Questions — Edexcel (10514 questions)

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Edexcel M1 Q5
15 marks Moderate -0.3
\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. The point \(A\) has position vector \(6\mathbf{j}\) m relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity \((5\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4\mathbf{i}\) ms\(^{-1}\).
  1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds. [3 marks]
  2. Show that the distance \(d\) m between \(P\) and \(Q\) at time \(t\) seconds is such that $$d^2 = 5t^2 - 24t + 36.$$ [5 marks]
  3. Find the value of \(t\) for which \(d^2\) is a minimum. [3 marks]
  4. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together. [4 marks]
Edexcel M1 Q6
15 marks Standard +0.3
\(A\), \(B\) and \(C\) are three small spheres of equal radii and masses \(2m\), \(m\) and \(5m\) respectively. They are placed in a straight line on a smooth horizontal surface. \(A\) is projected with speed 6 ms\(^{-1}\) towards \(B\), which is at rest. When \(A\) hits \(B\) it exerts an impulse of magnitude \(8m\) Ns on \(B\).
  1. Find the speed with which \(B\) starts to move. [2 marks]
  2. Show that the speed of \(A\) after it collides with \(B\) is 2 ms\(^{-1}\). [3 marks]
After travelling 3 m, \(B\) hits \(C\), which is then travelling towards \(B\) at \(2.2\) ms\(^{-1}\). \(C\) is brought to rest by this impact.
  1. Show that the direction of \(B\)'s motion is reversed and find its new speed. [3 marks]
  2. Find how far \(B\) now travels before it collides with \(A\) again. [6 marks]
  3. State a modelling assumption that you have made about the spheres. [1 mark]
Edexcel M1 Q7
16 marks Standard +0.3
A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at 30° to the horizontal as shown. A light string is attached to \(P\) and makes an angle of 30° with the plane. When the tension in this string has magnitude \(kmg\), \(P\) is just on the point of moving up the plane. \includegraphics{figure_7}
  1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac{k\sqrt{3} - 1}{\sqrt{3} - k}\). [7 marks]
  2. Given further that \(k = \frac{3\sqrt{3}}{7}\), deduce that \(\mu = \frac{\sqrt{3}}{6}\). [3 marks]
The string is now removed.
  1. Determine whether \(P\) will move down the plane and, if it does, find its acceleration. [5 marks]
  2. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made. [1 mark]
Edexcel M1 Q1
4 marks Easy -1.8
Briefly define the following terms used in modelling in Mechanics:
  1. lamina,
  2. uniform rod,
  3. smooth surface,
  4. particle.
[4 marks]
Edexcel M1 Q2
8 marks Moderate -0.3
Two forces \(\mathbf{F}\) and \(\mathbf{G}\) are given by \(\mathbf{F} = (6\mathbf{i} - 5\mathbf{j})\) N, \(\mathbf{G} = (3\mathbf{i} + 17\mathbf{j})\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm.
  1. Find the magnitude of \(\mathbf{R}\), the resultant of \(\mathbf{F}\) and \(\mathbf{G}\). [3 marks]
  2. Find the angle between the direction of \(\mathbf{R}\) and the positive \(x\)-axis. [2 marks]
\(\mathbf{R}\) acts through the point \(P(-4, 3)\). \(O\) is the origin \((0, 0)\).
  1. Use the fact that \(OP\) is perpendicular to the line of action of \(\mathbf{R}\) to calculate the moment of \(\mathbf{R}\) about an axis through the origin and perpendicular to the \(x\)-\(y\) plane. [3 marks]
Edexcel M1 Q3
12 marks Standard +0.3
A string is attached to a packing case of mass 12 kg, which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and the string makes an angle of 35° with the vertical as shown, the case is on the point of moving. \includegraphics{figure_3}
  1. Find the coefficient of friction between the case and the plane. [5 marks]
The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of 2 ms\(^{-1}\) after 4 seconds.
