Questions M1 (2067 questions)

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Edexcel M1 2016 June Q5
10 marks Moderate -0.5
Two forces, \(\mathbf{F}_1\) and \(\mathbf{F}_2\), act on a particle \(A\). \(\mathbf{F}_1 = (2i - 3j)\) N and \(\mathbf{F}_2 = (pi + qj)\) N, where \(p\) and \(q\) are constants. Given that the resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is parallel to \((\mathbf{i} + 2\mathbf{j})\),
  1. show that \(2p - q + 7 = 0\) [5] Given that \(q = 11\) and that the mass of \(A\) is 2 kg, and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only forces acting on \(A\),
  2. find the magnitude of the acceleration of \(A\). [5]
Edexcel M1 2016 June Q6
17 marks Moderate -0.3
\includegraphics{figure_2} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a\) m s\(^{-2}\) for 3.5 s, reaching a speed of 14 m s\(^{-1}\). Car \(A\) then moves with constant speed 14 m s\(^{-1}\).
  1. Find the value of \(a\). [2] Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with constant acceleration of 3 m s\(^{-2}\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
  2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). [3]
  3. Find the value of \(T\). [8]
  4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\). [1]
  5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). [3]
Edexcel M1 2016 June Q7
15 marks Standard +0.8
\includegraphics{figure_3} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m\) kg is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac{4}{3}\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac{1}{4}\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,
  1. find the value of \(m\), [10]
  2. find the magnitude of the force exerted on the pulley by the string, [4]
  3. find the direction of the force exerted on the pulley by the string. [1]
Edexcel M1 2017 October Q1
7 marks Moderate -0.3
A suitcase of mass 40 kg is being dragged in a straight line along a rough horizontal floor at constant speed using a thin strap. The strap is inclined at \(20°\) above the horizontal. The coefficient of friction between the suitcase and the floor is \(\frac{3}{4}\). The strap is modelled as a light inextensible string and the suitcase is modelled as a particle. Find the tension in the strap. [7]
Edexcel M1 2017 October Q2
11 marks Moderate -0.8
\includegraphics{figure_1} A metal girder \(AB\), of weight 1080 N and length 6 m, rests in equilibrium in a horizontal position on two supports, one at \(C\) and one at \(D\), where \(AC = 0.5\) m and \(BD = 2\) m, as shown in Figure 1. A boy of weight 400 N stands on the girder at \(B\) and the girder remains horizontal and in equilibrium. The boy is modelled as a particle and the girder is modelled as a uniform rod.
  1. Find
    1. the magnitude of the reaction on the girder at \(C\),
    2. the magnitude of the reaction on the girder at \(D\).
    [6]
The boy now stands at a point \(E\) on the girder, where \(AE = x\) metres, and the girder remains horizontal and in equilibrium. Given that the magnitude of the reaction on the girder at \(D\) is now 520 N greater than the magnitude of the reaction on the girder at \(C\),
  1. find the value of \(x\). [5]
Edexcel M1 2017 October Q3
6 marks Moderate -0.3
Two particles \(P\) and \(Q\) have masses \(4m\) and \(m\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(4u\). In the collision, the particles join together to form a single particle. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision. [6]
Edexcel M1 2017 October Q4
9 marks Moderate -0.3
Two forces \(\mathbf{F_1}\) and \(\mathbf{F_2}\) act on a particle. The force \(\mathbf{F_1}\) has magnitude 8 N and acts due east. The resultant of \(\mathbf{F_1}\) and \(\mathbf{F_2}\) is a force of magnitude 14 N acting in a direction whose bearing is \(120°\). Find
  1. the magnitude of \(\mathbf{F_2}\), [4]
  2. the direction of \(\mathbf{F_2}\), giving your answer as a bearing to the nearest degree. [5]
Edexcel M1 2017 October Q5
11 marks Moderate -0.8
A small ball is projected vertically upwards from a point \(O\) with speed 14.7 m s\(^{-1}\). The point \(O\) is 2.5 m above the ground. The motion of the ball is modelled as that of a particle moving freely under gravity. Find
  1. the maximum height above the ground reached by the ball, [4]
  2. the time taken for the ball to first reach a height of 1 m above the ground, [4]
  3. the speed of the ball at the instant before it strikes the ground for the first time. [3]
Edexcel M1 2017 October Q6
14 marks Moderate -0.3
An athlete goes for a run along a straight horizontal road. Starting from rest, she accelerates at 0.6 m s\(^{-2}\) up to a speed of \(V\) m s\(^{-1}\). She then maintains this constant speed of \(V\) m s\(^{-1}\) before finally decelerating at 0.2 m s\(^{-2}\) back to rest. She covers a total distance of 1500 m in 270 s.
