Questions — Edexcel M1 (599 questions)

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Edexcel M1 2009 June Q8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \(( 1.2 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. Find the speed of \(H\).
      (2)
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{05fd9db2-2ff3-4b84-99c2-f348ff567ebd-11_599_1057_521_445} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A horizontal field \(O A B C\) is rectangular with \(O A\) due east and \(O C\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100 \mathbf { j } \mathrm {~m}\), relative to the fixed origin \(O\).
  2. Write down the position vector of \(H\) at time \(t\) seconds after noon. At noon, another hiker \(K\) is at the point with position vector \(( 9 \mathbf { i } + 46 \mathbf { j } )\) m. Hiker \(K\) is moving with constant velocity \(( 0.75 \mathbf { i } + 1.8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  3. Show that, at time \(t\) seconds after noon, $$\overrightarrow { H K } = [ ( 9 - 0.45 t ) \mathbf { i } + ( 2.7 t - 54 ) \mathbf { j } ] \text { metres. }$$ Hence,
  4. show that the two hikers meet and find the position vector of the point where they meet.
Edexcel M1 2010 June Q1
  1. A particle \(P\) is moving with constant velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 6 \mathrm {~s} P\) is at the point with position vector \(( - 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the distance of \(P\) from the origin at time \(t = 2 \mathrm {~s}\).
    (5)
  2. Particle \(P\) has mass \(m \mathrm {~kg}\) and particle \(Q\) has mass \(3 m \mathrm {~kg}\). The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(k u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.
    1. Find the value of \(k\).
    2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-04_195_579_260_507} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac { 1 } { 2 }\). The box is pushed by a force of magnitude 100 N which acts at an angle of \(30 ^ { \circ }\) with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.
Edexcel M1 2010 June Q4
4. A beam \(A B\) has length 6 m and weight 200 N . The beam rests in a horizontal position on two supports at the points \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\). Two children, Sophie and Tom, each of weight 500 N , stand on the beam with Sophie standing twice as far from the end \(B\) as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at \(D\) is three times the magnitude of the reaction at \(C\). By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end \(B\).
Edexcel M1 2010 June Q5
5. Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 0 , P\) overtakes \(Q\) which is moving with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From \(t = T\) seconds, P decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25 \mathrm {~s}\), \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).
  1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\).
  2. Find the value of \(T\).
Edexcel M1 2010 June Q6
6. A ball is projected vertically upwards with a speed of \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
  1. the greatest height, above the ground, reached by the ball,
  2. the speed with which the ball first strikes the ground,
  3. the total time from when the ball is projected to when it first strikes the ground.
Edexcel M1 2010 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-10_275_712_269_612} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).
Given that the particle is on the point of sliding up the plane, find
  1. the magnitude of the normal reaction between the particle and the plane,
  2. the value of \(P\).
Edexcel M1 2010 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25300ba0-1e54-4242-8db4-a593f5d5a80e-12_890_428_237_754} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.
  1. Find the tension in the string immediately after the particles are released.
  2. Find the acceleration of \(A\) immediately after the particles are released. When the particles have been moving for 0.5 s , the string breaks.
  3. Find the further time that elapses until \(B\) hits the floor.
Edexcel M1 2011 June Q1
  1. At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.
    1. Show that the speed of projection is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Find the times, in seconds, when the ball is 33.6 m above \(O\).
    3. Particle \(P\) has mass 3 kg and particle \(Q\) has mass 2 kg . The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, \(P\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, both particles move in the same direction and the difference in their speeds is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    4. Find the speed of each particle after the collision.
    5. Find the magnitude of the impulse exerted on \(P\) by \(Q\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9a1ffe48-cea7-49aa-9b6f-f781568d0600-04_344_771_221_589} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N . The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1.
    The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
    Given that the particle is on the point of sliding down the plane,
    (i) show that the magnitude of the normal reaction between the particle and the plane is 20 N ,
    (ii) find the value of \(W\).