  1. Find the magnitude of the new force. [5 marks]
  2. State any modelling assumptions you have made about the case and the string. [2 marks]
Edexcel M1 Q4
12 marks Standard +0.3
A uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}
  1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
  2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
  3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 marks]
Edexcel M1 Q5
12 marks Standard +0.3
Two small smooth spheres \(A\) and \(B\), of equal radius but masses \(m\) kg and \(km\) kg respectively, where \(k > 1\), move towards each other along a straight line and collide directly. Immediately before the collision, \(A\) has speed 5 ms\(^{-1}\) and \(B\) has speed 3 ms\(^{-1}\). In the collision, the impulse exerted by \(A\) on \(B\) has magnitude \(7km\) Ns.
  1. Find the speed of \(B\) after the impact. [3 marks]
  2. Show that the speed of \(A\) immediately after the collision is \((7k - 5)\) ms\(^{-1}\) and deduce that the direction of \(A\)'s motion is reversed. [5 marks]
\(B\) is now given a further impulse of magnitude \(mu\) Ns, as a result of which a second collision between it and \(A\) occurs.
  1. Show that \(u > k(7k - 1)\). [4 marks]
Edexcel M1 Q6
13 marks Moderate -0.3
The velocity-time graph illustrates the motion of a particle which accelerates from rest to 8 ms\(^{-1}\) in \(x\) seconds and then to 24 ms\(^{-1}\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to 12 ms\(^{-1}\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. \includegraphics{figure_6} Given that the total distance travelled by the particle is 496 m,
  1. show that \(2x + 21y = 195\). [4 marks]
Given also that the average speed of the particle during its motion is 15.5 ms\(^{-1}\),
  1. show that \(x + 2y = 21\). [3 marks]
  2. Hence find the values of \(x\) and \(y\). [3 marks]
  3. Write down the acceleration for each section of the motion. [3 marks]
Edexcel M1 Q7
14 marks Standard +0.8
Two particles \(P\) and \(Q\), of masses \(2m\) and \(3m\) respectively, are connected by a light string. Initially, \(P\) is at rest on a smooth horizontal table. The string passes over a small smooth pulley and \(Q\) rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{4}{3}\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac{1}{6}\). \includegraphics{figure_7} The system is released from rest with \(Q\) at a distance of 0.8 metres above a horizontal floor.
  1. Show that the acceleration of \(P\) and \(Q\) is \(\frac{21g}{50}\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
  2. Find the speed with which \(Q\) hits the floor. [2 marks]
After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.
  1. Find the total time after the system is released before \(P\) hits the pulley. [5 marks]
Edexcel M1 Q1
7 marks Moderate -0.3
A boy holds a 30 cm metal ruler between three fingers of one hand, pushing down with the middle finger and up with the other two, at the points marked 5 cm, 10 cm and \(x\) cm and exerting forces of magnitude 11 N, 18 N and 8 N respectively. The ruler is in equilibrium in this position. Modelling the ruler as a uniform rod, find \includegraphics{figure_1}
  1. the mass of the ruler, in grams, \hfill [3 marks]
  2. the value of \(x\). \hfill [3 marks]
  3. State how you have used the modelling assumption that the ruler is a uniform rod. \hfill [1 mark]
Edexcel M1 Q2
7 marks Standard +0.8
\includegraphics{figure_2} A small packet of mass 0.3 kg rests on a rough horizontal surface. The coefficient of friction between the packet and the surface is \(\frac{1}{4}\). Two strings are attached to the packet, making angles of 45° and 30° with the horizontal, and when forces of magnitude 2 N and \(F\) N are exerted through the strings as shown, the packet is on the point of moving in the direction \(\overrightarrow{AB}\). Find the value of \(F\). \hfill [7 marks]
Edexcel M1 Q3
7 marks Moderate -0.8
A body moves in a straight line with constant acceleration. Its speed increases from 17 ms\(^{-1}\) to 33 ms\(^{-1}\) while it travels a distance of 250 m. Find
  1. the time taken to travel the 250 m, \hfill [3 marks]
  2. the acceleration of the body. \hfill [2 marks]
The body now decelerates at a constant rate from 33 ms\(^{-1}\) to rest in 6 seconds.
  1. Find the distance travelled in these 6 seconds. \hfill [2 marks]
Edexcel M1 Q4
12 marks Standard +0.3
A particle \(P\) of mass \(m\) kg, at rest on a smooth horizontal table, is connected to particles \(Q\) and \(R\), of mass 0.1 kg and 0.5 kg respectively, by strings which pass over fixed pulleys at the edges of the table. The system is released from rest with \(Q\) and \(R\) hanging freely and it is found that the tension in the section of the string between \(P\) and \(R\) is 2 N.
