Questions M1 (1912 questions)

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CAIE M1 2010 November Q5
5 Particles \(P\) and \(Q\) are projected vertically upwards, from different points on horizontal ground, with velocities of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(Q\) is projected 0.4 s later than \(P\). Find
  1. the time for which \(P\) 's height above the ground is greater than 15 m ,
  2. the velocities of \(P\) and \(Q\) at the instant when the particles are at the same height.
CAIE M1 2010 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-3_579_1518_258_315} The diagram shows the velocity-time graph for a particle \(P\) which travels on a straight line \(A B\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\). The graph consists of five straight line segments. The particle starts from rest when \(t = 0\) at a point \(X\) on the line between \(A\) and \(B\) and moves towards \(A\). The particle comes to rest at \(A\) when \(t = 2.5\).
  1. Given that the distance \(X A\) is 4 m , find the greatest speed reached by \(P\) during this stage of the motion. In the second stage, \(P\) starts from rest at \(A\) when \(t = 2.5\) and moves towards \(B\). The distance \(A B\) is 48 m . The particle takes 12 s to travel from \(A\) to \(B\) and comes to rest at \(B\). For the first 2 s of this stage \(P\) accelerates at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  2. the value of \(V\),
  3. the value of \(t\) at which \(P\) starts to decelerate during this stage,
  4. the deceleration of \(P\) immediately before it reaches \(B\).
    \(7 \quad\) A particle \(P\) travels in a straight line. It passes through the point \(O\) of the line with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\), where \(t\) is in seconds. \(P\) 's velocity after leaving \(O\) is given by $$\left( 0.002 t ^ { 3 } - 0.12 t ^ { 2 } + 1.8 t + 5 \right) \mathrm { m } \mathrm {~s} ^ { - 1 }$$ The velocity of \(P\) is increasing when \(0 < t < T _ { 1 }\) and when \(t > T _ { 2 }\), and the velocity of \(P\) is decreasing when \(T _ { 1 } < t < T _ { 2 }\).
  5. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) and the distance \(O P\) when \(t = T _ { 2 }\).
  6. Find the velocity of \(P\) when \(t = T _ { 2 }\) and sketch the velocity-time graph for the motion of \(P\).
CAIE M1 2010 November Q1
1 A particle \(P\) is released from rest at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Find the speed of \(P\)
  1. when it has travelled 0.9 m ,
  2. 0.8 s after it is released.
CAIE M1 2010 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_301_1267_616_440} The diagram shows the vertical cross-section \(A B C\) of a fixed surface. \(A B\) is a curve and \(B C\) is a horizontal straight line. The part of the surface containing \(A B\) is smooth and the part containing \(B C\) is rough. \(A\) is at a height of 1.8 m above \(B C\). A particle of mass 0.5 kg is released from rest at \(A\) and travels along the surface to \(C\).
  1. Find the speed of the particle at \(B\).
  2. Given that the particle reaches \(C\) with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the work done against the resistance to motion as the particle moves from \(B\) to \(C\).
CAIE M1 2010 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_368_853_1503_644} A small smooth pulley is fixed at the highest point \(A\) of a cross-section \(A B C\) of a triangular prism. Angle \(A B C = 90 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The prism is fixed with the face containing \(B C\) in contact with a horizontal surface. Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string, which passes over the pulley. The particles are in equilibrium with \(P\) hanging vertically below the pulley and \(Q\) in contact with \(A C\). The resultant force exerted on the pulley by the string is \(3 \sqrt { } 3 \mathrm {~N}\) (see diagram).
  1. Show that the tension in the string is 3 N . The coefficient of friction between \(Q\) and the prism is 0.75 .
  2. Given that \(Q\) is in limiting equilibrium and on the point of moving upwards, find its mass.
CAIE M1 2010 November Q4
4 A particle starts from rest at a point \(X\) and moves in a straight line until, 60 seconds later, it reaches a point \(Y\). At time \(t \mathrm {~s}\) after leaving \(X\), the acceleration of the particle is $$\begin{array} { r c c } 0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 0 < t < 4
0 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 4 < t < 54
- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 54 < t < 60 \end{array}$$
  1. Find the velocity of the particle when \(t = 4\) and when \(t = 60\), and sketch the velocity-time graph.
  2. Find the distance \(X Y\).
CAIE M1 2010 November Q5
5 A force of magnitude \(F \mathrm {~N}\) acts in a horizontal plane and has components 27.5 N and - 24 N in the \(x\)-direction and the \(y\)-direction respectively. The force acts at an angle of \(\alpha ^ { \circ }\) below the \(x\)-axis.
  1. Find the values of \(F\) and \(\alpha\). A second force, of magnitude 87.6 N , acts in the same plane at \(90 ^ { \circ }\) anticlockwise from the force of magnitude \(F \mathrm {~N}\). The resultant of the two forces has magnitude \(R \mathrm {~N}\) and makes an angle of \(\theta ^ { \circ }\) with the positive \(x\)-axis.
