Questions — Edexcel M1 (663 questions)

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Edexcel M1 Specimen Q6
13 marks Standard +0.3
\includegraphics{figure_4} A particle of mass \(m\) rests on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a small light smooth pulley \(P\) fixed at the top of the plane. The other end of the string is attached to a particle \(B\) of mass \(3m\), and \(B\) hangs freely below \(P\), as shown in Fig. 4. The particles are released from rest with the string taut. The particle \(B\) moves down with acceleration of magnitude \(\frac{1}{3}g\). Find
  1. the tension in the string, [4]
  2. the coefficient of friction between \(A\) and the plane. [9]
Edexcel M1 Specimen Q7
15 marks Moderate -0.3
Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \text{ m s}^{-1}\) due east, and the velocity of \(B\) is \((10\mathbf{i} + 10\mathbf{j}) \text{ m s}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300\mathbf{j}\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively.
  1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3]
  2. Hence write down an expression for \(\overrightarrow{AB}\) in terms of \(t\). [1]
  3. Find the time when the bearing of \(B\) from \(A\) is \(045°\). [5]
  4. Find the time when the cars are again 300 m apart. [6]
Edexcel M1 Q1
6 marks Easy -1.2
A bee flies in a straight line from \(A\) to \(B\), where \(\overrightarrow{AB} = (3\mathbf{i} - 12\mathbf{j})\) m, in 5 seconds at a constant speed. Find
  1. the straight-line distance \(AB\), [2 marks]
  2. the speed of the bee, [2 marks]
  3. the velocity vector of the bee. [2 marks]
Edexcel M1 Q2
7 marks Moderate -0.3
A small ball \(B\), of mass 0.8 kg, is suspended from a horizontal ceiling by two light inextensible strings. \(B\) is in equilibrium under gravity with both strings inclined at 30° to the horizontal, as shown. \includegraphics{figure_2}
  1. Find the tension, in N, in either string. [3 marks]
  2. Calculate the magnitude of the least horizontal force that must be applied to \(B\) in this position to cause one string to become slack. [4 marks]
Edexcel M1 Q3
7 marks Moderate -0.3
A particle \(P\) moves in a straight line through a fixed point \(O\) with constant acceleration \(a\) ms\(^{-2}\). 3 seconds after passing through \(O\), \(P\) is 6 m from \(O\). After a further 6 seconds, \(P\) has travelled a further 33 m in the same direction. Calculate
  1. the value of \(a\), [5 marks]
  2. the speed with which \(P\) passed through \(O\). [2 marks]
Edexcel M1 Q4
7 marks Moderate -0.8
A force of magnitude \(F\) N is applied to a block of mass \(M\) kg which is initially at rest on a horizontal plane. The block starts to move with acceleration 3 ms\(^{-2}\). Modelling the block as a particle, \includegraphics{figure_4}
  1. if the plane is smooth, find an expression for \(F\) in terms of \(M\). [2 marks]
If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),
  1. express \(F\) in terms of \(M\), \(\mu\) and \(g\). [2 marks]
  2. Calculate the value of \(\mu\) if \(F = \frac{1}{2}Mg\). [3 marks]
Edexcel M1 Q5
12 marks Standard +0.3
Two metal weights \(A\) and \(B\), of masses 2.4 kg and 1.8 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley so that the string hangs vertically on each side. The system is released from rest with the string taut.
  1. Calculate the acceleration of each weight and the tension in the string. [6 marks]
\(A\) is now replaced by a different weight of mass \(m\) kg, where \(m < 1.8\), and the system is again released from rest. The magnitude of the acceleration has half of its previous value.
