Questions M1 (2067 questions)

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Edexcel M1 2012 June Q7
15 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-12_150_1104_255_422} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(P\) and \(Q\), of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time \(t = 0\), a constant force \(\mathbf { F }\) of magnitude 4 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 3. The system moves under the action of this force until \(t = 6 \mathrm {~s}\). During the motion, the resistance to the motion of \(P\) has constant magnitude 1 N and the resistance to the motion of \(Q\) has constant magnitude 2 N . Find
  1. the acceleration of the particles as the system moves under the action of \(\mathbf { F }\),
  2. the speed of the particles at \(t = 6 \mathrm {~s}\),
  3. the tension in the rod as the system moves under the action of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , \mathbf { F }\) is removed and the system decelerates to rest. The resistances to motion are unchanged. Find
  4. the distance moved by \(P\) as the system decelerates,
  5. the thrust in the rod as the system decelerates.
Edexcel M1 2014 June Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-02_586_506_285_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of weight \(W\) newtons is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 5 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and with \(O P\) making an angle of \(25 ^ { \circ }\) to the downward vertical, as shown in Figure 1. Find
  1. the tension in the string,
  2. the value of \(W\).
Edexcel M1 2014 June Q2
10 marks Moderate -0.8
  1. Two forces \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) and \(( 2 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) act on a particle \(P\) of mass 1.5 kg . The resultant of these two forces is parallel to the vector \(( 2 \mathbf { i } + \mathbf { j } )\).
    1. Find the value of \(q\).
    At time \(t = 0 , P\) is moving with velocity \(( - 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Find the speed of \(P\) at time \(t = 2\) seconds.
Edexcel M1 2014 June Q3
13 marks Moderate -0.3
3. A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 20 seconds. It moves at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 30 seconds, then moves with constant deceleration \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves at speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 15 seconds and then moves with constant deceleration \(\frac { 1 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.
  1. Sketch, in the space below, a speed-time graph for this journey. In the first 20 seconds of this journey the car travels 140 m . Find
  2. the value of \(V\),
  3. the total time for this journey,
  4. the total distance travelled by the car.
Edexcel M1 2014 June Q4
8 marks Moderate -0.8
  1. At time \(t = 0\), a particle is projected vertically upwards with speed \(u\) from a point \(A\). The particle moves freely under gravity. At time \(T\) the particle is at its maximum height \(H\) above \(A\).
    1. Find \(T\) in terms of \(u\) and \(g\).
    2. Show that \(H = \frac { u ^ { 2 } } { 2 g }\)
    The point \(A\) is at a height \(3 H\) above the ground.
  2. Find, in terms of \(T\), the total time from the instant of projection to the instant when the particle hits the ground.
Edexcel M1 2014 June Q5
14 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-09_364_422_269_753} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(A\) and \(B\) are above a horizontal plane, as shown in Figure 2. The system is released from rest.
  1. Show that the tension in the string immediately after the particles are released is \(\frac { 12 } { 5 } m g\). After descending \(1.5 \mathrm {~m} , B\) strikes the plane and is immediately brought to rest. In the subsequent motion, \(A\) does not reach the pulley.
  2. Find the distance travelled by \(A\) between the instant when \(B\) strikes the plane and the instant when the string next becomes taut. Given that \(m = 0.5 \mathrm {~kg}\),
  3. find the magnitude of the impulse on \(B\) due to the impact with the plane.
Edexcel M1 2014 June Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-11_600_969_127_491} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A non-uniform beam \(A D\) has weight \(W\) newtons and length 4 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points \(B\) and \(C\) on the beam, where \(A B = 1 \mathrm {~m}\) and \(C D = 1 \mathrm {~m}\), as shown in Figure 3. The tension in the rope attached to \(C\) is double the tension in the rope attached to \(B\). The beam is modelled as a rod and the ropes are modelled as light inextensible strings.
  1. Find the distance of the centre of mass of the beam from \(A\). A small load of weight \(k W\) newtons is attached to the beam at \(D\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. Find
  2. an expression for the tension in the rope attached to \(B\), giving your answer in terms of \(k\) and \(W\),
  3. the set of possible values of \(k\) for which both ropes remain taut.
Edexcel M1 2014 June Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-13_364_833_269_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass 2.7 kg lies on a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find
  1. the magnitude of the normal reaction of the plane on \(P\),
  2. the coefficient of friction between \(P\) and the plane. The force of magnitude 15 N is removed.
  3. Determine whether \(P\) moves, justifying your answer.
Edexcel M1 2014 June Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-02_332_921_260_516} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of weight \(W\) newtons is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(B C\) is 6 N , find
  1. the tension in \(A C\),
  2. the value of \(W\).
Edexcel M1 2014 June Q2
7 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-03_435_840_269_561} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A rough plane is inclined at \(40 ^ { \circ }\) to the horizontal. Two points \(A\) and \(B\) are 3 metres apart and lie on a line of greatest slope of the inclined plane, with \(A\) above \(B\), as shown in Figure 2. A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the plane at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is released.