  2. Find the values of \(R\) and \(\theta\).
CAIE M1 2010 November Q6
6 A particle travels along a straight line. It starts from rest at a point \(A\) on the line and comes to rest again, 10 seconds later, at another point \(B\) on the line. The velocity \(t\) seconds after leaving \(A\) is $$\begin{array} { r l l } 0.72 t ^ { 2 } - 0.096 t ^ { 3 } & \text { for } & 0 \leqslant t \leqslant 5
2.4 t - 0.24 t ^ { 2 } & \text { for } & 5 \leqslant t \leqslant 10 \end{array}$$
  1. Show that there is no instantaneous change in the acceleration of the particle when \(t = 5\).
  2. Find the distance \(A B\).
CAIE M1 2010 November Q7
7 A car of mass 1250 kg travels along a horizontal straight road. The power of the car's engine is constant and equal to 24 kW and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). The car passes through the point \(A\) on the road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the value of \(R\). The car continues with increasing speed, passing through the point \(B\) on the road with speed \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car subsequently passes through the point \(C\).
  2. Find the acceleration of the car at \(B\), giving the answer in \(\mathrm { m } \mathrm { s } ^ { - 2 }\) correct to 3 decimal places.
  3. Show that, while the car's speed is increasing, it cannot reach \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Explain why the speed of the car is approximately constant between \(B\) and \(C\).
  5. State a value of the approximately constant speed, and the maximum possible error in this value at any point between \(B\) and \(C\). The work done by the car's engine during the motion from \(B\) to \(C\) is 1200 kJ .
  6. By assuming the speed of the car is constant from \(B\) to \(C\), find, in either order,
    (a) the approximate time taken for the car to travel from \(B\) to \(C\),
    (b) an approximation for the distance \(B C\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2011 November Q1
1 One end of a light inextensible string is attached to a block. The string is used to pull the block along a horizontal surface with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string makes an angle of \(20 ^ { \circ }\) with the horizontal and the tension in the string is 25 N . Find the work done by the tension in a period of 8 seconds.
CAIE M1 2011 November Q2
2 Particles \(A\) of mass 0.65 kg and \(B\) of mass 0.35 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(B\) is held at rest with the string taut and both of its straight parts vertical. The system is released from rest and the particles move vertically. Find the tension in the string and the magnitude of the resultant force exerted on the pulley by the string.
CAIE M1 2011 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_476_714_744_719} Three coplanar forces of magnitudes \(15 \mathrm {~N} , 12 \mathrm {~N}\) and 12 N act at a point \(A\) in directions as shown in the diagram.
  1. Find the component of the resultant of the three forces
    (a) in the direction of \(A B\),
    (b) perpendicular to \(A B\).
  2. Hence find the magnitude and direction of the resultant of the three forces.
CAIE M1 2011 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_449_1273_1829_438}
\(A , B\) and \(C\) are three points on a line of greatest slope of a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. \(A\) is higher than \(B\) and \(B\) is higher than \(C\), and the distances \(A B\) and \(B C\) are 1.76 m and 2.16 m respectively. A particle slides down the plane with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The speed of the particle at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The particle takes 0.8 s to travel from \(A\) to \(B\) and takes 1.4 s to travel from \(A\) to \(C\). Find
  1. the values of \(u\) and \(a\),
  2. the value of \(\theta\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_188_510_260_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_196_570_255_1187} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A block of mass 2 kg is at rest on a horizontal floor. The coefficient of friction between the block and the floor is \(\mu\). A force of magnitude 12 N acts on the block at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). When the applied force acts downwards as in Fig. 1 the block remains at rest.
CAIE M1 2011 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-3_218_1280_1146_431}
\(A B\) and \(B C\) are straight roads inclined at \(5 ^ { \circ }\) to the horizontal and \(1 ^ { \circ }\) to the horizontal respectively. \(A\) and \(C\) are at the same horizontal level and \(B\) is 45 m above the level of \(A\) and \(C\) (see diagram, which is not to scale). A car of mass 1200 kg travels from \(A\) to \(C\) passing through \(B\).
  1. For the motion from \(A\) to \(B\), the speed of the car is constant and the work done against the resistance to motion is 360 kJ . Find the work done by the car's engine from \(A\) to \(B\). The resistance to motion is constant throughout the whole journey.
  2. For the motion from \(B\) to \(C\) the work done by the driving force is 1660 kJ . Given that the speed of the car at \(B\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that its speed at \(C\) is \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  3. The car's driving force immediately after leaving \(B\) is 1.5 times the driving force immediately before reaching \(C\). Find, correct to 2 significant figures, the ratio of the power developed by the car's engine immediately after leaving \(B\) to the power developed immediately before reaching \(C\).
CAIE M1 2011 November Q7
7 A particle \(P\) starts from a point \(O\) and moves along a straight line. \(P\) 's velocity \(t\) s after leaving \(O\) is \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$v = 0.16 t ^ { \frac { 3 } { 2 } } - 0.016 t ^ { 2 } .$$ \(P\) comes to rest instantaneously at the point \(A\).