Edexcel M1 2011 June Q4
  1. A girl runs a 400 m race in a time of 84 s . In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s , reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She maintains this speed for 60 s and then moves with constant deceleration for 20 s , crossing the finishing line with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race.
    2. Find the distance run by the girl in the first 64 s of the race.
    3. Find the value of \(V\).
    4. Find the deceleration of the girl in the final 20 s of her race.
Edexcel M1 2011 June Q5
  1. A plank \(P Q R\), of length 8 m and mass 20 kg , is in equilibrium in a horizontal position on two supports at \(P\) and \(Q\), where \(P Q = 6 \mathrm {~m}\).
A child of mass 40 kg stands on the plank at a distance of 2 m from \(P\) and a block of mass \(M \mathrm {~kg}\) is placed on the plank at the end \(R\). The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at \(P\) is equal to the force exerted on the plank by the support at \(Q\). By modelling the plank as a uniform rod, and the child and the block as particles,
    1. find the magnitude of the force exerted on the plank by the support at \(P\),
    2. find the value of \(M\).
  2. State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.
Edexcel M1 2011 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a1ffe48-cea7-49aa-9b6f-f781568d0600-10_369_954_214_497} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(P\) and \(Q\) have masses 0.3 kg and \(m \mathrm {~kg}\) respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle \(P\) is held at rest on the inclined plane and the particle \(Q\) hangs freely below the pulley with the string taut, as shown in Figure 2. The system is released from rest and \(Q\) accelerates vertically downwards at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
  1. the magnitude of the normal reaction of the inclined plane on \(P\),
  2. the value of \(m\). When the particles have been moving for 0.5 s , the string breaks. Assuming that \(P\) does not reach the pulley,
  3. find the further time that elapses until \(P\) comes to instantaneous rest.
Edexcel M1 2011 June Q7
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).]
Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and ship \(Q\) moves with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Find, to the nearest degree, the bearing on which \(Q\) is moving. At 2 pm , ship \(P\) is at the point with position vector \(( \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(\operatorname { ship } Q\) is at the point with position vector \(( - 2 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours after 2 pm , the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\) and the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\).
  2. Write down expressions, in terms of \(t\), for
    1. \(\mathbf { p }\),
    2. \(\mathbf { q }\),
    3. \(\overrightarrow { P Q }\).
  3. Find the time when
    1. \(Q\) is due north of \(P\),
    2. \(Q\) is north-west of \(P\).
Edexcel M1 2012 June Q1
  1. Two particles \(A\) and \(B\), of mass \(5 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, the speed of \(A\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the speed of \(B\) immediately after the collision.
    In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is 3.3 N s .
  2. Find the value of \(m\).
Edexcel M1 2012 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-03_215_716_233_614} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform rod \(A B\) has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(P\) and at \(Q\), where \(A P = 0.8 \mathrm {~m}\) and \(Q B = 0.6 \mathrm {~m}\), as shown in Figure 1. The centre of mass of the rod is at \(G\). Given that the magnitude of the reaction of the support at \(P\) on the rod is twice the magnitude of the reaction of the support at \(Q\) on the rod, find
  1. the magnitude of the reaction of the support at \(Q\) on the rod,
  2. the distance \(A G\).
Edexcel M1 2012 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-04_432_780_210_584} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A box of mass 5 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N , as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane.
The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find
  1. the magnitude of the normal reaction of the plane on the box,
  2. the coefficient of friction between the box and the plane.
Edexcel M1 2012 June Q4
  1. A car is moving on a straight horizontal road. At time \(t = 0\), the car is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at the point \(A\). The car maintains the speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s . The car then moves with constant deceleration \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reducing its speed from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then moves with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 60 s . The car then moves with constant acceleration until it is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\).
    1. Sketch a speed-time graph to represent the motion of the car from \(A\) to \(B\).
    2. Find the time for which the car is decelerating.
    Given that the distance from \(A\) to \(B\) is 1960 m ,
  2. find the time taken for the car to move from \(A\) to \(B\).