  1. Find the acceleration of \(P\) down the plane.
  2. Find the speed of \(P\) at \(B\).
Edexcel M1 2014 June Q3
13 marks Moderate -0.8
  1. A ball of mass 0.3 kg is released from rest at a point which is 2 m above horizontal ground. The ball moves freely under gravity. After striking the ground, the ball rebounds vertically and rises to a maximum height of 1.5 m above the ground, before falling to the ground again. The ball is modelled as a particle.
    1. Find the speed of the ball at the instant before it strikes the ground for the first time.
    2. Find the speed of the ball at the instant after it rebounds from the ground for the first time.
    3. Find the magnitude of the impulse on the ball in the first impact with the ground.
    4. Sketch, in the space provided, a velocity-time graph for the motion of the ball from the instant when it is released until the instant when it strikes the ground for the second time.
    5. Find the time between the instant when the ball is released and the instant when it strikes the ground for the second time.
Edexcel M1 2014 June Q4
12 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-07_513_993_276_479} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A beam \(A B\) has weight \(W\) newtons and length 4 m . The beam is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\) and the other rope is attached to the point \(C\) on the beam, where \(A C = d\) metres, as shown in Figure 3. The beam is modelled as a uniform rod and the ropes as light inextensible strings. The tension in the rope attached at \(C\) is double the tension in the rope attached at \(A\).
  1. Find the value of \(d\). A small load of weight \(k W\) newtons is attached to the beam at \(B\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. The tension in the rope attached at \(C\) is now four times the tension in the rope attached at \(A\).
  2. Find the value of \(k\).
Edexcel M1 2014 June Q5
12 marks Moderate -0.3
5. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\).
  1. Show that the magnitude of the acceleration of \(P\) is \(2 \sqrt { 13 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Find the velocity of \(P\) at time \(t = 2\) seconds. Another particle \(Q\) moves with constant velocity \(\mathbf { v } = ( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  3. Find the distance moved by \(Q\) in 2 seconds.
  4. Show that at time \(t = 3.5\) seconds both particles are moving in the same direction.
Edexcel M1 2014 June Q6
9 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-11_472_908_285_520} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at \(O\). The angle between the lines of action of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(120 ^ { \circ }\) as shown in Figure 4. The force \(\mathbf { P }\) has magnitude 20 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\). Given that the magnitude of \(\mathbf { R }\) is \(3 X\) newtons, find, giving your answers to 3 significant figures
  1. the value of \(X\),
  2. the magnitude of \(( \mathbf { P } - \mathbf { Q } )\).
Edexcel M1 2014 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-13_490_316_267_815} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Three particles \(A , B\) and \(C\) have masses \(3 m , 2 m\) and \(2 m\) respectively. Particle \(C\) is attached to particle \(B\). Particles \(A\) and \(B\) are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 5. The system is released from rest and \(A\) moves upwards.
    1. Show that the acceleration of \(A\) is \(\frac { g } { 7 }\)
    2. Find the tension in the string as \(A\) ascends. At the instant when \(A\) is 0.7 m above its original position, \(C\) separates from \(B\) and falls away. In the subsequent motion, \(A\) does not reach the pulley.
  2. Find the speed of \(A\) at the instant when it is 0.7 m above its original position.
  3. Find the acceleration of \(A\) at the instant after \(C\) separates from \(B\).
  4. Find the greatest height reached by \(A\) above its original position. \includegraphics[max width=\textwidth, alt={}, center]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-14_115_161_2455_1784}
Edexcel M1 2015 June Q1
6 marks Moderate -0.5
  1. Particle \(P\) of mass \(m\) and particle \(Q\) of mass \(k m\) are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision the speed of \(P\) is \(5 u\) and the speed of \(Q\) is \(u\). Immediately after the collision the speed of each particle is halved and the direction of motion of each particle is reversed.
Find
  1. the value of \(k\),
  2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2015 June Q2
7 marks Moderate -0.8
2. A small stone is projected vertically upwards from a point \(O\) with a speed of \(19.6 \mathrm {~ms} ^ { - 1 }\). Modelling the stone as a particle moving freely under gravity,
  1. find the greatest height above \(O\) reached by the stone,
  2. find the length of time for which the stone is more than 14.7 m above \(O\).
Edexcel M1 2015 June Q3
7 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-04_540_958_116_482} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of mass 2 kg is suspended from a horizontal ceiling by two light inextensible strings, \(P R\) and \(Q R\). The particle hangs at \(R\) in equilibrium, with the strings in a vertical plane. The string \(P R\) is inclined at \(55 ^ { \circ }\) to the horizontal and the string \(Q R\) is inclined at \(35 ^ { \circ }\) to the horizontal, as shown in Figure 1. \section*{Find}
  1. the tension in the string \(P R\),
  2. the tension in the string \(Q R\).