  1. Verify that the value of \(t\) when \(P\) is at \(A\) is 100 .
  2. Find the maximum speed of \(P\) in the interval \(0 < t < 100\).
  3. Find the distance \(O A\).
  4. Find the value of \(t\) when \(P\) passes through \(O\) on returning from \(A\).
CAIE M1 2011 November Q1
1 A racing cyclist, whose mass with his cycle is 75 kg , works at a rate of 720 W while moving on a straight horizontal road. The resistance to the cyclist's motion is constant and equal to \(R \mathrm {~N}\).
  1. Given that the cyclist is accelerating at \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when his speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(R\).
  2. Given that the cyclist's acceleration is positive, show that his speed is less than \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2011 November Q2
2 A block of mass 6 kg is sliding down a line of greatest slope of a plane inclined at \(8 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 .
  1. Find the deceleration of the block.
  2. Given that the initial speed of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find how far the block travels.
CAIE M1 2011 November Q3
3 A particle \(P\) moves in a straight line. It starts from a point \(O\) on the line with velocity \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(0.8 t ^ { - 0.75 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the displacement of \(P\) from \(O\) when \(t = 16\).
CAIE M1 2011 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-2_608_723_1247_712} A particle \(P\) has weight 10 N and is in limiting equilibrium on a rough horizontal table. The forces shown in the diagram represent the weight of \(P\), an applied force of magnitude 4 N acting on \(P\) in a direction at \(30 ^ { \circ }\) above the horizontal, and the contact force exerted on \(P\) by the table (the resultant of the frictional and normal components) of magnitude \(C \mathrm {~N}\).
  1. Find the value of \(C\).
  2. Find the coefficient of friction between \(P\) and the table.
CAIE M1 2011 November Q5
5 Particles \(A\) and \(B\), of masses 0.9 kg and 0.6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical and with the particles at the same height above the horizontal floor. In the subsequent motion, \(B\) does not reach the pulley.
  1. Find the acceleration of \(A\) and the tension in the string during the motion before \(A\) hits the floor. After \(A\) hits the floor, \(B\) continues to move vertically upwards for a further 0.3 s .
  2. Find the height of the particles above the floor at the instant that they started to move.
CAIE M1 2011 November Q6
6 A lorry of mass 16000 kg climbs a straight hill \(A B C D\) which makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). For the motion from \(A\) to \(B\), the work done by the driving force of the lorry is 1200 kJ and the resistance to motion is constant and equal to 1240 N . The speed of the lorry is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) and \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\).
  1. Find the distance \(A B\). For the motion from \(B\) to \(D\) the gain in potential energy of the lorry is 2400 kJ .
  2. Find the distance \(B D\). For the motion from \(B\) to \(D\) the driving force of the lorry is constant and equal to 7200 N . From \(B\) to \(C\) the resistance to motion is constant and equal to 1240 N and from \(C\) to \(D\) the resistance to motion is constant and equal to 1860 N .
  3. Given that the speed of the lorry at \(D\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance \(B C\).
CAIE M1 2011 November Q7
7 A tractor travels in a straight line from a point \(A\) to a point \(B\). The velocity of the tractor is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\).

  1. \includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-4_668_1091_397_568} The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for
    (a) the distance \(A B\),
    (b) the acceleration of the tractor for \(0 < t < 400\) and for \(400 < t < 800\).
  2. The actual velocity of the tractor is given by \(v = 0.04 t - 0.00005 t ^ { 2 }\) for \(0 \leqslant t \leqslant 800\).
    (a) Find the values of \(t\) for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part (i). For the interval \(0 \leqslant t \leqslant 400\), the approximate velocity of the tractor in part (i) is denoted by \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (b) Express \(v _ { 1 }\) in terms of \(t\) and hence show that \(v _ { 1 } - v = 0.00005 ( t - 200 ) ^ { 2 } - 1\).
    (c) Deduce that \(- 1 \leqslant v _ { 1 } - v \leqslant 1\).
CAIE M1 2011 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-2_675_1380_255_379} A woman walks in a straight line. The woman's velocity \(t\) seconds after passing through a fixed point \(A\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The graph of \(v\) against \(t\) consists of 4 straight line segments (see diagram). The woman is at the point \(B\) when \(t = 60\). Find
  1. the woman's acceleration for \(0 < t < 30\) and for \(30 < t < 40\),
  2. the distance \(A B\),
  3. the total distance walked by the woman.
CAIE M1 2011 November Q2
6 marks
2
\includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-2_652_493_1457_826} Coplanar forces of magnitudes \(58 \mathrm {~N} , 31 \mathrm {~N}\) and 26 N act at a point in the directions shown in the diagram. Given that \(\tan \alpha = \frac { 5 } { 12 }\), find the magnitude and direction of the resultant of the three forces.
[0pt] [6]
CAIE M1 2011 November Q3
3 Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical, and with both particles at a height of 2 m above horizontal ground. \(P\) moves vertically downwards and does not rebound when it hits the ground. At the instant that \(P\) hits the ground, \(Q\) is at the point \(X\), from where it continues to move vertically upwards without reaching the pulley. Given that \(P\) has mass 0.9 kg and that the tension in the string is 7.2 N while \(P\) is moving, find the total distance travelled by \(Q\) from the instant it first reaches \(X\) until it returns to \(X\).