Questions — Edexcel M1 (663 questions)

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Edexcel M1 Q2
8 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows an aerial view of a revolving door consisting of 4 panels, each of width 1.2 m and set at 90° intervals, which are free to rotate about a fixed central column, \(O\). The revolving door is situated outside a lecture theatre and four students are trying to push the door. Two of the students are pushing panels \(OA\) and \(OD\) clockwise (as viewed from above) with horizontal forces of 70 N and 90 N respectively, whilst the other two are pushing panels \(OB\) and \(OC\) anti-clockwise with horizontal forces of 80 N and 60 N respectively.
  1. Calculate the total moment about \(O\) when the four students are pushing the panels at their outer edge, 1.2 m from \(O\). [3 marks]
The student at \(C\) moves her hand 0.2 m closer to \(O\) and the student at \(D\) moves his hand \(x\) m closer to \(O\). Given that the students all push in the same directions and with the same forces as in part (a), and that the door is in equilibrium,
  1. Find the value of \(x\). [5 marks]
Edexcel M1 Q3
10 marks Moderate -0.3
During a cricket match, the batsman hits the ball and begins running with constant velocity \(4\mathbf{i}\) m s\(^{-1}\) to try and score a run. When the batsman is at the fixed origin \(O\), the ball is thrown by a member of the opposing team with velocity \((^-8\mathbf{i} + 24\mathbf{j})\) m s\(^{-1}\) from the point with position vector \((30\mathbf{i} - 60\mathbf{j})\) m, where \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors. At time \(t\) seconds after the ball is thrown, the position vectors of the batsman and the ball are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively. In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.
  1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3 marks]
  2. Show that the ball hits the batsman and find the position vector of the batsman when this occurs. [5 marks]
  3. Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model. [2 marks]
Edexcel M1 Q4
10 marks Standard +0.3
In a physics experiment, two balls \(A\) and \(B\), of mass \(4m\) and \(3m\) respectively, are travelling towards one another on a straight horizontal track. Both balls are travelling with speed 2 m s\(^{-1}\) immediately before they collide. As a result of the impact, \(A\) is brought to rest and the direction of motion of \(B\) is reversed. Modelling the track as smooth and the balls as particles,
  1. find the speed of \(B\) immediately after the collision. [3 marks]
A student notices that after the collision, \(B\) comes to rest 0.2 m from \(A\).
  1. Show that the coefficient of friction between \(B\) and the track is 0.113, correct to 3 decimal places. [7 marks]
Edexcel M1 Q5
12 marks Standard +0.3
A cyclist is riding up a hill inclined at an angle of 5° to the horizontal. She produces a driving force of 50 N and experiences resistive forces which total 20 N. Given that the combined mass of the cyclist and her bicycle is 70 kg,
  1. find, correct to 2 decimal places, the magnitude of the deceleration of the cyclist. [4 marks]
When the cyclist reaches the top of the hill, her speed is 3 m s\(^{-1}\). She subsequently accelerates uniformly so that in the fifth second after she has reached the top of the hill, she travels 12 m.
  1. Find her speed at the end of the fifth second. [8 marks]
Edexcel M1 Q6
14 marks Challenging +1.2
\includegraphics{figure_2} Figure 2 shows a particle \(A\) of mass 5 kg, lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects \(A\) to a particle \(B\) of mass 2 kg. The string passes over a smooth pulley which is fixed to the edge of the table and \(B\) hangs vertically 0.4 m below the pulley. When the system is released from rest,
  1. show that the magnitude of the force exerted on the pulley is \(\frac{10\sqrt{5}}{7}\) g N. [7 marks]
  2. find the speed with which \(A\) hits the pulley. [3 marks]
When \(A\) hits the pulley, the string breaks and \(B\) subsequently falls freely under gravity.
  1. Find the speed with which \(B\) hits the ground. [4 marks]
Edexcel M1 Q7
14 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a block of mass 25 kg held in equilibrium on a plane inclined at an angle of 35° to the horizontal by means of a string which is at an angle of 15° to the line of greatest slope of the plane. In an initial model of the situation, the plane is assumed to be smooth. Giving your answers correct to 3 significant figures,
  1. show that the tension in the string is 145 N. [3 marks]
  2. find the magnitude of the reaction between the plane and the block. [4 marks]
In a more refined model, the plane is assumed to be rough. Given that the tension in the string can be increased to 200 N before the block begins to move up the slope,
  1. find, correct to 3 significant figures, the magnitude of the frictional force and state the direction in which it acts. [4 marks]
  2. Without performing any further calculations, state whether the reaction calculated in part (b) will increase, decrease or remain the same in the refined model. Give a reason for your answer. [3 marks]
Edexcel M1 Q1
7 marks Standard +0.3
Two particles \(P\) and \(Q\), of mass \(m\) and \(km\) respectively, are travelling in opposite directions on a straight horizontal path with speeds \(3u\) and \(2u\) respectively. \(P\) and \(Q\) collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.
