Questions M1 (2067 questions)

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Edexcel M1 2003 November Q6
12 marks Standard +0.3
\includegraphics{figure_2} A non-uniform rod \(AB\) has length 5 m and weight 200 N. The rod rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1.5\) m and \(DB = 1\) m, as shown in Fig. 2. The centre of mass of \(AB\) is \(x\) metres from \(A\). A particle of weight \(W\) newtons is placed on the rod at \(A\). The rod remains in equilibrium and the magnitude of the reaction of \(C\) on the rod is 160 N.
  1. Show that \(50x - W = 100\). [5]
The particle is now removed from \(A\) and placed on the rod at \(B\). The rod remains in equilibrium and the reaction of \(C\) on the rod now has magnitude 50 N.
  1. Obtain another equation connecting \(W\) and \(x\). [3]
  2. Calculate the value of \(x\) and the value of \(W\). [4]
Edexcel M1 2003 November Q7
16 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\), of mass \(m\) kg and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac{1}{8}g\).
  1. Write down an equation of motion for \(B\). [2]
  2. Find the tension in the string. [2]
  3. Prove that \(m = \frac{16}{35}\). [4]
  4. State where in the calculations you have used the information that \(P\) is a light smooth pulley. [1]
On release, \(B\) is at a height of one metre above the ground and \(AP = 1.4\) m. The particle \(B\) strikes the ground and does not rebound.
  1. Calculate the speed of \(B\) as it reaches the ground. [2]
  2. Show that \(A\) comes to rest as it reaches \(P\). [5]
END
Edexcel M1 2004 November Q1
5 marks Moderate -0.8
A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P\), 300 m from \(S\), its speed is \(30 \text{ m s}^{-1}\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).
  1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\). [2]
  2. Find the time taken by the car to travel from \(P\) to \(S\). [3]
Edexcel M1 2004 November Q2
7 marks Moderate -0.3
\includegraphics{figure_1} The particles have mass 3 kg and \(m\) kg, where \(m < 3\). They are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are held in position with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The particles are then released from rest. The initial acceleration of each particle has magnitude \(\frac{1}{2}g\). Find
  1. the tension in the string immediately after the particles are released, [3]
  2. the value of \(m\). [4]
Edexcel M1 2004 November Q3
8 marks Moderate -0.3
\includegraphics{figure_2} A plank of wood \(AB\) has mass 10 kg and length 4 m. It rests in a horizontal position on two smooth supports. One support is at the end \(A\). The other is at the point \(C\), 0.4 m from \(B\), as shown in Figure 2. A girl of mass 30 kg stands at \(B\) with the plank in equilibrium. By modelling the plank as a uniform rod and the girl as a particle,
  1. find the reaction on the plank at \(A\). [4]
The girl gets off the plank. A boulder of mass \(m\) kg is placed on the plank at \(A\) and a man of mass 80 kg stands on the plank at \(B\). The plank remains in equilibrium and is on the point of tilting about \(C\). By modelling the plank again as a uniform rod, and the man and the boulder as particles,
  1. find the value of \(m\). [4]
Edexcel M1 2004 November Q4
8 marks Moderate -0.8
A tent peg is driven into soft ground by a blow from a hammer. The tent peg has mass 0.2 kg and the hammer has mass 3 kg. The hammer strikes the peg vertically. Immediately before the impact, the speed of the hammer is \(16 \text{ m s}^{-1}\). It is assumed that, immediately after the impact, the hammer and the peg move together vertically downwards.
  1. Find the common speed of the peg and the hammer immediately after the impact. [3]
Until the peg and hammer come to rest, the resistance exerted by the ground is assumed to be constant and of magnitude \(R\) newtons. The hammer and peg are brought to rest 0.05 s after the impact.
  1. Find, to 3 significant figures, the value of \(R\). [5]
Edexcel M1 2004 November Q5
10 marks Moderate -0.8
A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \((-\mathbf{i} + 2\mathbf{j}) \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \((2\mathbf{i} - 3\mathbf{j}) \text{ m s}^{-1}\).
  1. Find, to the nearest degree, the angle between the vector \(\mathbf{j}\) and the direction of motion of \(P\) when \(t = 0\). [3]
At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). Find
  1. an expression for \(\mathbf{v}\) in terms of \(t\), in the form \(a\mathbf{i} + b\mathbf{j}\), [2]
  2. the speed of \(P\) when \(t = 3\), [3]
  3. the time when \(P\) is moving parallel to \(\mathbf{i}\). [2]
Edexcel M1 2004 November Q6
11 marks Moderate -0.8
Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. At time \(t = 0\), they are side by side, passing a point \(O\) on the road. Car \(A\) travels at a constant speed of \(30 \text{ m s}^{-1}\). Car \(B\) passes \(O\) with a speed of \(20 \text{ m s}^{-1}\), and has constant acceleration of \(4 \text{ m s}^{-2}\). Find
  1. the speed of \(B\) when it has travelled 78 m from \(O\), [2]
  2. the distance from \(O\) of \(A\) when \(B\) is 78 m from \(O\), [4]
  3. the time when \(B\) overtakes \(A\). [5]
Edexcel M1 2004 November Q7
12 marks Moderate -0.3
\includegraphics{figure_3} A sledge has mass 30 kg. The sledge is pulled in a straight line along horizontal ground by means of a rope. The rope makes an angle \(20°\) with the horizontal, as shown in Figure 3. The coefficient of friction between the sledge and the ground is 0.2. The sledge is modelled as a particle and the rope as a light inextensible string. The tension in the rope is 150 N. Find, to 3 significant figures,
  1. the normal reaction of the ground on the sledge, [3]
  2. the acceleration of the sledge. [3]
When the sledge is moving at \(12 \text{ m s}^{-1}\), the rope is released from the sledge.
