Questions M1 (2067 questions)

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Edexcel M1 2022 June Q8
17 marks Standard +0.3
8. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two boats, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(15 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  1. Find the direction in which \(Q\) is travelling, giving your answer as a bearing. The boats are modelled as particles.
    At time \(t = 0 , P\) is at the origin \(O\) and \(Q\) is at the point with position vector \(200 \mathbf { j } \mathrm {~m}\). At time \(t\) seconds, the position vector of \(P\) is \(\mathbf { p m }\) and the position vector of \(Q\) is \(\mathbf { q m }\).
  2. Show that $$\overrightarrow { P Q } = [ 5 t \mathbf { i } + ( 200 - 20 t ) \mathbf { j } ] \mathrm { m }$$
  3. Find the bearing of \(P\) from \(Q\) when \(t = 10\)
  4. Find the distance between \(P\) and \(Q\) when \(Q\) is north east of \(P\)
  5. Find the times when \(P\) and \(Q\) are 200 m apart.
Edexcel M1 2023 June Q1
7 marks Moderate -0.8
  1. A particle \(A\) has mass 4 kg and a particle \(B\) has mass 2 kg .
The particles move towards each other in opposite directions along the same straight line on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(2 u \mathrm {~ms} ^ { - 1 }\) and the speed of \(B\) is \(3 u \mathrm {~ms} ^ { - 1 }\) Immediately after the collision, the speed of \(B\) is \(2 u \mathrm {~ms} ^ { - 1 }\) The direction of motion of \(B\) is reversed by the collision.
  1. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
  2. State the direction of motion of \(A\) immediately after the collision.
  3. Find, in terms of \(u\), the magnitude of the impulse received by \(B\) in the collision. State the units of your answer. \section*{[In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]}
Edexcel M1 2023 June Q2
10 marks Moderate -0.8
  1. A particle \(P\) rests in equilibrium on a smooth horizontal plane.
A system of three forces, \(\mathbf { F } _ { 1 } \mathrm {~N} , \mathbf { F } _ { 2 } \mathrm {~N}\) and \(\mathbf { F } _ { 3 } \mathrm {~N}\) where $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 3 c \mathbf { i } + 4 c \mathbf { j } ) \\ & \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } + 7 \mathbf { j } ) \end{aligned}$$ is applied to \(P\).
Given that \(P\) remains in equilibrium,
  1. find \(\mathbf { F } _ { 3 }\) in terms of \(c\), \(\mathbf { i }\) and \(\mathbf { j }\). The force \(\mathbf { F } _ { 3 }\) is removed from the system.
    Given that \(c = 2\)
  2. find the size of the angle between the direction of \(\mathbf { i }\) and the direction of the resultant force acting on \(P\). The mass of \(P\) is \(m \mathrm {~kg}\).
    Given that the magnitude of the acceleration of \(P\) is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  3. find the value of \(m\).
Edexcel M1 2023 June Q3
8 marks Standard +0.3
  1. Two students observe a book of mass 0.2 kg fall vertically from rest from a shelf that is 1.5 m above the floor.
Student \(A\) suggests that the book is modelled as a particle falling freely under gravity.
  1. Use student \(A\) 's model to find the time taken for the book to reach the floor. Student \(B\) suggests an improved model where the book is modelled as a particle experiencing a constant resistance to motion of magnitude \(R\) newtons. Given that the time taken for the book to reach the floor is 0.6 seconds,
  2. use student \(B\) 's model to find the value of \(R\)
Edexcel M1 2023 June Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-08_625_1488_246_287} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a beam \(A B\), of mass \(m \mathrm {~kg}\) and length 2 m , suspended by two light vertical ropes.
The ropes are attached to the points \(C\) and \(D\) on the beam, where \(A C = 0.6 \mathrm {~m}\) and \(D B = 0.2 \mathrm {~m}\) The beam is in equilibrium in a horizontal position.
A particle of mass pmkg is attached to the beam at \(A\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod.
  1. Given that the tension in the rope attached at \(C\) is four times the tension in the rope attached at \(D\), use the model to find the exact value of \(p\). The particle of mass \(p m \mathrm {~kg}\) at \(A\) is removed and replaced by a particle of mass \(q m \mathrm {~kg}\) at \(A\).
    The beam remains in equilibrium in a horizontal position but is now on the point of tilting.
  2. Using the model, find the exact value of \(q\)
  3. State how you have used the modelling assumption that the beam is uniform.
Edexcel M1 2023 June Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-12_629_1251_244_406} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The speed-time graph in Figure 2 illustrates the motion of a car travelling along a straight horizontal road.
At time \(t = 0\), the car starts from rest and accelerates uniformly for 30 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until time \(t = T\) seconds.
  1. Show that the distance travelled by the car between \(t = 0\) and \(t = T\) seconds is \(V ( T - 15 )\) metres. A motorbike also travels along the same road.
