\). [4]
  2. Calculate the value of \(k\). [3]
Edexcel M2 2003 January Q2
8 marks Moderate -0.3
A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(f\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.
  1. Calculate the value of \(f\). [4]
When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,
  1. use the work-energy principle to calculate the value of \(d\). [3]
  2. Give a reason why the model used for the resistance to motion may not be realistic. [1]
Edexcel M2 2003 January Q3
9 marks Standard +0.3
\includegraphics{figure_1} A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{2}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]
Edexcel M2 2003 January Q4
9 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 8a\) and \(AE = 6a\). The point \(X\) is the mid-point of \(BD\) and \(XC = 4a\). The centre of mass of the lamina is at \(G\).
  1. Show that \(GX = \frac{14}{15}a\). [6]
The mass of the lamina is \(M\). A particle of mass \(\lambda M\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.
  1. Find the value of \(\lambda\). [3]
Edexcel M2 2003 January Q5
11 marks Standard +0.3
A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((4t - 8)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). Given that \(v = 6\) when \(t = 0\), find
  1. \(v\) in terms of \(t\), [4]
  2. the distance between the two points where \(P\) is instantaneously at rest. [7]
Edexcel M2 2003 January Q6
15 marks Standard +0.3
A smooth sphere \(P\) of mass \(2m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. Another smooth sphere \(Q\) of mass \(m\) is at rest on the table. The sphere \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). The spheres are modelled as particles.
  1. Show that, immediately after the collision, the speeds of \(P\) and \(Q\) are \(\frac{2}{9}u\) and \(\frac{8}{9}u\) respectively. [7]
After the collision, \(Q\) strikes a fixed vertical wall which is perpendicular to the direction of motion of \(P\) and \(Q\). The coefficient of restitution between \(Q\) and the wall is \(e\). When \(P\) and \(Q\) collide again, \(P\) is brought to rest.
  1. Find the value of \(e\). [7]
  2. Explain why there must be a third collision between \(P\) and \(Q\). [1]
Edexcel M2 2003 January Q7
16 marks Standard +0.3
\includegraphics{figure_3} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \((-20\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\). Immediately after being struck it has velocity \((15\mathbf{i} + 16\mathbf{j})\) m s\(^{-1}\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.
  1. Calculate the magnitude of the impulse exerted by the bat on \(B\). [4]
  2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\). [6]
  3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\). [4]
  4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic. [2]
Edexcel M2 2006 January Q1
6 marks Moderate -0.8
A brick of mass 3 kg slides in a straight line on a horizontal floor. The brick is modelled as a particle and the floor as a rough plane. The initial speed of the brick is 8 m s\(^{-1}\). The brick is brought to rest after moving 12 m by the constant frictional force between the brick and the floor.
  1. Calculate the kinetic energy lost by the brick in coming to rest, stating the units of your answer. [2]
  2. Calculate the coefficient of friction between the brick and the floor. [4]
Edexcel M2 2006 January Q2
8 marks Moderate -0.3
A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by $$\mathbf{r} = (t^2 + 4t)\mathbf{i} + (3t - t^3)\mathbf{j}.$$
  1. Calculate the speed of \(P\) when \(t = 3\). [5]
When \(t = 3\), the particle \(P\) is given an impulse \((8\mathbf{i} - 12\mathbf{j})\) N s.
  1. Find the velocity of \(P\) immediately after the impulse. [3]
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]