  1. Find, to 3 significant figures, the distance travelled by the sledge from the moment when the rope is released to the moment when the sledge comes to rest. [6]
Edexcel M1 2004 November Q8
14 marks Moderate -0.3
\includegraphics{figure_4} A heavy package is held in equilibrium on a slope by a rope. The package is attached to one end of the rope, the other end being held by a man standing at the top of the slope. The package is modelled as a particle of mass 20 kg. The slope is modelled as a rough plane inclined at \(60°\) to the horizontal and the rope as a light inextensible string. The string is assumed to be parallel to a line of greatest slope of the plane, as shown in Figure 4. At the contact between the package and the slope, the coefficient of friction is 0.4.
  1. Find the minimum tension in the rope for the package to stay in equilibrium on the slope. [8]
The man now pulls the package up the slope. Given that the package moves at constant speed,
  1. find the tension in the rope. [4]
  2. State how you have used, in your answer to part (b), the fact that the package moves
    1. up the slope,
    2. at constant speed.
    [2]
Edexcel M1 Specimen Q1
7 marks Moderate -0.8
\includegraphics{figure_1} A tennis ball \(P\) is attached to one end of a light inextensible string, the other end of the string being attached to a the top of a fixed vertical pole. A girl applies a horizontal force of magnitude 50 N to \(P\), and \(P\) is in equilibrium under gravity with the string making an angle of \(40°\) with the pole, as shown in Fig. 1. By modelling the ball as a particle find, to 3 significant figures,
  1. the tension in the string, [3]
  2. the weight of \(P\). [4]
Edexcel M1 Specimen Q2
7 marks Moderate -0.8
A car starts from rest at a point \(O\) and moves in a straight line. The car moves with constant acceleration \(4 \text{ m s}^{-2}\) until it passes the point \(A\) when it is moving with speed \(10 \text{ m s}^{-1}\). It then moves with constant acceleration \(3 \text{ m s}^{-2}\) for 6 s until it reaches the point \(B\). Find
  1. the speed of the car at \(B\), [2]
  2. the distance \(OB\). [5]
Edexcel M1 Specimen Q3
9 marks Moderate -0.3
\includegraphics{figure_2} A non-uniform plank of wood \(AB\) has length 6 m and mass 90 kg. The plank is smoothly supported at its two ends \(A\) and \(B\), with \(A\) and \(B\) at the same horizontal level. A woman of mass 60 kg stands on the plank at the point \(C\), where \(AC = 2\) m, as shown in Fig. 2. The plank is in equilibrium and the magnitudes of the reactions on the plank at \(A\) and \(B\) are equal. The plank is modelled as a non-uniform rod and the woman as a particle. Find
  1. the magnitude of the reaction on the plank at \(B\), [2]
  2. the distance of the centre of mass of the plank from \(A\). [5]
  3. State briefly how you have used the modelling assumption that
    1. the plank is a rod,
    2. the woman is a particle.
    [2]
Edexcel M1 Specimen Q4
12 marks Moderate -0.8
A train \(T_1\) moves from rest at Station \(A\) with constant acceleration \(2 \text{ m s}^{-2}\) until it reaches a speed of \(36 \text{ m s}^{-1}\). In maintains this constant speed for 90 s before the brakes are applied, which produce constant retardation \(3 \text{ m s}^{-2}\). The train \(T_1\) comes to rest at station \(B\).
  1. Sketch a speed-time graph to illustrate the journey of \(T_1\) from \(A\) to \(B\). [3]
  2. Show that the distance between \(A\) and \(B\) is 3780 m. [5]
\includegraphics{figure_3} A second train \(T_2\) takes 150 s to move form rest at \(A\) to rest at \(B\). Figure 3 shows the speed-time graph illustrating this journey.
  1. Explain briefly one way in which \(T_1\)'s journey differs from \(T_2\)'s journey. [1]
  2. Find the greatest speed, in m s\(^{-1}\), attained by \(T_2\) during its journey. [3]
Edexcel M1 Specimen Q5
12 marks Moderate -0.3
A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \text{ m s}^{-1}\), the speed of \(B\) is \(4 \text{ m s}^{-1}\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.
  1. Find the speed and direction of \(C\) after the collision. [4]
  2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision. [3]
  3. State a modelling assumption which you have made about the trucks in your solution [1]
Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.
  1. Find the value of \(d\). [4]
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]