    At time \(t = T\) seconds, the distance travelled by each vehicle is the same.
  2. Find the value of \(T\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-15_643_1266_1882_402} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
Edexcel M1 2023 June Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-16_314_815_246_625} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle of weight \(W\) newtons lies at rest on a rough horizontal surface, as shown in Figure 3.
A force of magnitude \(P\) newtons is applied to the particle.
The force acts at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\) The coefficient of friction between the particle and the surface is \(\frac { 1 } { 4 }\) Given that the particle does not move, show that $$P \leqslant \frac { 5 W } { 8 }$$
Edexcel M1 2023 June Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-18_326_1107_246_479} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A car of mass 1200 kg is towing a trailer of mass 600 kg up a straight road, as shown in Figure 4. The road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The driving force produced by the engine of the car is 3000 N .
The car moves with acceleration \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The non-gravitational resistance to motion of
  • the car is modelled as a constant force of magnitude \(2 R\) newtons
  • the trailer is modelled as a constant force of magnitude \(R\) newtons
The car and the trailer are modelled as particles.
The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion. Using the model,
  1. show that the value of \(R\) is 60
  2. find the tension in the tow bar. When the car and trailer are moving at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the tow bar breaks.
    Given that the non-gravitational resistance to motion of the trailer remains unchanged,
  3. use the model to find the further distance moved by the trailer before it first comes to rest.
Edexcel M1 2023 June Q8
9 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-22_792_841_246_612} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A square floor space \(A B C D\), with centre \(O\), is modelled as a flat horizontal surface measuring 50 m by 50 m , as shown in Figure 5 .
The horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the direction of \(\overrightarrow { A B }\) and \(\overrightarrow { A D }\) respectively.
All position vectors are given relative to \(O\).
A small robot \(R\) is programmed to travel across the floor at a constant velocity.
  • At time \(t = 0 , R\) is at the point with position vector ( \(- 2 \mathbf { i } + \mathbf { j }\) ) m
  • At time \(t = 11 \mathrm {~s} , R\) is at the point with position vector \(( 9 \mathbf { i } + 23 \mathbf { j } ) \mathrm { m }\)
  • At time \(t\) seconds, the position vector of \(R\) is \(\mathbf { r }\) metres
    1. Find, in terms of \(t\), i and j, an expression for \(\mathbf { r }\)
A second robot \(S\) is at the point \(C\).
$$\overrightarrow { S R } = [ ( 2 t - 27 ) \mathbf { i } + ( 3 t - 24 ) \mathbf { j } ] \mathbf { m }$$
  • Find the time when the distance between \(R\) and \(S\) is a minimum.
    •
    Edexcel M1 2024 June Q1
    5 marks Moderate -0.3
    1. Two particles, \(A\) and \(B\), have masses \(m\) and \(3 m\) respectively. The particles are connected by a light inextensible string. Initially \(A\) and \(B\) are at rest on a smooth horizontal plane with the string slack.
    Particle \(A\) is then projected along the plane away from \(B\) with speed \(U\).
    Given that the common speed of the particles immediately after the string becomes taut is \(S\)
    1. find \(S\) in terms of \(U\).
    2. Find, in terms of \(m\) and \(U\), the magnitude of the impulse exerted on \(A\) immediately after the string becomes taut.
    Edexcel M1 2024 June Q2
    4 marks Moderate -0.8
    1. Two forces, \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle.
    • \(\mathbf { P }\) has magnitude 10 N and acts due west
    • Q has magnitude 8 N and acts on a bearing of \(330 ^ { \circ }\)
    Given that \(\mathbf { F } = \mathbf { P } + \mathbf { Q }\), find the magnitude of \(\mathbf { F }\).
    Edexcel M1 2024 June Q3
    6 marks Moderate -0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-06_472_208_343_1102} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Two particles, \(P\) and \(Q\), have masses \(4 m\) and \(2 m\) respectively. The particles are connected by a light inextensible string. A second light inextensible string has one end attached to \(Q\). Both strings are taut and vertical, as shown in Figure 1. The particles are accelerating vertically downwards.
    Given that the tension in the string connecting the two particles is \(3 m g\), find, in terms of \(m\) and \(g\), the tension in the upper string.
    Edexcel M1 2024 June Q4
    6 marks Standard +0.3
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-08_417_1745_378_258} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A non-uniform rod \(A B\) has length 6.5 m and mass 1.2 kg . The centre of mass of the rod is 3 m from \(A\). The rod rests on a horizontal step and overhangs the end of the step \(C\) by 1.5 m , as shown in Figure 2. The rod is perpendicular to the edge of the step.
    A particle of mass 4 kg is placed on the rod at \(B\) and another particle, whose mass is \(M \mathrm {~kg}\), is placed on the rod at \(D\), where \(A D = 0.5 \mathrm {~m}\). The rod remains in equilibrium in a horizontal position.
