Questions M1 (2067 questions)

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OCR MEI M1 2008 January Q1
6 marks Easy -1.3
A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of 15 m s\(^{-1}\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.
  1. Sketch a velocity-time graph for the motion. [3]
  2. Calculate the distance travelled by the cyclist. [3]
OCR MEI M1 2008 January Q2
7 marks Moderate -0.8
The force acting on a particle of mass 1.5 kg is given by the vector \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) N.
  1. Give the acceleration of the particle as a vector. [2]
  2. Calculate the angle that the acceleration makes with the direction \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). [2]
  3. At a certain point of its motion, the particle has a velocity of \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\) m s\(^{-1}\). Calculate the displacement of the particle over the next two seconds. [3]
OCR MEI M1 2008 January Q3
8 marks Moderate -0.8
\includegraphics{figure_3} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at 40° to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N.
  1. The sphere has a mass of \(m\) kg. Calculate the value of \(m\). [2]
  2. Calculate the frictional force acting on the block. [3]
  3. Calculate the normal reaction of the plane on the block. [3]
OCR MEI M1 2008 January Q4
7 marks Easy -1.2
Force \(\mathbf{F}\) is \(\begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}\) N and force \(\mathbf{G}\) is \(\begin{pmatrix} -6 \\ 2 \\ 4 \end{pmatrix}\) N.
  1. Find the resultant of \(\mathbf{F}\) and \(\mathbf{G}\) and calculate its magnitude. [4]
  2. Forces \(\mathbf{F}\), \(2\mathbf{G}\) and \(\mathbf{H}\) act on a particle which is in equilibrium. Find \(\mathbf{H}\). [3]
OCR MEI M1 2008 January Q5
8 marks Standard +0.3
\includegraphics{figure_5} A toy car is moving along the straight line \(Ox\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at A, 3 m from O as shown in Fig. 5. The velocity of the car, \(v\) m s\(^{-1}\), is given by $$v = 2 + 12t - 3t^2.$$ Calculate the distance of the car from O when its acceleration is zero. [8]
OCR MEI M1 2008 January Q6
17 marks Moderate -0.3
A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1. \includegraphics{figure_6_1}
  1. When the man is descending with an acceleration 1.5 m s\(^{-2}\) downwards, how much time does it take for his speed to increase from 0.5 m s\(^{-1}\) downwards to 3.5 m s\(^{-1}\) downwards? How far does he descend in this time? [4]
The man has a mass of 80 kg. All resistances to motion may be neglected.
  1. Calculate the tension in the wire when the man is being lowered
    1. with an acceleration of 1.5 m s\(^{-2}\) downwards,
    2. with an acceleration of 1.5 m s\(^{-2}\) upwards. [5]
Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.
  1. For safety reasons, the tension in the wire should not exceed 2500 N. What is the maximum acceleration allowed when the man is being raised? [4]
At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N. \includegraphics{figure_6_2}
  1. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the tension in the wire. [4]
OCR MEI M1 2008 January Q7
19 marks Moderate -0.3
A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O, as shown in Fig. 7. Air resistance is negligible.
[diagram]
The initial horizontal component of the velocity of the firework is 21 m s\(^{-1}\).
  1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is 28 m s\(^{-1}\). [4]
  2. Show that the firework is \((28t - 4.9t^2)\) m above the ground \(t\) seconds after its projection. [1]
When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of 21 m s\(^{-1}\) and the other with a quarter of this speed.
  1. State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of \(t\) for the distance between the parts \(t\) seconds after the explosion. [3]
  2. Find the distance between these parts of the firework
    1. when they reach the ground, [2]
    2. when they are 10 m above the ground. [5]
  3. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac{4}{90}(120x - x^2)\), referred to the coordinate axes shown in Fig. 7. [4]
Edexcel M1 Q1
5 marks Moderate -0.3
Two particles, \(P\) and \(Q\), of mass 2 kg and 1.5 kg respectively are at rest on a smooth, horizontal surface. They are connected by a light, inelastic string which is initially slack. Particle \(P\) is projected away from \(Q\) with a speed of 7 ms\(^{-1}\).
  1. Find the common speed of the particles after the string becomes taut. [3 marks]
  2. Calculate the impulse in the string when it jerks tight. [2 marks]
Edexcel M1 Q2
6 marks Moderate -0.8
Particle \(A\) has velocity \((8\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) and particle \(B\) has velocity \((15\mathbf{i} - 8\mathbf{j})\) ms\(^{-1}\) where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors.