  1. Calculate the value of \(m\). [6 marks]
Edexcel M1 Q6
12 marks Moderate -0.8
The diagram shows the speed-time graph for a particle during a period of \(9T\) seconds. \includegraphics{figure_6}
  1. If \(T = 5\), find
    1. the acceleration for each section of the motion, [2 marks]
    2. the total distance travelled by the particle. [2 marks]
  2. Sketch, for this motion,
    1. an acceleration-time graph, [2 marks]
    2. a displacement-time graph. [2 marks]
  3. Calculate the value of \(T\) for which the distance travelled over the \(9T\) seconds is 3.708 km. [4 marks]
Edexcel M1 Q7
12 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of masses 60 grams and 90 grams respectively, are at rest on a smooth horizontal table. \(A\) is projected towards \(B\) with speed 4 ms\(^{-1}\) and the particles collide. After the collision, \(A\) and \(B\) move in the same direction as each other, with speeds \(u\) ms\(^{-1}\) and \(6u\) ms\(^{-1}\) respectively. Calculate
  1. the value of \(u\), [4 marks]
  2. the magnitude of the impulse exerted by \(A\) on \(B\), stating the units of your answer. [3 marks]
\(A\) and \(B\) are now replaced in their original positions and projected towards each other with speeds 2 ms\(^{-1}\) and 8 ms\(^{-1}\) respectively. They collide again, after which \(A\) moves with speed 7 ms\(^{-1}\), its direction of motion being reversed.
  1. Find the speed of \(B\) after this collision and state whether its direction of motion has been reversed. [5 marks]
Edexcel M1 Q8
12 marks Standard +0.3
In a theatre, three lights \(A\), \(B\) and \(C\) are suspended from a horizontal beam \(XY\) of length 4.5 m. \(A\) and \(C\) are each of mass 8 kg and \(B\) is of mass 6 kg. The beam \(XY\) is held in place by vertical ropes \(PX\) and \(QY\), as shown. \includegraphics{figure_8} In a simple mathematical model of this situation, \(XY\) is modelled as a light rod.
  1. Calculate the tension in each of \(PX\) and \(QY\). [6 marks]
In a refined model, \(XY\) is modelled as a uniform rod of mass \(m\) kg.
  1. If the tension in \(PX\) is 1.5 times that in \(QY\), calculate the value of \(m\). [6 marks]
Edexcel M1 Q1
5 marks Moderate -0.3
A plank of wood \(AB\), of mass 8 kg and length 6 m, rests on a support at \(P\), where \(AP = 4\) m. When particles of mass 1 kg and \(k\) kg are suspended from \(A\) and \(B\) respectively, the plank rests horizontally in equilibrium. Modelling the plank as a uniform rod, find
  1. the value of \(k\), [3 marks]
  2. the magnitude of the force exerted by the support on the plank at \(P\). [2 marks]
Edexcel M1 Q2
6 marks Moderate -0.8
Forces of magnitude 4 N and 6 N act in directions which make an angle of \(40°\) with each other, as shown. Calculate
  1. the magnitude of the resultant of the two forces, [4 marks]
  2. the angle, in degrees, between the resultant and the 4 N force. [2 marks]
\includegraphics{figure_1}
Edexcel M1 Q3
9 marks Moderate -0.8
A stone is dropped from rest at a height of 7 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards with half the speed with which it hit the ground. Calculate
  1. the time taken for the stone to fall to the ground, [2 marks]
  2. the speed with which the stone hits the ground, [2 marks]
  3. the height to which the stone rises before it comes to instantaneous rest. [3 marks]
State two modelling assumptions that you have made. [2 marks]
Edexcel M1 Q4
12 marks Moderate -0.3
A boy starts at the corner \(O\) of a rectangular playing field and runs across the field with constant velocity vector \((\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions of two perpendicular sides of the field. After 40 seconds, at the point \(P\) in the field, he changes speed and direction so that his new velocity vector is \((2.4\mathbf{i} - 1.8\mathbf{j})\) ms\(^{-1}\) and maintains this velocity until he reaches the point \(Q\), where \(PQ = 75\) m. Calculate
  1. the distance \(OP\), [3 marks]
  2. the time taken to travel from \(P\) to \(Q\), [2 marks]
  3. the position vector of \(Q\) relative to \(O\). [3 marks]
Another boy travels directly from \(O\) to \(Q\) with constant velocity \((a\mathbf{i} + b\mathbf{j})\) ms\(^{-1}\), leaving \(O\) and reaching \(Q\) at the same times as the first boy.