    1. Find the smallest possible value of \(M\). The particle at \(B\) and the particle at \(D\) are now removed.
      A new particle is placed on the rod at the point \(E\), where \(E B = 0.9 \mathrm {~m}\).
      The rod remains in equilibrium in a horizontal position but is on the point of tilting about \(C\).
    2. Find the magnitude of the force acting on the rod at \(C\).
    Edexcel M1 2024 June Q5
    14 marks Standard +0.3
    1. A parachute is used to deliver a box of supplies. The parachute is attached to the box.
    • the parachute and box are dropped from rest from a helicopter that is hovering at a height of 520 m above the ground
    • the parachute and box fall vertically and freely under gravity for 5 seconds, then the parachute opens
    • from the instant the parachute opens, it provides a resistance to motion of magnitude 3200 N
    • the parachute and box continue to fall vertically downwards after the parachute opens
    • the parachute and box are modelled throughout the motion as a particle \(P\) of mass 250 kg
      1. Find the distance fallen by \(P\) in the first 5 seconds.
      2. Find the speed with which \(P\) lands on the ground.
      3. Find the total time from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.
      4. Sketch a speed-time graph for the motion of \(P\) from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.
    VJYV SIHI NI JIIYM ION OCVayv sthin NI JLIYM ION OAVJYV SIHI NI JAIVM ION OC
    Edexcel M1 2024 June Q6
    12 marks Moderate -0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-16_272_1391_336_436} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A box of mass \(m\) lies on a rough horizontal plane. The box is pulled along the plane in a straight line at constant speed by a light rope. The rope is inclined at an angle \(\theta\) to the plane, as shown in Figure 3.
    The coefficient of friction between the box and the plane is \(\frac { 1 } { 3 }\) The box is modelled as a particle.
    Given that \(\tan \theta = \frac { 3 } { 4 }\)
    1. find, in terms of \(m\) and \(g\), the tension in the rope. The rope is now removed and the box is placed at rest on the plane.
      The box is then projected horizontally along the plane with speed \(u\).
      The box is again modelled as a particle.
      When the box has moved a distance \(d\) along the plane, the speed of the box is \(\frac { 1 } { 2 } u\).
    2. Find \(d\) in terms of \(u\) and \(g\).
      VJYV SIHI NI JIIIM ION OCvauv sthin NI JLHMA LON OOV34V SIHI NI IIIIM ION OC
    Edexcel M1 2024 June Q7
    13 marks Moderate -0.3
    1. \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
    Two speedboats, \(A\) and \(B\), are each moving with constant velocity.
    • the velocity of \(A\) is \(40 \mathrm { kmh } ^ { - 1 }\) due east
    • the velocity of \(B\) is \(20 \mathrm { kmh } ^ { - 1 }\) on a bearing of angle \(\alpha \left( 0 ^ { \circ } < \alpha < 90 ^ { \circ } \right)\), where \(\tan \alpha = \frac { 4 } { 3 }\) The boats are modelled as particles.
      1. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) in \(\mathrm { km } \mathrm { h } ^ { - 1 }\)
    At noon
    • the position vector of \(A\) is \(20 \mathbf { j } \mathrm {~km}\)
    • the position vector of \(B\) is \(( 10 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\)
    At time \(t\) hours after noon
    • the position vector of \(A\) is \(\mathbf { r k m }\), where \(\mathbf { r } = 20 \mathbf { j } + 40 t \mathbf { i }\)
    • the position vector of \(B\) is \(\mathbf { s }\) km
    • Find an expression for \(\mathbf { s }\) in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
    • Show that at time \(t\) hours after noon,
    $$\overrightarrow { A B } = [ ( 10 - 24 t ) \mathbf { i } + ( 12 t - 15 ) \mathbf { j } ] \mathrm { km }$$
  • Show that the boats will never collide.
  • Find the distance between the boats when the bearing of \(B\) from \(A\) is \(225 ^ { \circ }\)
    •
    Edexcel M1 2024 June Q8
    15 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7a65555e-1bb2-4947-8e70-50f267017bfd-24_442_1167_341_548} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). Particle \(A\) is held at rest on a rough plane which is inclined to horizontal ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. Particle \(B\) hangs vertically below \(P\) with the string taut, at a height \(h\) above the ground, as shown in Figure 4. The part of the string between \(A\) and \(P\) lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane.
    The coefficient of friction between \(A\) and the plane is \(\frac { 11 } { 36 }\) The particle \(A\) is released from rest and begins to move up the plane.
    1. Show that the frictional force acting on \(A\) as it moves up the plane is \(\frac { 22 m g } { 39 }\)
    2. Write down an equation of motion for \(B\).
    3. Show that the acceleration of \(A\) immediately after its release is \(\frac { 1 } { 3 } g\) In the subsequent motion, \(A\) comes to rest before it reaches the pulley.