  1. Find the speed of \(B\). [2 marks]
  2. Find the velocity of \(B\) relative to \(A\). [2 marks]
  3. Find the acute angle between the relative velocity found in part (b) and the vector \(\mathbf{i}\), giving your answer in degrees correct to 1 decimal place. [2 marks]
Edexcel M1 Q3
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a uniform plank \(AB\) of length 8 m and mass 30 kg. It is supported in a horizontal position by two pivots, one situated at \(A\) and the other 2 m from \(B\). A man whose mass is 80 kg is standing on the plank 2 m from \(A\) when his dog steps onto the plank at \(B\). Given that the plank remains in equilibrium and that the magnitude of the forces exerted by each of the pivots on the plank are equal,
  1. calculate the magnitude of the force exerted on the plank by the pivot at \(A\), [5 marks]
  2. find the dog's mass. [3 marks]
If the dog was heavier and the plank was on the point of tilting,
  1. explain how the force exerted on the plank by each of the pivots would be changed. [2 marks]
Edexcel M1 Q4
10 marks Moderate -0.3
A cyclist and her bicycle have a combined mass of 78 kg. While riding on level ground and using her greatest driving force, she is able to accelerate uniformly from rest to 10 ms\(^{-1}\) in 15 seconds against constant resistive forces that total 60 N.
  1. Show that her maximum driving force is 112 N. [4 marks]
The cyclist begins to ascend a hill, inclined at an angle \(\alpha\) to the horizontal, riding with her maximum driving force and against the same resistive forces. In this case, she is able to maintain a steady speed.
  1. Find the angle \(\alpha\), giving your answer to the nearest degree. [4 marks]
  2. Comment on the assumption that the resistive force remains constant
    1. in the case when the cyclist is accelerating,
    2. in the case when she is maintaining a steady speed. [2 marks]
Edexcel M1 Q5
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a large block of mass 50 kg being pulled on rough horizontal ground by means of a rope attached to the block. The tension in the rope is 200 N and it makes an angle of 40° with the horizontal. Under these conditions, the block is on the point of moving. Modelling the block as a particle,
  1. show that the coefficient of friction between the block and the ground is 0.424 correct to 3 significant figures. [6 marks]
The angle with the horizontal at which the rope is being pulled is reduced to 30°. Ignoring air resistance and assuming that the tension in the rope and the coefficient of friction remain unchanged,
  1. find the acceleration of the block. [6 marks]
Edexcel M1 Q6
14 marks Moderate -0.3
Anila is practising catching tennis balls. She uses a mobile computer-controlled machine which fires tennis balls vertically upwards from a height of 2.5 metres above the ground. Once it has fired a ball, the machine is programmed to move position rapidly to allow Anila time to get into a suitable position to catch the ball. The machine fires a ball at 24 ms\(^{-1}\) vertically upwards and Anila catches the ball just before it touches the ground.
  1. Draw a speed-time graph for the motion of the ball from the time it is fired by the machine to the instant before Anila catches it. [3 marks]
  2. Find, to the nearest centimetre, the maximum height which the ball reaches above the ground. [4 marks]
  3. Calculate the speed at which the ball is travelling when Anila catches it. [4 marks]
  4. Calculate the length of time that the ball is in the air. [3 marks]
Edexcel M1 Q7
18 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a particle \(X\) of mass 3 kg on a smooth plane inclined at an angle 30° to the horizontal, and a particle \(Y\) of mass 2 kg on a smooth plane inclined at an angle 60° to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at \(C\) which is the highest point of the two planes. Initially, \(Y\) is at a point just below \(C\) touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, \(AC\) in the case of \(X\) and \(BC\) in the case of \(Y\), of their respective planes. \(A\) and \(B\) are the points where the planes meet the horizontal ground and \(AB = 4\) metres.
  1. Show that the initial acceleration of the system is given by \(\frac{g}{10}\left(2\sqrt{3} - 3\right)\) ms\(^{-2}\). [7 marks]
  2. By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical. [7 marks]
  3. Find, correct to 2 decimal places, the speed with which \(Y\) hits the ground. [4 marks]
Edexcel M1 Q1
7 marks Moderate -0.8
A constant force, \(\mathbf{F}\), acts on a particle, \(P\), of mass 5 kg causing its velocity to change from \((-2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) to \((4\mathbf{i} - 7\mathbf{j})\) m s\(^{-1}\) in 2 seconds.
  1. Find, in the form \(a\mathbf{i} + b\mathbf{j}\), the acceleration of \(P\). [2 marks]
  2. Show that the magnitude of \(\mathbf{F}\) is 25 N and find, to the nearest degree, the acute angle between the line of action of \(\mathbf{F}\) and the vector \(\mathbf{j}\). [5 marks]
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]