Questions M1 (1912 questions)

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CAIE M1 2013 November Q2
Standard +0.3
2
\includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).
  1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
  2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.
CAIE M1 2013 November Q3
Moderate -0.3
3 A cyclist exerts a constant driving force of magnitude \(F \mathrm {~N}\) while moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 36 } { 325 }\). A constant resistance to motion of 32 N acts on the cyclist. The total weight of the cyclist and his bicycle is 780 N . The cyclist's acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the value of \(F\). The cyclist's speed is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill.
  2. Find how far up the hill the cyclist travels before coming to rest.
CAIE M1 2013 November Q4
Standard +0.3
4 Particles \(P\) and \(Q\) are moving in a straight line on a rough horizontal plane. The frictional forces are the only horizontal forces acting on the particles.
  1. Find the deceleration of each of the particles given that the coefficient of friction between \(P\) and the plane is 0.2 , and between \(Q\) and the plane is 0.25 . At a certain instant, \(P\) passes through the point \(A\) and \(Q\) passes through the point \(B\). The distance \(A B\) is 5 m . The velocities of \(P\) and \(Q\) at \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively, both in the direction \(A B\).
  2. Find the speeds of \(P\) and \(Q\) immediately before they collide.
CAIE M1 2013 November Q5
Standard +0.3
5 A lorry of mass 15000 kg climbs from the bottom to the top of a straight hill, of length 1440 m , at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N .
  1. Find the work done by the driving force. On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point \(P\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and is now equal to 1600 N . The work done by the lorry's engine from the top of the hill to the point \(P\) is 5030 kJ .
  2. Find the distance from the top of the hill to the point \(P\).
CAIE M1 2013 November Q6
Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.
  1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
  2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor.
    \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
  3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
  4. Find the distance \(O A\).
CAIE M1 2013 November Q1
Standard +0.3
1
\includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_346_583_255_781} A small block of weight 5.1 N rests on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 8 } { 17 }\). The block is held in equilibrium by means of a light inextensible string. The string makes an angle \(\beta\) above the line of greatest slope on which the block rests, where \(\sin \beta = \frac { 7 } { 25 }\) (see diagram). Find the tension in the string.
CAIE M1 2013 November Q2
Standard +0.3
2 A box of mass 25 kg is pulled in a straight line along a horizontal floor. The box starts from rest at a point \(A\) and has a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches a point \(B\). The distance \(A B\) is 15 m . The pulling force has magnitude 220 N and acts at an angle of \(\alpha ^ { \circ }\) above the horizontal. The work done against the resistance to motion acting on the box, as the box moves from \(A\) to \(B\), is 3000 J . Find the value of \(\alpha\).
CAIE M1 2013 November Q3
Moderate -0.3
3 The resistance to motion acting on a runner of mass 70 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the runner's speed and \(k\) is a constant. The greatest power the runner can exert is 100 W . The runner's greatest steady speed on horizontal ground is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 6.25\).
  2. Find the greatest steady speed of the runner while running uphill on a straight path inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).
CAIE M1 2013 November Q4
8 marks Standard +0.8
4
\includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_365_493_1749_826} A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = 2.4\). A small block of mass 0.6 kg is held at rest on the plane by a horizontal force of magnitude \(P \mathrm {~N}\). This force acts in a vertical plane through a line of greatest slope (see diagram). The coefficient of friction between the block and the plane is 0.4 . The block is on the point of slipping down the plane. By resolving forces parallel to and perpendicular to the inclined plane, or otherwise, find the value of \(P\).
[0pt] [8]
CAIE M1 2013 November Q5
Standard +0.3
5 A particle \(P\) moves in a straight line. \(P\) starts from rest at \(O\) and travels to \(A\) where it comes to rest, taking 50 seconds. The speed of \(P\) at time \(t\) seconds after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v\) is defined as follows. $$\begin{aligned} \text { For } 0 \leqslant t \leqslant 5 , & v = t - 0.1 t ^ { 2 } \\ \text { for } 5 \leqslant t \leqslant 45 , & v \text { is constant } \\ \text { for } 45 \leqslant t \leqslant 50 , & v = 9 t - 0.1 t ^ { 2 } - 200 \end{aligned}$$
  1. Find the distance travelled by \(P\) in the first 5 seconds.
  2. Find the total distance from \(O\) to \(A\), and deduce the average speed of \(P\) for the whole journey from \(O\) to \(A\).
