Questions M1 (2067 questions)

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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]
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]
OCR MEI M1 Q1
19 marks Moderate -0.3
The displacement, \(x\) m, from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36t + 3t^2 - 2t^3,$$ where \(t\) is the time in seconds and \(-4 \leqslant t \leqslant 6\).
  1. Write down the displacement of the particle when \(t = 0\). [1]
  2. Find an expression in terms of \(t\) for the velocity, \(v\) ms\(^{-1}\), of the particle. [2]
  3. Find an expression in terms of \(t\) for the acceleration of the particle. [2]
  4. Find the maximum value of \(v\) in the interval \(-4 \leqslant t \leqslant 6\). [3]
  5. Show that \(v = 0\) only when \(t = -2\) and when \(t = 3\). Find the values of \(x\) at these times. [5]
  6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [3]
  7. Determine how many times the particle passes through O in the interval \(-4 \leqslant t \leqslant 6\). [3]
OCR MEI M1 Q2
8 marks Moderate -0.8
A particle moves along the \(x\)-axis with velocity, \(v\) ms\(^{-1}\), at time \(t\) given by $$v = 24t - 6t^2.$$ The positive direction is in the sense of \(x\) increasing.
  1. Find an expression for the acceleration of the particle at time \(t\). [2]
  2. Find the times, \(t_1\) and \(t_2\), at which the particle has zero speed. [2]
  3. Find the distance travelled between the times \(t_1\) and \(t_2\). [4]
OCR MEI M1 Q3
8 marks Moderate -0.3
Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6\) ms\(^{-1}\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}
  1. Show that Nina's speed, \(v\) ms\(^{-1}\), is given by \begin{align} v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4,
    v &= 8 \quad \text{for } t > 4. \end{align} [3]
  2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t < 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5\frac{1}{4}\). [4]
  3. Show that Nina catches up with Marie when \(t = 5\frac{1}{4}\). [1]
OCR MEI M1 Q4
7 marks Moderate -0.3
Two cars, P and Q, are being crashed as part of a film 'stunt'. At the start
  • P is travelling directly towards Q with a speed of \(8\) ms\(^{-1}\),
  • Q is instantaneously at rest and has an acceleration of \(4\) ms\(^{-2}\) directly towards P.
P continues with the same velocity and Q continues with the same acceleration. The cars collide \(T\) seconds after the start.
  1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start. [2]
At the start, P is 90 m from Q.
  1. Show that \(T^2 + 4T - 45 = 0\) and hence find \(T\). [5]
OCR MEI M1 Q5
8 marks Moderate -0.8
The velocity, \(v\) ms\(^{-1}\), of a particle moving along a straight line is given by $$v = 3t^2 - 12t + 14,$$ where \(t\) is the time in seconds.
  1. Find an expression for the acceleration of the particle at time \(t\). [2]
  2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\). [4]
  3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times. [2]
OCR MEI M1 Q1
8 marks Standard +0.3
A rock of mass 8 kg is acted on by just the two forces \(-80\)k N and \((-\mathbf{i} + 16\mathbf{j} + 72\)k\()\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane and k is a unit vector vertically upward.
  1. Show that the acceleration of the rock is \(\left(\frac{1}{8}\mathbf{i} + 2\mathbf{j}\right)\) k\()\) ms\(^{-2}\). [2]
The rock passes through the origin of position vectors, O, with velocity \((\mathbf{i} - 4\mathbf{j} + 3\)k\()\) m s\(^{-1}\) and 4 seconds later passes through the point A.
  1. Find the position vector of A. [3]
  2. Find the distance OA. [1]
  3. Find the angle that OA makes with the horizontal. [2]
OCR MEI M1 Q2
8 marks Moderate -0.3
Fig. 4 shows the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the directions of the cartesian axes \(Ox\) and \(Oy\), respectively. O is the origin of the axes and of position vectors. \includegraphics{figure_1} The position vector of a particle is given by \(\mathbf{r} = 3t\mathbf{i} + (18t^2 - 11)\mathbf{j}\) for \(t \geq 0\), where \(t\) is time.
  1. Show that the path of the particle cuts the \(x\)-axis just once. [2]
  2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the \(\mathbf{j}\) direction. [3]
  3. Find the cartesian equation of the path of the particle, simplifying your answer. [3]