Questions — Edexcel M1 (663 questions)

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Edexcel M1 Q2
7 marks Moderate -0.3
Forces of magnitude \(4\) N, \(5\) N and \(8\) N act on a particle in directions whose bearings are \(000°\), \(090°\) and \(210°\) respectively. Find the magnitude of the resultant force and the bearing of the direction in which it acts. [7 marks] \includegraphics{figure_2}
Edexcel M1 Q3
9 marks Moderate -0.3
A packing-case, of mass \(60\) kg, is standing on the floor of a lift. The mass of the lift-cage is \(200\) kg. The lift-cage is raised and lowered by means of a cable attached to its roof. In each of the following cases, find the magnitude of the force exerted by the floor of the lift-cage on the packing-case and the tension in the cable supporting the lift:
  1. The lift is descending with constant speed. [3 marks]
  2. The lift is ascending and accelerating at \(1.2 \text{ ms}^{-2}\). [4 marks]
  3. State any modelling assumptions you have made. [2 marks]
Edexcel M1 Q4
12 marks Standard +0.3
\(AB\) is a light rod. Forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\), of magnitudes \(3\) N, \(2\) N and \(6\) N respectively, act upwards at right angles to the rod in a vertical plane at points dividing \(AB\) in the ratio \(1:4:2:4\), as shown. \includegraphics{figure_4} A single force \(\mathbf{P}\) is applied downwards at the point \(C\) to keep the rod horizontal in equilibrium.
  1. State the magnitude of \(\mathbf{P}\). [1 mark]
  2. Show that \(AC:CB = 5:6\). [5 marks]
Two particles, of weights \(3\) N and \(k\) N, are now placed on the rod at \(A\) and \(B\) respectively, while the same upward forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\) act as before. It is found that a single downward force at the same point \(C\) as before keeps \(AB\) horizontal under gravity.
  1. Find the value of \(k\). [6 marks]
Edexcel M1 Q5
13 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\) and the table being \(\frac{2}{7}\). \(B\) hangs freely on the end of the vertical portion of the string. \includegraphics{figure_5} \(A\) is now given an impulse, directed away from the pulley, of magnitude \(5m\) Ns.
  1. Show that the system starts to move with speed \(2.5 \text{ ms}^{-1}\). [1 mark]
  2. State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal. [1 mark]
Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,
  1. find the time taken for the system to come to rest. [7 marks]
  2. Find the distance travelled by \(A\) before it first comes to rest. [4 marks]
Edexcel M1 Q6
14 marks Standard +0.3
The diagram shows the velocity-time graph for a cyclist's journey. Each section has constant acceleration or deceleration and the three sections are of equal duration \(x\) seconds each. \includegraphics{figure_6} Given that the total distance travelled is \(792\) m,
  1. find the value of \(x\) and the acceleration for the first section of the journey. [6 marks]
Another cyclist covers the same journey in three sections of equal duration, accelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the first section, travelling at constant speed for the second section and decelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the third section.
  1. Find the time taken by this cyclist to complete the journey. [6 marks]
  2. Show that the maximum speeds of both cyclists are the same. [2 marks]
Edexcel M1 Q7
14 marks Moderate -0.3
Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors.
  1. Find the distance \(XY\). [2 marks]
A particle \(P\) of mass \(2\) kg moves from \(X\) to \(Y\) in \(4\) seconds, in a straight line at a constant speed.
  1. Show that the velocity vector of \(P\) is \((2\mathbf{i} + 1.5\mathbf{j}) \text{ ms}^{-1}\). [3 marks]
The particle continues beyond \(Y\) with the same constant velocity.
  1. Write down an expression for the position vector of \(P\) \(t\) seconds after leaving \(X\). [2 marks]
  2. Find the value of \(t\) when \(P\) is at the point with position vector \((16\mathbf{i} + 4\mathbf{j})\) m. [2 marks]
When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass \(4\) kg, which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.
  1. Find the velocity vector of the combined particle after the collision. [5 marks]
Edexcel M1 Q1
5 marks Moderate -0.8
Two forces, both of magnitude 5 N, act on a particle in the directions with bearings 000° and 070°, as shown. \includegraphics{figure_1} Calculate
  1. the magnitude of the resultant force on the particle, [3 marks]
  2. the bearing on which this resultant force acts. [2 marks]
Edexcel M1 Q2
6 marks Standard +0.3
A uniform plank \(XY\) has length 7 m and mass 2 kg. It is placed with the portion \(ZY\) in contact with a horizontal surface, where \(ZY = 2.8\) m. To prevent the plank from toppling, a stone is placed on the plank at \(Y\). \includegraphics{figure_2}
  1. Find the smallest possible mass of the stone. [4 marks]
  2. State, with a reason, whether your answer to part (a) would be greater or smaller if a shorter portion of the plank were in contact with the surface. [2 marks]
Edexcel M1 Q3
7 marks Moderate -0.8
A car, of mass 1800 kg, pulls a trailer of mass 350 kg along a straight horizontal road. When the car is accelerating at \(0.2\) ms\(^{-2}\), the resistances to the motion of the car and trailer have magnitudes 300 N and 100 N respectively. Find, at this time,
  1. the driving force produced by the engine of the car, [3 marks]
  2. the tension in the tow-bar between the car and the trailer. [4 marks]
Edexcel M1 Q4
11 marks Moderate -0.8
A train starts from rest at a station \(S\) and accelerates at a constant rate for \(2x\) seconds to a speed of \(5x\) ms\(^{-1}\). It maintains this speed until 126 seconds after it left \(S\) and then decelerates at a constant rate until it comes to rest at another station \(T\), \(20x\) seconds after it left \(S\).
  1. Sketch a velocity-time graph for this journey. [4 marks]
Given that the distance between \(S\) and \(T\) is \(5.4\) km,
  1. show that \(x^2 + 7x = 120\). [4 marks]
  2. Find the value of \(x\). [3 marks]
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]