  1. Find the values of the constants \(a\) and \(b\). [4 marks]
Edexcel M1 Q5
12 marks Standard +0.3
Two railway trucks \(A\) and \(B\), of masses 10 000 kg and 7 000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84 000 Ns. Immediately after the collision, the trucks move together with speed 10 ms\(^{-1}\). Modelling the trucks as particles,
  1. find the speed of each truck immediately before the collision. [6 marks]
When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15. Assuming that no other resisting forces act on the trucks, calculate
  1. the magnitude of the resisting force on the trucks, [3 marks]
  2. the time taken after the collision for the trucks to come to rest. [3 marks]
Edexcel M1 Q6
15 marks Standard +0.3
A small package \(P\), of mass 1 kg, is initially at rest on the rough horizontal top surface of a wooden packing case which is 1.5 m long and 1 m high and stands on a horizontal floor. The coefficient of friction between \(P\) and the case is 0.2. \(P\) is attached by a light inextensible string, which passes over a smooth fixed pulley, to a weight \(Q\) of mass \(M\) kg which rests against the smooth vertical side of the case. The system is released from rest with \(P\) 0.75 m from the pulley and \(Q\) 0.5 m from the pulley. \(P\) and \(Q\) start to move with acceleration 0.4 ms\(^{-2}\). Calculate
  1. the tension in the string, in N, [3 marks]
  2. the value of \(M\), [3 marks]
  3. the time taken for \(Q\) to hit the floor. [3 marks]
Given that \(Q\) does not rebound from the floor,
  1. calculate the distance of \(P\) from the pulley when it comes to rest. [6 marks]
\includegraphics{figure_2}
Edexcel M1 Q7
16 marks Standard +0.8
A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 ms\(^{-2}\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m.
  1. Sketch a graph of velocity against time for the total period of the car's motion. [3 marks]
  2. Find the car's average speed for the whole journey. [6 marks]
In reality the car's acceleration \(a\) ms\(^{-2}\) in the first 10 seconds is not constant, but increases from 0 to 4 ms\(^{-2}\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k(mt - t^2)\) for \(0 \leq t \leq 10\).
  1. Calculate the values of the constants \(k\) and \(m\). [5 marks]
  2. Find the acceleration of the car when \(t = 2\) according to this model. [2 marks]
Edexcel M1 Q1
7 marks Moderate -0.8
A particle \(P\), of mass \(2.5\) kg, initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \((\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due East and due North respectively. Find
  1. the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\), \hfill [2 marks]
  2. the magnitude and direction of the velocity of \(P\) when \(t = 7\), \hfill [3 marks]
  3. the magnitude, in N, of the force acting on \(P\). \hfill [2 marks]
Edexcel M1 Q2
7 marks Standard +0.3
An iron bar \(AB\), of length \(4\) m, is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(PB = 0.85\) m. The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(AG = 1.4\) m, and the wire is modelled as a light inextensible string. Given that the tension in the wire is \(12\) N, calculate
  1. the weight of the bar, \hfill [4 marks]
  2. the magnitude of the reaction on the bar at \(A\). \hfill [2 marks]
  3. State briefly how you have used the given modelling assumption about the bar. \hfill [1 mark]
Edexcel M1 Q3
10 marks Standard +0.3
\includegraphics{figure_3} A small packet, of mass \(1.2\) kg, is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the packet and the plane is \(\frac{1}{8}\). When a force of magnitude \(8.4\) N, acting parallel to the plane, is applied to the packet as shown, the packet is just on the point of moving up the plane. Modelling the packet as a particle,