    4. Find, in terms of \(h\), the total distance travelled by \(A\) from when it was released from rest to when it first comes to rest again.
      VJYV SIHI NI JIIIM ION OCvauv sthin NI BLIYM ION OCV34V SIHI NI IIIIMM ION OC
      VJYV SIHI NI JIIIM ION OCvauv sthin NI BLIYM ION OOV34V SIHI NI IIIIMM ION OC
    Edexcel M1 2016 October Q1
    7 marks Moderate -0.8
    Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other, in opposite directions, along the same straight line, on a smooth horizontal plane. The particles collide. Immediately before they collide the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). In the collision the magnitude of the impulse exerted on \(P\) by \(Q\) is \(5 m u\).
    1. Find the speed of \(P\) immediately after the collision.
    2. State whether the direction of motion of \(P\) has been reversed by the collision.
    3. Find the speed of \(Q\) immediately after the collision.
    Edexcel M1 2016 October Q2
    9 marks Moderate -0.3
    2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] Three forces, \(( - 10 \mathbf { i } + a \mathbf { j } ) \mathrm { N } , ( b \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( 2 a \mathbf { i } + 7 \mathbf { j } ) \mathrm { N }\), where \(a\) and \(b\) are constants, act on a particle \(P\) of mass 3 kg . The acceleration of \(P\) is \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
    1. Find the value of \(a\) and the value of \(b\). At time \(t = 0\) seconds the speed of \(P\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at time \(t = 4\) seconds the velocity of \(P\) is \(( 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
    2. Find the value of \(u\).
    Edexcel M1 2016 October Q3
    7 marks Moderate -0.3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-06_267_1092_254_428} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A plank \(A B\) has length 8 m and mass 12 kg . The plank rests on two supports. One support is at \(C\), where \(A C = 3 \mathrm {~m}\) and the other support is at \(D\), where \(A D = x\) metres. A block of mass 3 kg is placed on the plank at \(B\), as shown in Figure 1. The plank rests in equilibrium in a horizontal position. The magnitude of the force exerted on the plank by the support at \(D\) is twice the magnitude of the force exerted on the plank by the support at \(C\). The plank is modelled as a uniform rod and the block is modelled as a particle. Find the value of \(x\).
    Edexcel M1 2016 October Q4
    10 marks Moderate -0.8
    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 \(O\) ]
    A particle \(P\) is moving with velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t = 0\) hours, the position vector of \(P\) is \(( - 5 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).
    1. Find an expression for \(\mathbf { p }\) in terms of \(t\). The point \(A\) has position vector ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) km.
    2. Find the position vector of \(P\) when \(P\) is due west of \(A\). Another particle \(Q\) is moving with velocity \([ ( 2 b - 1 ) \mathbf { i } + ( 5 - 2 b ) \mathbf { j } ] \mathrm { km } \mathrm { h } ^ { - 1 }\) where \(b\) is a constant. Given that the particles are moving along parallel lines,
    3. find the value of \(b\).
    Edexcel M1 2016 October Q5
    9 marks Standard +0.3
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-10_419_933_123_525} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A particle \(P\) of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at \(30 ^ { \circ }\). The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of \(40 ^ { \circ }\) to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).
    Edexcel M1 2016 October Q6
    9 marks Standard +0.3
    6. Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. Car \(A\) is moving with uniform acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\) and car \(B\) is moving with uniform acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant when \(B\) is 200 m behind \(A\), the speed of \(A\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(44 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(B\) when it overtakes \(A\).
    (9)
    Edexcel M1 2016 October Q7
    11 marks Standard +0.3
    7. A train moves on a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(1 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train maintains this speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next \(T\) seconds before slowing down with constant deceleration \(0.5 \mathrm {~ms} ^ { - 2 }\), coming to rest at \(B\). The journey from \(A\) to \(B\) takes 180 s and the distance between the stations is 4800 m .
    1. Sketch a speed-time graph for the motion of the train from \(A\) to \(B\).
    2. Show that \(T = 180 - 3 V\).
    3. Find the value of \(V\).
    Edexcel M1 2016 October Q8
    13 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6978be48-561b-49a0-a297-c8886ca66c19-20_312_1068_230_438} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Two particles \(P\) and \(Q\) have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\). Initially \(P\) is held at rest on the inclined plane with the part of the string from \(P\) to the pulley parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between \(P\) and the plane is \(\mu\). The system is released from rest, with the string taut, and \(Q\) strikes the ground before \(P\) reaches the pulley. The speed of \(Q\) at the instant when it strikes the ground is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. For the motion before \(Q\) strikes the ground, find the tension in the string.
    2. Find the value of \(\mu\).
      END