CAIE M1 2013 November Q6
Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-3_526_519_902_813} Particles \(A\) of mass 0.4 kg and \(B\) of mass 1.6 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 1.2 m above the floor (see diagram). \(A\) is released and both particles start to move.
  1. Find the work done on \(B\) by the tension in the string, as \(B\) moves to the floor. When particle \(B\) reaches the floor it remains at rest. Particle \(A\) continues to move upwards.
  2. Find the greatest height above the floor reached by particle \(A\).
CAIE M1 2013 November Q7
Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-4_492_1365_255_392} An elevator is pulled vertically upwards by a cable. The velocity-time graph for the motion is shown above. Find
  1. the distance travelled by the elevator,
  2. the acceleration during the first stage and the deceleration during the third stage. The mass of the elevator is 800 kg and there is a box of mass 100 kg on the floor of the elevator.
  3. Find the tension in the cable in each of the three stages of the motion.
  4. Find the greatest and least values of the magnitude of the force exerted on the box by the floor of the elevator.
CAIE M1 2013 November Q1
Moderate -0.3
1 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).
  1. Show that the acceleration of the particle is \(- 6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that the particle's initial speed is \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance that the particle travels up the plane.
CAIE M1 2013 November Q2
Standard +0.3
2
\includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_569_519_676_813} Particle \(A\) of mass 0.2 kg and particle \(B\) of mass 0.6 kg are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(B\) is held at rest at a height of 1.6 m above the floor. \(A\) hangs freely at a height of \(h \mathrm {~m}\) above the floor. Both straight parts of the string are vertical (see diagram). \(B\) is released and both particles start to move. When \(B\) reaches the floor it remains at rest, but \(A\) continues to move vertically upwards until it reaches a height of 3 m above the floor. Find the speed of \(B\) immediately before it hits the floor, and hence find the value of \(h\).
CAIE M1 2013 November Q3
Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_307_857_1695_644} A particle \(P\) of mass 1.05 kg is attached to one end of each of two light inextensible strings, of lengths 2.6 m and 1.25 m . The other ends of the strings are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. \(P\) hangs in equilibrium at a point 1 m below the level of \(A\) and \(B\) (see diagram). Find the tensions in the strings.
CAIE M1 2013 November Q4
Standard +0.3
4 A box of mass 30 kg is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\), acted on by a force of magnitude 40 N . The force acts upwards and parallel to a line of greatest slope of the plane. The box is on the point of slipping up the plane.
  1. Find the coefficient of friction between the box and the plane. The force of magnitude 40 N is removed.
  2. Determine, giving a reason, whether or not the box remains in equilibrium.
CAIE M1 2013 November Q5
Standard +0.3
5 A car travels in a straight line from \(A\) to \(B\), a distance of 12 km , taking 552 seconds. The car starts from rest at \(A\) and accelerates for \(T _ { 1 } \mathrm {~s}\) at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then continues to move at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T _ { 2 } \mathrm {~s}\). It then decelerates for \(T _ { 3 } \mathrm {~s}\) at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).
  1. Sketch the velocity-time graph for the motion and express \(T _ { 1 }\) and \(T _ { 3 }\) in terms of \(V\).
  2. Express the total distance travelled in terms of \(V\) and show that \(13 V ^ { 2 } - 3312 V + 72000 = 0\). Hence find the value of \(V\).
CAIE M1 2013 November Q6
Standard +0.3
6 A lorry of mass 12500 kg travels along a road from \(A\) to \(C\) passing through a point \(B\). The resistance to motion of the lorry is 4800 N for the whole journey from \(A\) to \(C\).
  1. The section \(A B\) of the road is straight and horizontal. On this section of the road the power of the lorry's engine is constant and equal to 144 kW . The speed of the lorry at \(A\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration at \(B\) is \(0.096 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the acceleration of the lorry at \(A\) and show that its speed at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. The section \(B C\) of the road has length 500 m , is straight and inclined upwards towards \(C\). On this section of the road the lorry's driving force is constant and equal to 5800 N . The speed of the lorry at \(C\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the height of \(C\) above the level of \(A B\).