Edexcel M1 2015 June Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-06_428_373_246_788} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A lift of mass 200 kg is being lowered into a mineshaft by a vertical cable attached to the top of the lift. A crate of mass 55 kg is on the floor inside the lift, as shown in Figure 2. The lift descends vertically with constant acceleration. There is a constant upwards resistance of magnitude 150 N on the lift. The crate experiences a constant normal reaction of magnitude 473 N from the floor of the lift.
  1. Find the acceleration of the lift.
  2. Find the magnitude of the force exerted on the lift by the cable.
Edexcel M1 2015 June Q5
12 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-08_582_1230_271_374} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A beam \(A B\) has length 5 m and mass 25 kg . The beam is suspended in equilibrium in a horizontal position by two vertical ropes. One rope is attached to the beam at \(A\) and the other rope is attached to the point \(C\) on the beam where \(C B = 0.5 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 60 kg is attached to the beam at \(B\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod and the ropes are modelled as light strings.
  1. Find
    1. the tension in the rope attached to the beam at \(A\),
    2. the tension in the rope attached to the beam at \(C\). Particle \(P\) is removed and replaced by a particle \(Q\) of mass \(M \mathrm {~kg}\) at \(B\). Given that the beam remains in equilibrium in a horizontal position,
  2. find
    1. the greatest possible value of \(M\),
    2. the greatest possible tension in the rope attached to the beam at \(C\).
Edexcel M1 2015 June Q6
8 marks Easy -1.3
  1. A particle \(P\) is moving with constant velocity. The position vector of \(P\) at time \(t\) seconds \(( t \geqslant 0 )\) is \(\mathbf { r }\) metres, relative to a fixed origin \(O\), and is given by
$$\mathbf { r } = ( 2 t - 3 ) \mathbf { i } + ( 4 - 5 t ) \mathbf { j }$$
  1. Find the initial position vector of \(P\). The particle \(P\) passes through the point with position vector \(( 3.4 \mathbf { i } - 12 \mathbf { j } )\) m at time \(T\) seconds.
  2. Find the value of \(T\).
  3. Find the speed of \(P\).
Edexcel M1 2015 June Q7
13 marks Standard +0.3
7. A train travels along a straight horizontal track between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~ms} ^ { - 1 } , ( V < 50 )\). The train then travels at this constant speed before it moves with constant deceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at \(B\).
  1. Sketch in the space below a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). The total time for the journey from \(A\) to \(B\) is 5 minutes.
  2. Find, in terms of \(V\), the length of time, in seconds, for which the train is
    1. accelerating,
    2. decelerating,
    3. moving with constant speed. Given that the distance between the two stations \(A\) and \(B\) is 6.3 km ,
  3. find the value of \(V\).
Edexcel M1 2015 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de3245a7-cf6e-423e-8689-9a074bdbc23b-14_643_931_118_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(P\) and \(Q\) have mass 4 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough plane, which is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 4 } { 3 }\). The coefficient of friction between \(P\) and the plane is 0.5 . The string lies along the plane and passes over a small smooth light pulley which is fixed at the top of the plane. Particle \(Q\) hangs freely at rest vertically below the pulley. The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 4. Particle \(P\) is released from rest with the string taut and slides down the plane. Given that \(Q\) has not hit the pulley, find
  1. the tension in the string during the motion,
  2. the magnitude of the resultant force exerted by the string on the pulley.
Edexcel M1 2016 June Q1
10 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
Two cars \(P\) and \(Q\) are moving on straight horizontal roads with constant velocities. The velocity of \(P\) is \(( 15 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  1. Find the direction of motion of \(Q\), giving your answer as a bearing to the nearest degree. At time \(t = 0\), the position vector of \(P\) is \(400 \mathbf { i }\) metres and the position vector of \(Q\) is 800j metres. At time \(t\) seconds, the position vectors of \(P\) and \(Q\) are \(\mathbf { p }\) metres and \(\mathbf { q }\) metres respectively.
  2. Find an expression for
    1. \(\mathbf { p }\) in terms of \(t\),
    2. \(\mathbf { q }\) in terms of \(t\).
  3. Find the position vector of \(Q\) when \(Q\) is due west of \(P\).
Edexcel M1 2016 June Q2
6 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d408dd83-c5b2-4e55-b5c1-3e7f3faadbcb-04_327_255_283_847} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A vertical rope \(A B\) has its end \(B\) attached to the top of a scale pan. The scale pan has mass 0.5 kg and carries a brick of mass 1.5 kg , as shown in Figure 1. The scale pan is raised vertically upwards with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) using the rope \(A B\). The rope is modelled as a light inextensible string.
  1. Find the tension in the rope \(A B\).
  2. Find the magnitude of the force exerted on the scale pan by the brick.