Questions — Edexcel (10514 questions)

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Edexcel M2 2003 June Q7
15 marks Standard +0.3
A uniform sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another uniform sphere \(B\) of mass \(2m\) which is at rest on the table. The spheres are of equal radius and the coefficient of restitution between them is \(e\). The direction of motion of \(A\) is unchanged by the collision.
  1. Find the speeds of \(A\) and \(B\) immediately after the collision. [7]
  2. Find the range of possible values of \(e\). [2]
After being struck by \(A\), the sphere \(B\) collides directly with another sphere \(C\), of mass \(4m\) and of the same size as \(B\). The sphere \(C\) is at rest on the table immediately before being struck by \(B\). The coefficient of restitution between \(B\) and \(C\) is also \(e\).
  1. Show that, after \(B\) has struck \(C\), there will be a further collision between \(A\) and \(B\). [6]
Edexcel M2 2006 June Q1
6 marks Moderate -0.8
A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, its acceleration is \((5 - 2t)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. When \(t = 0\), its velocity is 6 m s\(^{-1}\) measured in the direction of \(x\) increasing. Find the time when \(P\) is instantaneously at rest in the subsequent motion. [6]
Edexcel M2 2006 June Q2
6 marks Moderate -0.3
A car of mass 1200 kg moves along a straight horizontal road with a constant speed of 24 m s\(^{-1}\). The resistance to motion of the car has magnitude 600 N.
  1. Find, in kW, the rate at which the engine of the car is working. [2]
The car now moves up a hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{20}\). The resistance to motion of the car from non-gravitational forces remains of magnitude 600 N. The engine of the car now works at a rate of 30 kW.
  1. Find the acceleration of the car when its speed is 20 m s\(^{-1}\). [4]
Edexcel M2 2006 June Q3
8 marks Moderate -0.8
A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \((-30\mathbf{i})\) m s\(^{-1}\). Immediately after being struck, the velocity of the ball is \((16\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\).
  1. Find the magnitude of the impulse exerted on the ball by the bat. [4]
In the subsequent motion, the position vector of the ball is \(\mathbf{r}\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf{r} = [16t\mathbf{i} + (20t - 5t^2)\mathbf{j}]\). Using this model,
  1. find the speed of the ball when \(t = 3\). [4]
Edexcel M2 2006 June Q4
10 marks Standard +0.3
Figure 1 \includegraphics{figure_1} Figure 1 shows four uniform rods joined to form a rigid rectangular framework \(ABCD\), where \(AB = CD = 2a\), and \(BC = AD = 3a\). Each rod has mass \(m\). Particles, of mass \(6m\) and \(2m\), are attached to the framework at points \(C\) and \(D\) respectively.
  1. Find the distance of the centre of mass of the loaded framework from
    1. \(AB\),
    2. \(AD\).
    [7]
The loaded framework is freely suspended from \(B\) and hangs in equilibrium.
  1. Find the angle which \(BC\) makes with the vertical. [3]
Edexcel M2 2006 June Q5
8 marks Standard +0.3
A vertical cliff is 73.5 m high. Two stones \(A\) and \(B\) are projected simultaneously. Stone \(A\) is projected horizontally from the top of the cliff with speed 28 m s\(^{-1}\). Stone \(B\) is projected from the bottom of the cliff with speed 35 m s\(^{-1}\) at an angle \(\alpha\) above the horizontal. The stones move freely under gravity in the same vertical plane and collide in mid-air. By considering the horizontal motion of each stone,
  1. prove that \(\cos \alpha = \frac{4}{5}\). [4]
  1. Find the time which elapses between the instant when the stones are projected and the instant when they collide. [4]
Edexcel M2 2006 June Q6
10 marks Standard +0.3
Figure 2 \includegraphics{figure_2} A wooden plank \(AB\) has mass \(4m\) and length \(4a\). The end \(A\) of the plank lies on rough horizontal ground. A small stone of mass \(m\) is attached to the plank at \(B\). The plank is resting on a small smooth horizontal peg \(C\), where \(BC = a\), as shown in Figure 2. The plank is in equilibrium making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between the plank and the ground is \(\mu\). The plank is modelled as a uniform rod lying in a vertical plane perpendicular to the peg, and the stone as a particle. Show that
  1. the reaction of the peg on the plank has magnitude \(\frac{16}{5}mg\), [3]
  1. \(\mu \geq \frac{48}{61}\). [6]
  1. State how you have used the information that the peg is smooth. [1]
Edexcel M2 2006 June Q7
12 marks Standard +0.3
A particle \(P\) has mass 4 kg. It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\frac{2}{5}\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(AB = 2.5\) m. It then moves back down the plane to \(A\).