  1. Find the value of \(k\). [4 marks]
  2. Write down an expression in terms of \(m\) and \(u\) for the magnitude of the impulse which \(P\) exerts on \(Q\) during the collision. [3 marks]
Edexcel M1 Q2
9 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows a plank \(AB\) of mass 40 kg and length 6 m, which rests on supports at each of its ends. The plank is wedge-shaped, being thicker at end \(A\) than at end \(B\). A woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\).
  1. Suggest suitable modelling assumptions which can be made about
    1. the plank,
    2. the woman. [3 marks]
    Given that the reactions at each support are of equal magnitude,
  2. find the magnitude of the reaction on the support at \(A\), [2 marks]
  3. calculate the distance of the centre of mass of the plank from \(A\). [4 marks]
Edexcel M1 Q3
9 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a cable car \(C\) of mass 1 tonne which has broken down. The cable car is suspended in equilibrium by two perpendicular cables \(AC\) and \(BC\) which are attached to fixed points \(A\) and \(B\), at the same horizontal level on either side of a valley. The cable \(AC\) is inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac{3}{4}\).
  1. Show that the tension in the cable \(AC\) is 5880 N and find the tension in the cable \(BC\). [7 marks] A gust of wind then blows along the valley.
  2. Explain the effect that this will have on the tension in the two cables. [2 marks]
Edexcel M1 Q4
10 marks Moderate -0.3
Andrew hits a tennis ball vertically upwards towards his sister Barbara who is leaning out of a window 7.5 m above the ground to try to catch it. When the ball leaves Andrew's racket, it is 1.9 m above the ground and travelling at \(21 \text{ m s}^{-1}\). Barbara fails to catch the ball on its way up but succeeds as the ball comes back down. Modelling the ball as a particle and assuming that air resistance can be neglected,
  1. find the maximum height above the ground which the ball reaches. [4 marks]
  2. find how long Barbara has to wait from the moment that the ball first passes her until she catches it. [6 marks]
Edexcel M1 Q5
11 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively, connected by a light inextensible string which passes over a smooth fixed pulley. When the system is released from rest with both particles 0.5 m above the ground, particle \(A\) moves vertically upwards with acceleration \(\frac{1}{4} g \text{ m s}^{-2}\).
  1. Write down, with a brief justification, the magnitude and direction of the acceleration of \(B\). [2 marks]
  2. Find the value of \(k\). [6 marks] Given that \(A\) does not hit the pulley,
  3. calculate, correct to 3 significant figures, the speed with which \(B\) hits the ground. [3 marks]
Edexcel M1 Q6
12 marks Moderate -0.3
Two trains \(A\) and \(B\) leave the same station, \(O\), at 10 a.m. and travel along straight horizontal tracks. \(A\) travels with constant speed \(80 \text{ km h}^{-1}\) due east and \(B\) travels with constant speed \(52 \text{ km h}^{-1}\) in the direction \((5\mathbf{i} + 12\mathbf{j})\) where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively.
  1. Show that the velocity of \(B\) is \((20\mathbf{i} + 48\mathbf{j}) \text{ km h}^{-1}\). [3 marks]
  2. Find the displacement vector of \(B\) from \(A\) at 10:15 a.m. [3 marks] Given that the trains are 23 km apart \(t\) minutes after 10 a.m.
  3. find the value of \(t\) correct to the nearest whole number. [6 marks]
Edexcel M1 Q7
17 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows two golf balls \(P\) and \(Q\) being held at the top of planes inclined at \(30°\) and \(60°\) to the vertical respectively. Both planes slope down to a common hole at \(H\), which is 3 m vertically below \(P\) and \(Q\). \(P\) is released from rest and travels down the line of greatest slope of the plane it is on which is assumed to be smooth.
  1. Find the acceleration of \(P\) down the slope. [3 marks]
  2. Show that the time taken for \(P\) to reach the hole is 0.904 seconds, correct to 3 significant figures. [5 marks] \(Q\) travels down the line of greatest slope of the plane it is on which is rough. The coefficient of friction between \(Q\) and the plane is \(\mu\). Given that the acceleration of \(Q\) down the slope is \(3 \text{ m s}^{-2}\),
  3. find, correct to 3 significant figures, the value of \(\mu\). [5 marks] In order for the two balls to arrive at the hole at the same time, \(Q\) must be released \(t\) seconds before \(P\).
  4. Find the value of \(t\) correct to 2 decimal places. [4 marks]