  1. Show that the acceleration of the particles has magnitude 5.8 ms\(^{-2}\). \hfill [3 marks]
  2. Find the value of \(m\). \hfill [5 marks]
Modelling assumptions have been made about the pulley and the strings.
  1. Briefly describe these two assumptions. For each one, state how the mathematical model would be altered if the assumption were not made. \hfill [4 marks]
Edexcel M1 Q5
12 marks Standard +0.3
Two trucks \(P\) and \(Q\), of masses 18 000 kg and 16 000 kg respectively, collide while moving towards each other in a straight line. Immediately before the collision, both trucks are travelling at the same speed, \(u\) ms\(^{-1}\). Immediately after the collision, \(P\) is moving at half its original speed, its direction of motion having been reversed.
  1. Find, in terms of \(u\), the speed of \(Q\) immediately after the collision. \hfill [5 marks]
  2. State, with a reason, whether the direction of \(Q\)'s motion has been reversed. \hfill [1 mark]
  3. Find, in terms of \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision, stating the units of your answer. \hfill [3 marks]
The force exerted by each truck on the other in the impact has magnitude \(108000u\) N.
  1. Find the time for which the trucks are in contact. \hfill [3 marks]
Edexcel M1 Q6
13 marks Moderate -0.8
A particle \(P\) moves in a straight line such that its displacement from a fixed point \(O\) at time \(t\) s is \(y\) metres. The graph of \(y\) against \(t\) is as shown.
[diagram]
  1. Write down the velocity of \(P\) when
    1. \(t = 1\), \quad (ii) \(t = 10\). \hfill [2 marks]
  2. State the total distance travelled by \(P\). \hfill [2 marks]
  3. Write down a formula for \(y\) in terms of \(t\) when \(2 \leq t < 4\). \hfill [3 marks]
  4. Sketch a velocity-time graph for the motion of \(P\) during the twelve seconds. \hfill [3 marks]
  5. Find the maximum speed of \(P\) during the motion. \hfill [3 marks]
Edexcel M1 Q7
17 marks Standard +0.3
Two trains \(S\) and \(T\) are moving with constant speeds on straight tracks which intersect at the point \(O\). At 9.00 a.m. \(S\) has position vector \((-10\mathbf{i} + 24\mathbf{j})\) km and \(T\) has position vector \(25\mathbf{j}\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due east and due north respectively. \(S\) is moving with speed 52 km h\(^{-1}\) and \(T\) is moving with speed 50 km h\(^{-1}\), both towards \(O\).
  1. Show that the velocity vector of \(S\) is \((20\mathbf{i} - 48\mathbf{j})\) km h\(^{-1}\) and find the velocity vector of \(T\). \hfill [5 marks]
  2. Find expressions for the position vectors of \(S\) and \(T\) at time \(t\) minutes after 9.00 a.m. \hfill [5 marks]
  3. Show that the bearing of \(T\) from \(S\) remains constant during the motion, and find this bearing. \hfill [5 marks]
  4. Show that if the trains continue at the given speeds they will collide. \hfill [2 marks]
Edexcel M1 Q1
5 marks Moderate -0.3
Two particles, \(P\) and \(Q\), of mass 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle \(P\) is projected away from \(Q\) with a speed of 7 ms\(^{-1}\).
  1. Find the common speed of the particles after the string becomes taut. [3 marks]
  2. Calculate the impulse in the string when it jerks tight. [2 marks]
Edexcel M1 Q2
6 marks Moderate -0.8
Particle \(A\) has velocity \((8\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) and particle \(B\) has velocity \((15\mathbf{i} - 8\mathbf{j})\) ms\(^{-1}\) where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors.
