Questions — Edexcel (10514 questions)

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Edexcel M1 Q2
7 marks Moderate -0.3
A particle \(A\) of mass \(3m\) is moving along a straight line with constant speed \(u\) m s\(^{-1}\). It collides with a particle \(B\) of mass \(2m\) moving at the same speed but in the opposite direction. As a result of the collision, \(A\) is brought to rest.
  1. Show that, after the collision, \(B\) has changed its direction of motion and that its speed has been halved. [4 marks]
Given that the magnitude of the impulse exerted by \(A\) on \(B\) is \(9m\) Ns,
  1. find the value of \(u\). [3 marks]
Edexcel M1 Q3
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows two window cleaners, Alan and Baber, of mass 60 kg and 100 kg respectively standing on a platform \(PQ\) of length 3 metres and mass 20 kg. The platform is suspended by two vertical cables attached to the ends \(P\) and \(Q\). Alan is standing at the point \(A\), 1.25 metres from \(P\), Baber is standing at the point \(B\) and the tension in the cable at \(P\) is twice the tension in the cable at \(Q\). Modelling the platform as a uniform rod and Alan and Baber as particles,
  1. find the tension in the cable at \(P\), [2 marks]
  2. find the distance \(BP\). [5 marks]
  3. State how you have used the modelling assumptions that
    1. the platform is uniform,
    2. the platform is a rod.
    [2 marks]
Edexcel M1 Q4
9 marks Standard +0.3
A sports car is being driven along a straight test track. It passes the point \(O\) at time \(t = 0\) at which time it begins to decelerate uniformly. The car passes the points \(L\) and \(M\) at times \(t = 1\) and \(t = 4\) respectively. Given that \(OL\) is 54 m and \(LM\) is 90 m,
  1. find the rate of deceleration of the car. [5 marks]
The car subsequently comes to rest at \(N\).
  1. Find the distance \(MN\). [4 marks]
Edexcel M1 Q5
11 marks Standard +0.3
\includegraphics{figure_2} A particle \(P\), of mass 2 kg, lies on a rough plane inclined at an angle of 30° to the horizontal. A force \(H\), whose line of action is parallel to the line of greatest slope of the plane, is applied to the particle as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{\sqrt{3}}\). Given that the particle is on the point of moving up the plane,
  1. draw a diagram showing all the forces acting on the particle, [2 marks]
  2. show that the ratio of the magnitude of the frictional force to the magnitude of \(H\) is equal to \(1 : 2\) [7 marks]
The force \(H\) is now removed but \(P\) remains at rest.
  1. Use the principle of friction to explain how this is possible. [2 marks]
Edexcel M1 Q6
15 marks Standard +0.3
A car of mass 1.25 tonnes tows a caravan of mass 0.75 tonnes along a straight, level road. The total resistance to motion experienced by the car and the caravan is 1200 N. The car and caravan accelerate uniformly from rest to 25 m s\(^{-1}\) in 20 seconds.
  1. Calculate the driving force produced by the car's engine. [4 marks]
Given that the resistance to motion experienced by the car and by the caravan are in the same ratio as their masses,
  1. find these resistances and the tension in the towbar. [4 marks]
When the car and caravan are travelling at a steady speed of 25 m s\(^{-1}\), the towbar snaps. Assuming that the caravan experiences the same resistive force as before,
  1. calculate the distance travelled by the caravan before it comes to rest, [5 marks]
  2. give a reason why your answer to \((c)\) may be unrealistic. [2 marks]
Edexcel M1 Q7
17 marks Standard +0.3
Two ramblers, Alison and Bill, are out walking. At midday, Alison is at the fixed origin \(O\), and Bill is at the point with position vector \((-5\mathbf{i} + 12\mathbf{j})\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors. They are both walking with constant velocity – Alison at \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\), and Bill at a speed of \(2\sqrt{10}\) km h\(^{-1}\) in a direction parallel to the vector \((3\mathbf{i} + \mathbf{j})\).
  1. Find the distance between the two ramblers at midday. [2 marks]
  2. Show that the velocity of Bill is \((6\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). [3 marks]
  3. Show that, at time \(t\) hours after midday, the position vector of Bill relative to Alison is $$[(4t - 5)\mathbf{i} + (12 - 3t)\mathbf{j}] \text{ km.}$$ [5 marks]
  4. Show that the distance, \(d\) km, between the two ramblers is given by $$d^2 = 25t^2 - 112t + 169.$$ [2 marks]
  5. Using your answer to part \((d)\), find the length of time to the nearest minute for which the distance between the Alison and Bill is less than 11 km. [5 marks]
Edexcel M1 Q1
5 marks Moderate -0.8
A particle, \(P\), of mass 5 kg moves with speed 3 m s\(^{-1}\) along a smooth horizontal track. It strikes a particle \(Q\) of mass 2 kg which is at rest on the track. Immediately after the collision, \(P\) and \(Q\) move in the same direction with speeds \(v\) and 2v m s\(^{-1}\) respectively.