Edexcel M1 2012 June Q5
  1. A particle \(P\) is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is 17.5 m above horizontal ground. The particle \(P\) moves freely under gravity until it reaches the ground with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(u = 21\)
    At time \(t\) seconds after projection, \(P\) is 19 m above \(A\).
  2. Find the possible values of \(t\). The ground is soft and, after \(P\) reaches the ground, \(P\) sinks vertically downwards into the ground before coming to rest. The mass of \(P\) is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on \(P\).
  3. Find the vertical distance that \(P\) sinks into the ground before coming to rest.
Edexcel M1 2012 June Q6
6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
  2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
  3. Find the distance of \(S\) from \(B\) when \(t = 3\)
  4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).
Edexcel M1 2012 June Q7
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 2013 June Q1
  1. Two particles \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately before the collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse exerted on \(B\) by \(A\) is 14 N s . Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{278c8424-38aa-48c2-bc82-af4be9234f71-03_359_1298_219_413} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle of weight 8 N is attached at \(C\) to the ends of two light inextensible strings \(A C\) and \(B C\). The other ends, \(A\) and \(B\), are attached to a fixed horizontal ceiling. The particle hangs at rest in equilibrium, with the strings in a vertical plane. The string \(A C\) is inclined at \(35 ^ { \circ }\) to the horizontal and the string \(B C\) is inclined at \(25 ^ { \circ }\) to the horizontal, as shown in Figure 1. Find
    (i) the tension in the string \(A C\),
    (ii) the tension in the string \(B C\).
Edexcel M1 2013 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{278c8424-38aa-48c2-bc82-af4be9234f71-04_589_1027_248_440} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A fixed rough plane is inclined at \(30 ^ { \circ }\) to the horizontal. A small smooth pulley \(P\) is fixed at the top of the plane. Two particles \(A\) and \(B\), of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley \(P\). The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs freely below \(P\), as shown in Figure 2. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { \sqrt { } 3 }\). Initially \(A\) is held at rest on the plane. The particles are released from rest with the string taut and \(A\) moves up the plane. Find the tension in the string immediately after the particles are released.
Edexcel M1 2013 June Q4
4. At time \(t = 0\), two balls \(A\) and \(B\) are projected vertically upwards. The ball \(A\) is projected vertically upwards with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 50 m above the horizontal ground. The ball \(B\) is projected vertically upwards from the ground with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = T\) seconds, the two balls are at the same vertical height, \(h\) metres, above the ground. The balls are modelled as particles moving freely under gravity. Find
  1. the value of \(T\),
  2. the value of \(h\).
Edexcel M1 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{278c8424-38aa-48c2-bc82-af4be9234f71-07_520_1143_116_406} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at \(25 ^ { \circ }\) to the horizontal. The particle passes through two points \(A\) and \(B\), where \(A B = 10 \mathrm {~m}\), as shown in Figure 3. The speed of \(P\) at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle \(P\) takes 3.5 s to move from \(A\) to \(B\). Find
  1. the speed of \(P\) at \(B\),
  2. the acceleration of \(P\),
  3. the coefficient of friction between \(P\) and the plane.
Edexcel M1 2013 June Q6
6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin \(O\).] A ship \(S\) is moving with constant velocity \(( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t = 0\), the position vector of \(S\) is \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }\).
  1. Find the position vector of \(S\) at time \(t\) hours. A ship \(T\) is moving with constant velocity \(( - 2 \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t = 0\), the position vector of \(T\) is \(( 6 \mathbf { i } + \mathbf { j } ) \mathrm { km }\). The two ships meet at the point \(P\).
  2. Find the value of \(n\).
  3. Find the distance \(O P\).
Edexcel M1 2013 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{278c8424-38aa-48c2-bc82-af4be9234f71-11_216_1335_207_306} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle \(\theta\) to the road, as shown in Figure 4. The vehicles are travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they enter a zone where the speed limit is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The truck's brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is 100 m .
  1. Find the deceleration of the truck and the car. The constant braking force on the truck has magnitude \(R\) newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that \(\cos \theta = 0.9\), find
  2. the force in the towbar,
  3. the value of \(R\).