  1. Sketch a speed-time graph to represent the athlete's run. [2]
  2. Show that she accelerates for \(\frac{5V}{3}\) seconds. [2]
  3. Show that \(V^2 - kV + 450 = 0\), where \(k\) is a constant to be found. [6]
  4. Find the value of \(V\), justifying your answer. [4]
Edexcel M1 2017 October Q7
17 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest on the surface of a fixed rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth light pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley. For the period before \(B\) hits the ground,
  1. write down an equation of motion for each particle. [4]
  2. Hence show that the acceleration of \(B\) is \(\frac{8}{35}g\). [5]
  3. Explain how you have used the fact that the string is inextensible in your calculation. [1]
When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.
  1. Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest. [7]
Edexcel M1 2022 October Q1
5 marks Moderate -0.8
A railway truck \(S\) of mass 20 tonnes is moving along a straight horizontal track when it collides with another railway truck \(T\) of mass 30 tonnes which is at rest. Immediately before the collision the speed of \(S\) is \(4\text{ ms}^{-1}\) As a result of the collision, the two railway trucks join together. Find
  1. the common speed of the railway trucks immediately after the collision, [2]
  2. the magnitude of the impulse exerted on \(S\) in the collision, stating the units of your answer. [3]
Edexcel M1 2022 October Q2
6 marks Moderate -0.3
\includegraphics{figure_1} A uniform rod \(AB\) has length \(2a\) and mass \(M\). The rod is held in equilibrium in a horizontal position by two vertical light strings which are attached to the rod at \(C\) and \(D\), where \(AC = \frac{2}{5}a\) and \(DB = \frac{3}{5}a\), as shown in Figure 1. A particle \(P\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.
  1. Find, in terms of \(M\), the largest possible mass of the particle \(P\) [3] Given that the mass of \(P\) is \(\frac{1}{2}M\)
  2. Find, in terms of \(M\) and \(g\), the tension in the string that is attached to the rod at \(C\). [3]
Edexcel M1 2022 October Q3
11 marks Standard +0.3
\includegraphics{figure_2} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) A particle \(P\) of mass 2 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.
  1. Show that when \(X = 14.7\) there is no frictional force acting on \(P\) [3] The coefficient of friction between \(P\) and the plane is 0.5
  2. Find the smallest possible value of \(X\). [8]
Edexcel M1 2022 October Q4
6 marks Moderate -0.8
\includegraphics{figure_3} Two children, Alan and Bhavana, are standing on the horizontal floor of a lift, as shown in Figure 3. The lift has mass 250 kg. The lift is raised vertically upwards with constant acceleration by a vertical cable which is attached to the top of the lift. The cable is modelled as being light and inextensible. While the lift is accelerating upwards, the tension in the cable is 3616 N. As the lift accelerates upwards, the floor of the lift exerts a force of magnitude 565 N on Alan and a force of magnitude 226 N on Bhavana. Air resistance is modelled as being negligible and Alan and Bhavana are modelled as particles.
  1. By considering the forces acting on the lift only, find the acceleration of the lift. [3]
  2. Find the mass of Alan. [3]
Edexcel M1 2022 October Q5
9 marks Moderate -0.3
A small ball is projected vertically upwards with speed \(29.4\text{ ms}^{-1}\) from a point \(A\) which is \(19.6\text{ m}\) above horizontal ground. The ball is modelled as a particle moving freely under gravity until it hits the ground. It is assumed that the ball does not rebound.