CAIE M1 2013 November Q7
Standard +0.8
7 A vehicle starts from rest at a point \(O\) and moves in a straight line. Its speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after leaving \(O\) is defined as follows. $$\begin{aligned} \text { For } 0 & \leqslant t \leqslant 60 , \quad v = k _ { 1 } t - 0.005 t ^ { 2 } \\ \text { for } t \geqslant 60 , \quad v & = \frac { k _ { 2 } } { \sqrt { } t } \end{aligned}$$ The distance travelled by the vehicle during the first 60 s is 540 m .
  1. Find the value of the constant \(k _ { 1 }\) and show that \(k _ { 2 } = 12 \sqrt { } ( 60 )\).
  2. Find an expression in terms of \(t\) for the total distance travelled when \(t \geqslant 60\).
  3. Find the speed of the vehicle when it has travelled a total distance of 1260 m .
CAIE M1 2014 November Q1
Moderate -0.3
1 A car of mass 800 kg is moving on a straight horizontal road with its engine working at a rate of 22.5 kW . Find the resistance to the car's motion at an instant when the car's speed is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2014 November Q2
Moderate -0.5
2
\includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-2_385_621_488_762} Small blocks \(A\) and \(B\) are held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Each is held in equilibrium by a force of magnitude 18 N . The force on \(A\) acts upwards parallel to a line of greatest slope of the plane, and the force on \(B\) acts horizontally in the vertical plane containing a line of greatest slope (see diagram). Find the weight of \(A\) and the weight of \(B\).
CAIE M1 2014 November Q3
Standard +0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffefbc81-402f-4048-8741-23c8bae30d5a-2_231_485_1238_486} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffefbc81-402f-4048-8741-23c8bae30d5a-2_206_485_1263_1174} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A block of weight 7.5 N is at rest on a plane which is inclined to the horizontal at angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\). The coefficient of friction between the block and the plane is \(\mu\). A force of magnitude 7.2 N acting parallel to a line of greatest slope is applied to the block. When the force acts up the plane (see Fig. 1) the block remains at rest.
  1. Show that \(\mu \geqslant \frac { 17 } { 24 }\). When the force acts down the plane (see Fig. 2) the block slides downwards.
  2. Show that \(\mu < \frac { 31 } { 24 }\).
CAIE M1 2014 November Q4
Standard +0.8
4 Particles \(P\) and \(Q\) move on a straight line \(A O B\). The particles leave \(O\) simultaneously, with \(P\) moving towards \(A\) and with \(Q\) moving towards \(B\). The initial speed of \(P\) is \(1.3 \mathrm {~ms} ^ { - 1 }\) and its acceleration in the direction \(O A\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 } . Q\) moves with acceleration in the direction \(O B\) of \(0.016 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(t\) seconds is the time elapsed since the instant that \(P\) and \(Q\) started to move from \(O\). When \(t = 20\), particle \(P\) passes through \(A\) and particle \(Q\) passes through \(B\).
  1. Given that the speed of \(Q\) at \(B\) is the same as the speed of \(P\) at \(A\), find the speed of \(Q\) at time \(t = 0\).
  2. Find the distance \(A B\).
CAIE M1 2014 November Q5
Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_250_846_260_648} A small block \(B\) of mass 0.25 kg is attached to the mid-point of a light inextensible string. Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of the string. The string passes over two smooth pulleys fixed at opposite sides of a rough table, with \(B\) resting in limiting equilibrium on the table between the pulleys and particles \(P\) and \(Q\) and block \(B\) are in the same vertical plane (see diagram).
  1. Find the coefficient of friction between \(B\) and the table.
    \(Q\) is now removed so that \(P\) and \(B\) begin to move.
  2. Find the acceleration of \(P\) and the tension in the part \(P B\) of the string.
CAIE M1 2014 November Q6
Standard +0.3
6 A particle of mass 3 kg falls from rest at a point 5 m above the surface of a liquid which is in a container. There is no instantaneous change in speed of the particle as it enters the liquid. The depth of the liquid in the container is 4 m . The downward acceleration of the particle while it is moving in the liquid is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the resistance to motion of the particle while it is moving in the liquid.
  2. Sketch the velocity-time graph for the motion of the particle, from the time it starts to move until the time it reaches the bottom of the container. Show on your sketch the velocity and the time when the particle enters the liquid, and when the particle reaches the bottom of the container.