  1. Find the work done by friction as \(P\) moves from \(A\) to \(B\). [4]
  1. Using the work-energy principle, find the speed with which \(P\) is projected from \(A\). [4]
  1. Find the speed of \(P\) when it returns to \(A\). [4]
Edexcel M2 2006 June Q8
15 marks Standard +0.3
Two particles \(A\) and \(B\) move on a smooth horizontal table. The mass of \(A\) is \(m\), and the mass of \(B\) is \(4m\). Initially \(A\) is moving with speed \(u\) when it collides directly with \(B\), which is at rest on the table. As a result of the collision, the direction of motion of \(A\) is reversed. The coefficient of restitution between the particles is \(e\).
  1. Find expressions for the speed of \(A\) and the speed of \(B\) immediately after the collision. [7]
In the subsequent motion, \(B\) strikes a smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{4}{5}\). Given that there is a second collision between \(A\) and \(B\),
  1. show that \(\frac{1}{4} < e < \frac{9}{16}\). [5]
Given that \(e = \frac{1}{2}\),
  1. find the total kinetic energy lost in the first collision between \(A\) and \(B\). [3]
Edexcel M2 2010 June Q1
Moderate -0.8
A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds, \(t \geq 0\), is \((3t + 5)\) m s\(^{-2}\) in the positive \(x\)-direction. When \(t = 0\), the velocity of \(P\) is 2 m s\(^{-1}\) in the positive \(x\)-direction. When \(t = T\), the velocity of \(P\) is 6 m s\(^{-1}\) in the positive \(x\)-direction. Find the value of \(T\). (6)
Edexcel M2 2010 June Q2
Moderate -0.3
A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at 30° to the horizontal. When \(P\) has moved 12 m, its speed is 4 m s\(^{-1}\). Given that friction is the only non-gravitational resistive force acting on \(P\), find
  1. the work done against friction as the speed of \(P\) increases from 0 m s\(^{-1}\) to 4 m s\(^{-1}\), (4)
  2. the coefficient of friction between the particle and the plane. (4)
Edexcel M2 2010 June Q3
Standard +0.3
\includegraphics{figure_1} A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle \(ABC\), where \(AB = AC = 10\) cm and \(BC = 12\) cm, as shown in Figure 1.
  1. Find the distance of the centre of mass of the frame from \(BC\). (5)
The frame has total mass \(M\). A particle of mass \(M\) is attached to the frame at the mid-point of \(BC\). The frame is then freely suspended from \(B\) and hangs in equilibrium.
  1. Find the size of the angle between \(BC\) and the vertical. (4)
Edexcel M2 2010 June Q4
Moderate -0.3
A car of mass 750 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{15}\). The resistance to motion of the car from non-gravitational forces has constant magnitude \(R\) newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of 20 m s\(^{-1}\).
  1. Show that \(R = 260\). (4)
The power developed by the car's engine is now increased to 18 kW. The magnitude of the resistance to motion from non-gravitational forces remains at 260 N. At the instant when the car is moving up the road at 20 m s\(^{-1}\) the car's acceleration is \(a\) m s\(^{-2}\).
  1. Find the value of \(a\). (4)
Edexcel M2 2010 June Q5
Moderate -0.3
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane.] A ball of mass 0.5 kg is moving with velocity \((10\mathbf{i} + 24\mathbf{j})\) m s\(^{-1}\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20\mathbf{i}\) m s\(^{-1}\). Find
  1. the magnitude of the impulse of the bat on the ball, (4)
  2. the size of the angle between the vector \(\mathbf{i}\) and the impulse exerted by the bat on the ball, (2)
  3. the kinetic energy lost by the ball in the impact. (3)
Edexcel M2 2010 June Q6
Standard +0.3
\includegraphics{figure_2} Figure 2 shows a uniform rod \(AB\) of mass \(m\) and length \(4a\). The end \(A\) of the rod is freely hinged to a point on a vertical wall. A particle of mass \(m\) is attached to the rod at \(B\). One end of a light inextensible string is attached to the rod at \(C\), where \(AC = 3a\). The other end of the string is attached to the wall at \(D\), where \(AD = 2a\) and \(D\) is vertically above \(A\). The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is \(T\).
  1. Show that \(T = mg\sqrt{13}\). (5)
The particle of mass \(m\) at \(B\) is removed from the rod and replaced by a particle of mass \(M\) which is attached to the rod at \(B\). The string breaks if the tension exceeds \(2mg\sqrt{13}\). Given that the string does not break,
  1. show that \(M \leq \frac{5}{2}m\). (3)
Edexcel M2 2010 June Q7
Standard +0.3
\includegraphics{figure_3} A ball is projected with speed 40 m s\(^{-1}\) from a point \(P\) on a cliff above horizontal ground. The point \(O\) on the ground is vertically below \(P\) and \(OP\) is 36 m. The ball is projected at an angle \(\theta°\) to the horizontal. The point \(Q\) is the highest point of the path of the ball and is 12 m above the level of \(P\). The ball moves freely under gravity and hits the ground at the point \(R\), as shown in Figure 3. Find
  1. the value of \(\theta\), (3)
  2. the distance \(OR\), (6)
  3. the speed of the ball as it hits the ground at \(R\). (3)
Edexcel M2 2010 June Q8
Standard +0.3
A small ball \(A\) of mass \(3m\) is moving with speed \(u\) in a straight line on a smooth horizontal table. The ball collides directly with another small ball \(B\) of mass \(m\) moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\). The balls have the same radius and can be modelled as particles.