  1. Calculate the value of \(v\). [3 marks]
  2. Calculate the magnitude of the impulse received by \(Q\) on impact. [2 marks]
Edexcel M1 Q2
6 marks Moderate -0.8
A particle \(P\) moves with a constant velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) with respect to a fixed origin \(O\). It passes through the point \(A\) whose position vector is \((2\mathbf{i} + 11\mathbf{j})\) m at \(t = 0\).
  1. Find the angle in degrees that the velocity vector of \(P\) makes with the vector \(\mathbf{i}\). [2 marks]
  2. Calculate the distance of \(P\) from \(O\) when \(t = 2\). [4 marks]
Edexcel M1 Q3
7 marks Moderate -0.3
A car of mass 1250 kg is moving at constant speed up a hill, inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\). The driving force produced by the engine is 1800 N.
  1. Calculate the resistance to motion which the car experiences. [4 marks]
At the top of the hill, the road becomes horizontal.
  1. Find the initial acceleration of the car. [3 marks]
Edexcel M1 Q4
10 marks Moderate -0.3
A non-uniform plank \(AB\) of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\), the magnitude of the reaction at \(A\) is 35g N.
  1. Suggest a suitable model for
    1. the plank, [2 marks]
    2. the woman.
  2. Calculate the magnitude of the reaction at \(B\), giving your answer in terms of \(g\). [2 marks]
  3. Explain briefly, in the context of the problem, the term 'non-uniform'. [2 marks]
  4. Find the distance of the centre of mass of the plank from \(A\). [4 marks]
Edexcel M1 Q5
10 marks Standard +0.8
\includegraphics{figure_1} The points \(A\), \(O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x\) m from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of 3 m s\(^{-2}\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of 3 m s\(^{-1}\) in the direction of \(O\) with uniform acceleration of 4 m s\(^{-2}\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\). [10 marks]
Edexcel M1 Q6
11 marks Standard +0.3
A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle 10° to the horizontal. By modelling the sledge as a particle,
  1. show that the coefficient of friction, \(\mu\), between the sledge and the ground is 0.176 correct to 3 significant figures. [6 marks]
The sledge is placed on a steeper part of the slope which is inclined at an angle 30° to the horizontal. The value of \(\mu\) remains unchanged.
  1. Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill. [5 marks]
Edexcel M1 Q7
12 marks Standard +0.3
Whilst looking over the edge of a vertical cliff, 122.5 metres in height, Jim dislodges a stone. The stone falls freely from rest towards the sea below. Ignoring the effect of air resistance,
  1. calculate the time it would take for the stone to reach the sea, [3 marks]
  2. find the speed with which the stone would hit the water. [2 marks]
Two seconds after the stone begins to fall, Jim throws a tennis ball downwards at the stone. The tennis ball's initial speed is \(u\) m s\(^{-1}\) and it hits the stone before they both reach the water.
  1. Find the minimum value of \(u\). [5 marks]
  2. If you had taken air resistance into account in your calculations, what effect would this have had on your answer to part (c)? Explain your answer. [2 marks]
Edexcel M1 Q8
14 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with \(P\) and \(Q\) at the same level 1.5 metres above the ground and 2 metres below the pulley.
  1. Show that the initial acceleration of the system is \(\frac{g}{5}\) m s\(^{-2}\). [4 marks]
  2. Find the tension in the string. [2 marks]
  3. Find the speed with which \(P\) hits the ground. [3 marks]
When \(P\) hits the ground, it does not rebound.
  1. What is the closest that \(Q\) gets to the pulley. [5 marks]
Edexcel M1 Q1
7 marks Moderate -0.8
In a safety test, a car of mass 800 kg is driven directly at a wall at a constant speed of 15 m s\(^{-1}\). The constant force exerted by the wall on the car in bringing it to rest is 60 kN.
  1. Calculate the magnitude of the impulse exerted by the wall on the car. [2 marks]
  2. Find the time it takes for the car to come to rest. [2 marks]
  3. Show that the deceleration of the car is 75 m s\(^{-2}\). [3 marks]
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]