  1. Find the speed of \(B\). [2 marks]
  2. Find the velocity of \(B\) relative to \(A\). [2 marks]
  3. Find the acute angle between the relative velocity found in part (b) and the vector \(\mathbf{i}\), giving your answer in degrees correct to 1 decimal place. [2 marks]
Edexcel M1 Q3
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a uniform plank \(AB\) of length 8 m and mass 30 kg. It is supported in a horizontal position by two pivots, one situated at \(A\) and the other 2 m from \(B\). A man whose mass is 80 kg is standing on the plank 2 m from \(A\) when his dog steps onto the plank at \(B\). Given that the plank remains in equilibrium and that the magnitude of the forces exerted by each of the pivots on the plank are equal,
  1. calculate the magnitude of the force exerted on the plank by the pivot at \(A\), [5 marks]
  2. find the dog's mass. [3 marks]
If the dog was heavier and the plank was on the point of tilting,
  1. explain how the force exerted on the plank by each of the pivots would be changed. [2 marks]
Edexcel M1 Q4
10 marks Moderate -0.3
A cyclist and her bicycle have a combined mass of 78 kg. While riding on level ground and using her greatest driving force, she is able to accelerate uniformly from rest to 10 ms\(^{-1}\) in 15 seconds against constant resistive forces that total 60 N.
  1. Show that her maximum driving force is 112 N. [4 marks]
The cyclist begins to ascend a hill, inclined at an angle \(\alpha\) to the horizontal, riding with her maximum driving force and against the same resistive forces. In this case, she is able to maintain a steady speed.
  1. Find the angle \(\alpha\), giving your answer to the nearest degree. [4 marks]
  2. Comment on the assumption that the resistive force remains constant
    1. in the case when the cyclist is accelerating,
    2. in the case when she is maintaining a steady speed. [2 marks]
Edexcel M1 Q5
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a large block of mass 50 kg being pulled on rough horizontal ground by means of a rope attached to the block. The tension in the rope is 200 N and it makes an angle of 40° with the horizontal. Under these conditions, the block is on the point of moving. Modelling the block as a particle,
  1. show that the coefficient of friction between the block and the ground is 0.424 correct to 3 significant figures. [6 marks]
The angle with the horizontal at which the rope is being pulled is reduced to 30°. Ignoring air resistance and assuming that the tension in the rope and the coefficient of friction remain unchanged,
  1. find the acceleration of the block. [6 marks]
Edexcel M1 Q6
14 marks Moderate -0.3
Anila is practising catching tennis balls. She uses a mobile computer-controlled machine which fires tennis balls vertically upwards from a height of 2.5 metres above the ground. Once it has fired a ball, the machine is programmed to move position rapidly to allow Anila time to get into a suitable position to catch the ball. The machine fires a ball at 24 ms\(^{-1}\) vertically upwards and Anila catches the ball just before it touches the ground.
  1. Draw a speed-time graph for the motion of the ball from the time it is fired by the machine to the instant before Anila catches it. [3 marks]
  2. Find, to the nearest centimetre, the maximum height which the ball reaches above the ground. [4 marks]
  3. Calculate the speed at which the ball is travelling when Anila catches it. [4 marks]
  4. Calculate the length of time that the ball is in the air. [3 marks]
Edexcel M1 Q7
18 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a particle \(X\) of mass 3 kg on a smooth plane inclined at an angle 30° to the horizontal, and a particle \(Y\) of mass 2 kg on a smooth plane inclined at an angle 60° to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at \(C\) which is the highest point of the two planes. Initially, \(Y\) is at a point just below \(C\) touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, \(AC\) in the case of \(X\) and \(BC\) in the case of \(Y\), of their respective planes. \(A\) and \(B\) are the points where the planes meet the horizontal ground and \(AB = 4\) metres.
  1. Show that the initial acceleration of the system is given by \(\frac{g}{10}\left(2\sqrt{3} - 3\right)\) ms\(^{-2}\). [7 marks]
  2. By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical. [7 marks]
  3. Find, correct to 2 decimal places, the speed with which \(Y\) hits the ground. [4 marks]
Edexcel M1 Q1
7 marks Moderate -0.8
A constant force, \(\mathbf{F}\), acts on a particle, \(P\), of mass 5 kg causing its velocity to change from \((-2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) to \((4\mathbf{i} - 7\mathbf{j})\) m s\(^{-1}\) in 2 seconds.
  1. Find, in the form \(a\mathbf{i} + b\mathbf{j}\), the acceleration of \(P\). [2 marks]
  2. Show that the magnitude of \(\mathbf{F}\) is 25 N and find, to the nearest degree, the acute angle between the line of action of \(\mathbf{F}\) and the vector \(\mathbf{j}\). [5 marks]