  1. Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision.
    (7)
After the collision \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\).
  1. Find the speed of \(B\) immediately after hitting the wall. (2)
The first collision between \(A\) and \(B\) occurred at a distance \(4a\) from the wall. The balls collide again \(T\) seconds after the first collision.
  1. Show that \(T = \frac{112a}{15u}\). (6)
Edexcel M2 2011 June Q1
5 marks Moderate -0.3
A car of mass 1000 kg moves with constant speed \(V\) m s\(^{-1}\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{30}\). The engine of the car is working at a rate of 12 kW. The resistance to motion from non-gravitational forces has magnitude 500 N. Find the value of \(V\). [5]
Edexcel M2 2011 June Q2
8 marks Standard +0.3
A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision. Show that \(e > \frac{1}{3}\). [8]
Edexcel M2 2011 June Q3
8 marks Moderate -0.8
A ball of mass 0.5 kg is moving with velocity \(12\mathbf{i}\) m s\(^{-1}\) when it is struck by a bat. The impulse received by the ball is \((-4\mathbf{i} + 7\mathbf{j})\) N s. By modelling the ball as a particle, find
  1. the speed of the ball immediately after the impact, [4]
  2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf{i}\), [2]
  3. the kinetic energy gained by the ball as a result of the impact. [2]
Edexcel M2 2011 June Q4
7 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 4a\) and \(AE = 2a\). The point \(F\) is the midpoint of \(BD\) and \(FC = a\).
  1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(AE\). [4]
The lamina is freely suspended from \(A\) and hangs in equilibrium.
  1. Find the angle between \(AB\) and the downward vertical. [3]
Edexcel M2 2011 June Q5
10 marks Standard +0.3
\includegraphics{figure_2} A particle \(P\) of mass 0.5 kg is projected from a point \(A\) up a line of greatest slope \(AB\) of a fixed plane. The plane is inclined at 30° to the horizontal and \(AB = 2\) m with \(B\) above \(A\), as shown in Figure 2. The particle \(P\) passes through \(B\) with speed 5 m s\(^{-1}\). The plane is smooth from \(A\) to \(B\).
  1. Find the speed of projection. [4]
The particle \(P\) comes to instantaneous rest at the point \(C\) on the plane, where \(C\) is above \(B\) and \(BC = 1.5\) m. From \(B\) to \(C\) the plane is rough and the coefficient of friction between \(P\) and the plane is \(\mu\). By using the work-energy principle,
  1. find the value of \(\mu\). [6]
Edexcel M2 2011 June Q6
11 marks Moderate -0.3
A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((t - 4)\) m s\(^{-2}\) in the positive \(x\)-direction. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). When \(t = 0\), \(v = 6\). Find
  1. \(v\) in terms of \(t\), [4]
  2. the values of \(t\) when \(P\) is instantaneously at rest, [3]
  3. the distance between the two points at which \(P\) is instantaneously at rest. [4]
Edexcel M2 2011 June Q7
13 marks Standard +0.3
\includegraphics{figure_3} A uniform rod \(AB\), of mass \(3m\) and length \(4a\), is held in a horizontal position with the end \(A\) against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\) vertically above \(A\), where \(AD = 3a\). A particle of mass \(3m\) is attached to the rod at \(C\), where \(AC = x\). The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is \(\frac{25}{4}mg\). Show that
  1. \(x = 3a\), [5]
  2. the horizontal component of the force exerted by the wall on the rod has magnitude \(5mg\). [3]
The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is about to slip,
  1. find the value of \(\mu\). [5]
Edexcel M2 2011 June Q8
13 marks Standard +0.3
A particle is projected from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).
  1. Show that $$y = x \tan \alpha - \frac{gx^2}{2u^2 \cos^2 \alpha}$$ [4]
A girl throws a ball from a point \(A\) at the top of a cliff. The point \(A\) is 8 m above a horizontal beach. The ball is projected with speed 7 m s\(^{-1}\) at an angle of elevation of 45°. By modelling the ball as a particle moving freely under gravity,
  1. find the horizontal distance of the ball from \(A\) when the ball is 1 m above the beach. [5]
A boy is standing on the beach at the point \(B\) vertically below \(A\). He starts to run in a straight line with speed \(v\) m s\(^{-1}\), leaving \(B\) 0.4 seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.
  1. Find the value of \(v\). [4]