  1. show that \(7(\cos \alpha + 8 \sin \alpha) = 40\). \hfill [6 marks]
Given that the solution of this equation is \(\alpha = 38°\),
  1. find the acceleration with which the packet moves down the plane when it is released from rest with no external force applied. \hfill [4 marks]
Edexcel M1 Q4
11 marks Moderate -0.8
A car moves in a straight line from \(P\) to \(Q\), a distance of \(420\) m, with constant acceleration. At \(P\) the speed of the car is \(8\) ms\(^{-1}\). At \(Q\) the speed of the car is \(20\) ms\(^{-1}\). Find
  1. the time taken to travel from \(P\) to \(Q\), \hfill [2 marks]
  2. the acceleration of the car, \hfill [2 marks]
  3. the time taken for the car to travel \(240\) m from \(P\). \hfill [4 marks]
Given that the mass of the car is \(1200\) kg and the tractive force of the car is \(900\) N,
  1. find the magnitude of the resistance to the car's motion. \hfill [3 marks]
Edexcel M1 Q5
11 marks Standard +0.3
Two smooth spheres \(X\) and \(Y\), of masses \(x\) kg and \(y\) kg respectively, are free to move in a smooth straight groove in a horizontal table. \(X\) is projected with speed \(6\) ms\(^{-1}\) towards \(Y\), which is stationary. After the collision \(X\) moves with speed \(2\) ms\(^{-1}\) and \(Y\) moves with speed \(3\) ms\(^{-1}\).
  1. Calculate the two possible values of the ratio \(x : y\). \hfill [6 marks]
  2. State a modelling assumption that you have made concerning \(X\) and \(Y\). \hfill [1 mark]
\(Y\) now strikes a vertical barrier and rebounds along the groove with speed \(k\) ms\(^{-1}\), colliding again with \(X\) which is still moving at \(2\) ms\(^{-1}\). Given that in this impact \(Y\) is brought to rest and the direction of motion of \(X\) is reversed,
  1. show that \(k > 1.5\). \hfill [4 marks]
Edexcel M1 Q6
14 marks Standard +0.3
Two particles \(P\) and \(Q\), of masses \(3\) kg and \(2\) kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30°\), \(60°\) and \(90°\), as shown. \(P\) and \(Q\) are connected by a light string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge. \includegraphics{figure_6} The system is released from rest with \(P\) \(0.8\) m from the pulley and \(Q\) \(1\) m from the bottom of the wedge, and \(Q\) starts to move down. Calculate
  1. the acceleration of either particle, \hfill [5 marks]
  2. the tension in the string, \hfill [2 marks]
  3. the speed with which \(P\) reaches the pulley. \hfill [3 marks]
Two modelling assumptions have been made about the string and the pulley.
  1. State these two assumptions and briefly describe how you have used each one in your solution. \hfill [4 marks]
Edexcel M1 Q7
15 marks Standard +0.8
Two stones are projected simultaneously from a point \(O\) on horizontal ground. Stone \(A\) is thrown vertically upwards with speed \(98\) ms\(^{-1}\). Stone \(B\) is projected along the smooth ground in a straight line at \(24.5\) ms\(^{-1}\).
  1. Find the distances of the two stones from \(O\) after \(t\) seconds, where \(0 \leq t \leq 20\). \hfill [3 marks]
  2. Show that the distance \(d\) m between the two stones after \(t\) seconds is given by $$d^2 = 24.01(t^2 - 40t^2 + 425t^2).$$ \hfill [6 marks]
  3. Hence find the range of values of \(t\) for which the distance between the stones is decreasing. \hfill [6 marks]
Edexcel M1 Q1
6 marks Moderate -0.8
A tennis ball, moving horizontally, hits a wall at \(25 \text{ ms}^{-1}\) and rebounds along the same straight line at \(15 \text{ ms}^{-1}\). The impulse exerted by the wall on the ball has magnitude \(12\) Ns.
  1. Calculate the mass of the ball. [4 marks]
  2. State any modelling assumptions that you have made. [2 marks]