  1. Find the distance travelled by the ball while its speed is less than \(14.7\text{ ms}^{-1}\) [3]
  2. Find the time for which the ball is moving with a speed of more than \(29.4\text{ ms}^{-1}\) [3]
  3. Sketch a speed-time graph for the motion of the ball from the instant when it is projected from \(A\) to the instant when it hits the ground. Show clearly where your graph meets the axes. [3]
Edexcel M1 2022 October Q6
9 marks Moderate -0.3
[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors.] A particle \(A\) of mass 0.5 kg is at rest on a smooth horizontal plane. At time \(t = 0\), two forces, \(\mathbf{F}_1 = (-3\mathbf{i} + 2\mathbf{j})\) N and \(\mathbf{F}_2 = (p\mathbf{i} + q\mathbf{j})\) N, where \(p\) and \(q\) are constants, are applied to \(A\). Given that \(A\) moves in the direction of the vector \((\mathbf{i} - 2\mathbf{j})\),
  1. show that \(2p + q - 4 = 0\) [4] Given that \(p = 5\)
  2. Find the speed of \(A\) at time \(t = 4\) seconds. [5]
Edexcel M1 2022 October Q7
13 marks Standard +0.3
\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. Another particle \(Q\), also of mass \(m\), is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, \(P\) and \(Q\) all lie in the same vertical plane. The coefficient of friction between \(Q\) and the table is \(\mu\), where \(\mu < 1\) Particle \(Q\) is released from rest. The tension in the string before \(Q\) hits the pulley is \(kmg\), where \(k\) is a constant.
  1. Find \(k\) in terms of \(\mu\). [7] Given that \(Q\) is initially a distance \(d\) from the pulley,
  2. find, in terms of \(d\), \(g\) and \(\mu\), the time taken by \(Q\), after release, to reach the pulley. [4]
  3. Describe what would happen if \(\mu \geqslant 1\), giving a reason for your answer. [2]
Edexcel M1 2022 October Q8
16 marks Moderate -0.3
[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(A\) and \(B\), are moving with constant velocities. The velocity of \(A\) is \((3\mathbf{i} + 12\mathbf{j})\text{ kmh}^{-1}\) and the velocity of \(B\) is \((p\mathbf{i} + q\mathbf{j})\text{ kmh}^{-1}\)
  1. Find the speed of \(A\). [2] The ships are modelled as particles. At 12 noon, \(A\) is at the point with position vector \((-9\mathbf{i} + 6\mathbf{j})\) km and \(B\) is at the point with position vector \((16\mathbf{i} + 6\mathbf{j})\) km. At time \(t\) hours after 12 noon, $$\overrightarrow{AB} = [(25 - 12t)\mathbf{i} - 9t\mathbf{j}] \text{ km}$$
  2. Find the value of \(p\) and the value of \(q\). [7]
  3. Find the bearing of \(A\) from \(B\) when the ships are 15 km apart, giving your answer to the nearest degree. [7]
Edexcel M1 Specimen Q1
5 marks Moderate -0.8
A particle \(P\) is moving with constant velocity \((-3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 6\) s \(P\) is at the point with position vector \((-4\mathbf{i} - 7\mathbf{j})\) m. Find the distance of \(P\) from the origin at time \(t = 2\) s. [5]
Edexcel M1 Specimen Q2
7 marks Moderate -0.3
Particle \(P\) has mass \(m\) kg and particle \(Q\) has mass \(3m\) kg. The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4u\) m s\(^{-1}\) and \(Q\) has speed \(ku\) m s\(^{-1}\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.
  1. Find the value of \(k\). [4]
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\). [3]
Edexcel M1 Specimen Q3
7 marks Standard +0.3
\includegraphics{figure_1} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac{1}{2}\). The box is pushed by a force of magnitude 100 N which acts at an angle of 30° with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box. [7]
Edexcel M1 Specimen Q4
7 marks Standard +0.3
A beam \(AB\) has length 6 m and weight 200 N. The beam rests in a horizontal position on two supports at the points \(C\) and \(D\), where \(AC = 1\) m and \(DB = 1\) m. Two children, Sophie and Tom, each of weight 500 N, stand on the beam with Sophie standing twice as far from the end \(B\) as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at \(D\) is three times the magnitude of the reaction at \(C\). By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end \(B\). [7]
Edexcel M1 Specimen Q5
12 marks Standard +0.8
Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = 7\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).
  1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
  2. Find the value of \(T\). [8]
Edexcel M1 Specimen Q6
10 marks Moderate -0.8
A ball is projected vertically upwards with a speed of 14.7 m s\(^{-1}\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
  1. the greatest height, above the ground, reached by the ball, [4]
  2. the speed with which the ball first strikes the ground, [3]
  3. the total time from when the ball is projected to when it first strikes the ground. [3]
Edexcel M1 Specimen Q7
10 marks Standard +0.3
\includegraphics{figure_2} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{3}\). Given that the particle is on the point of sliding up the plane, find
  1. the magnitude of the normal reaction between the particle and the plane, [5]
  2. the value of \(P\). [5]