Questions — Edexcel M1 (663 questions)

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Edexcel M1 2016 January Q2
8 marks Moderate -0.3
Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) is moving due east and particle \(Q\) is moving due west. Particle \(P\) has mass \(2m\) and particle \(Q\) has mass \(3m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(4u\) and the speed of \(Q\) is \(u\). The magnitude of the impulse in the collision is \(\frac{33}{5}mu\).
  1. Find the speed and direction of motion of \(P\) immediately after the collision. [4]
  2. Find the speed and direction of motion of \(Q\) immediately after the collision. [4]
Edexcel M1 2016 January Q3
8 marks Standard +0.3
\includegraphics{figure_1} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at 30° to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac{1}{5}\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope. [8]
Edexcel M1 2016 January Q4
13 marks Moderate -0.3
A small stone is projected vertically upwards from the point \(O\) and moves freely under gravity. The point \(A\) is 3.6 m vertically above \(O\). When the stone first reaches \(A\), the stone is moving upwards with speed 11.2 m s\(^{-1}\). The stone is modelled as a particle.
  1. Find the maximum height above \(O\) reached by the stone. [4]
  2. Find the total time between the instant when the stone was projected from \(O\) and the instant when it returns to \(O\). [5]
  3. Sketch a velocity-time graph to represent the motion of the stone from the instant when it passes through \(A\) moving upwards to the instant when it returns to \(O\). Show, on the axes, the coordinates of the points where your graph meets the axes. [4]
Edexcel M1 2016 January Q5
10 marks Moderate -0.3
\includegraphics{figure_2} A non-uniform rod \(AB\) has length 4 m and weight 120 N. The centre of mass of the rod is at the point \(G\) where \(AG = 2.2\) m. The rod is suspended in a horizontal position by two vertical light inextensible strings, one at each end, as shown in Figure 2. A particle of weight 40 N is placed on the rod at the point \(P\), where \(AP = x\) metres. The rod remains horizontal and in equilibrium.
  1. Find, in terms of \(x\),
    1. the tension in the string at \(A\), [6]
    2. the tension in the string at \(B\).
    Either string will break if the tension in it exceeds 84 N.
  2. Find the range of possible values of \(x\). [4]
Edexcel M1 2016 January Q6
13 marks Moderate -0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm, the position vector of ship \(P\) is \((5\mathbf{i} - 3\mathbf{j})\) km and the position vector of ship \(Q\) is \((7\mathbf{i} + 5\mathbf{j})\) km.
  1. Find the distance between \(P\) and \(Q\) at 2 pm. [3]
Ship \(P\) is moving with constant velocity \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) is moving with constant velocity \((-3\mathbf{i} - 15\mathbf{j})\) km h\(^{-1}\).
  1. Find the position vector of \(P\) at time \(t\) hours after 2 pm. [2]
  2. Find the position vector of \(Q\) at time \(t\) hours after 2 pm. [1]
  3. Show that \(Q\) will meet \(P\) and find the time at which they meet. [5]
  4. Find the position vector of the point at which they meet. [2]
Edexcel M1 2016 January Q7
16 marks Standard +0.3
\includegraphics{figure_3} A particle \(P\) of mass 2 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) is at rest on the table, a distance \(d\) metres from the pulley. Particle \(Q\) is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between \(P\) and the table is \(\frac{1}{4}\). The system is released from rest and \(P\) slides along the table towards the pulley. Assuming that \(P\) has not reached the pulley and that \(Q\) remains on the inclined plane,
  1. write down an equation of motion for \(P\), [2]
  2. write down an equation of motion for \(Q\), [2]
    1. find the acceleration of \(P\),
    2. find the tension in the string. [5]
When \(P\) has moved a distance 0.5 m from its initial position, the string breaks. Given that \(P\) comes to rest just as it reaches the pulley,
  1. find the value of \(d\). [7]
Edexcel M1 2016 June Q1
7 marks Moderate -0.3
A car is moving along a straight horizontal road with constant acceleration \(a\) m s\(^{-2}\) (\(a > 0\)). At time \(t = 0\) the car passes the point \(P\) moving with speed \(u\) m s\(^{-1}\). In the next 4 s, the car travels 76 m and then in the following 6 s it travels a further 219 m. Find
  1. the value of \(u\),
  2. the value of \(a\).
[7]
Edexcel M1 2016 June Q2
6 marks Moderate -0.3
Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(km\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2u\). 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 \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision. [2]
Edexcel M1 2016 June Q3
10 marks Standard +0.3
A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.
  1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
  2. find the value of \(R\). [8]
Edexcel M1 2016 June Q4
10 marks Moderate -0.3
\includegraphics{figure_1} A diving board \(AB\) consists of a wooden plank of length 4 m and mass 30 kg. The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(AC = 0.6\) m, as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.
  1. Find
    1. the magnitude of the force acting on the plank at \(A\),
    2. the magnitude of the force acting on the plank at \(C\).
    [6] The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N.
  2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\). [3]
  3. Explain how you have used the fact that the diver is modelled as a particle. [1]
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]