Questions M1 (1912 questions)

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Edexcel M1 2019 January Q1
6 marks Moderate -0.8
  1. Two particles, \(A\) and \(B\), 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 when they collide directly. Immediately before they collide, the speed of \(A\) is \(3 u\) and the speed of \(B\) is \(u\). As a result of the collision, the speed of \(A\) is halved and the direction of motion of each particle is reversed.
    1. Find the speed of \(B\) immediately after the collision.
    2. Find the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2019 January Q2
13 marks Standard +0.3
2. [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 \(O\).] At time \(t = 0\), a bird \(A\) leaves its nest, that is located at the point with position vector \(( 20 \mathbf { i } - 17 \mathbf { j } ) \mathrm { m }\), and flies with constant velocity \(( - 6 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the same time a second bird \(B\) leaves its nest which is located at the point with position vector \(( - 8 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m }\) and flies with constant velocity ( \(p \mathbf { i } + 2 p \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), where \(p\) is a constant. At time \(t = 4 \mathrm {~s}\), bird \(B\) is south west of bird \(A\).
  1. Find the direction of motion of \(A\), giving your answer as a bearing to the nearest degree.
  2. Find the speed of \(B\).
Edexcel M1 2019 January Q3
7 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d5a56ba-6a33-4dc8-b612-d2957211124f-08_387_204_251_872} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A lift of mass \(M \mathrm {~kg}\) is being raised by a vertical cable attached to the top of the lift. A person of mass \(m \mathrm {~kg}\) stands on the floor inside the lift, as shown in Figure 1. The lift ascends vertically with constant acceleration \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the cable is 2800 N and the person experiences a constant normal reaction of magnitude 560 N from the floor of the lift. The cable is modelled as being light and inextensible, the person is modelled as a particle and air resistance is negligible.
  1. Write down an equation of motion for the person only.
  2. Write down an equation of motion for the lift only.
  3. Hence, or otherwise, find
    1. the value of \(m\),
    2. the value of \(M\).
Edexcel M1 2019 January Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d5a56ba-6a33-4dc8-b612-d2957211124f-10_410_1143_258_404} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A boy sees a box on the end \(Q\) of a plank \(P Q\) which overhangs a swimming pool. The plank has mass 30 kg , is 5 m long and rests in a horizontal position on two bricks. The bricks are modelled as smooth supports, one acting on the rod at \(P\) and one acting on the rod at \(R\), where \(P R = 3 \mathrm {~m}\). The support at \(R\) is on the edge of the swimming pool, as shown in Figure 2. The boy has mass 40 kg and the box has mass 2.5 kg . The plank is modelled as a uniform rod and the boy and the box are modelled as particles. The boy steps on to the plank at \(P\) and begins to walk slowly along the plank towards the box.
  1. Find the distance he can walk along the plank from \(P\) before the plank starts to tilt.
  2. State how you have used, in your working, the fact that the box is modelled as a particle. A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(P\). The boy is then able to walk slowly along the plank to the box at the end \(Q\) without the plank tilting. The rock is modelled as a particle.
  3. Find the smallest possible value of \(M\).
Edexcel M1 2019 January Q5
10 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d5a56ba-6a33-4dc8-b612-d2957211124f-14_451_551_255_699} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small metal box of mass 6 kg is attached at \(B\) to two ropes \(B P\) and \(B Q\). The fixed points \(P\) and \(Q\) are on a horizontal ceiling and \(P Q = 3.5 \mathrm {~m}\). The box hangs in equilibrium at a vertical distance of 2 m below the line \(P Q\), with the ropes in a vertical plane and with angle \(B Q P = 45 ^ { \circ }\), as shown in Figure 3. The box is modelled as a particle and the ropes are modelled as light inextensible strings. Find
  1. the tension in \(B P\),
  2. the tension in \(B Q\).
Edexcel M1 2019 January Q6
14 marks Standard +0.3
6. A train travels for a total of 270 s along a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration for 60 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train then travels at this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it moves with constant deceleration for 30 s , coming to rest at \(B\).
  1. Sketch below a speed-time graph for the journey of the train between the two stations \(A\) and \(B\). Given that the distance between the two stations is 4.5 km ,
  2. find the value of \(V\),
  3. find how long it takes the train to travel from station \(A\) to the point that is exactly halfway between the two stations. The train is travelling at speed \(\frac { 1 } { 4 } V \mathrm {~ms} ^ { - 1 }\) at times \(T _ { 1 }\) seconds and \(T _ { 2 }\) seconds after leaving station \(A\).
  4. Find the value of \(T _ { 1 }\) and the value of \(T _ { 2 }\)
Edexcel M1 2019 January Q7
16 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0d5a56ba-6a33-4dc8-b612-d2957211124f-20_410_1091_258_440} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(A\) and \(B\) have masses \(m\) and \(3 m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a rough horizontal table. The coefficient of friction between particle \(A\) and the table is \(\frac { 1 } { 5 }\). The string lies along the table and passes over a small smooth light pulley that is fixed at the edge of the table. Particle \(B\) is at rest on a rough plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 4. The coefficient of friction between particle \(B\) and the inclined plane is \(\frac { 1 } { 3 }\). The string lies in the vertical plane that contains the pulley and a line of greatest slope of the inclined plane. The system is released from rest with the string taut and \(B\) slides down the inclined plane. Given that \(A\) does not reach the pulley,
  1. find the tension in the string,
  2. state where in your working you have used the fact that the string is modelled as being light,
  3. find the magnitude of the force exerted on the pulley by the string.
    \includegraphics[max width=\textwidth, alt={}, center]{0d5a56ba-6a33-4dc8-b612-d2957211124f-24_172_1824_2581_123}
    \includegraphics[max width=\textwidth, alt={}, center]{0d5a56ba-6a33-4dc8-b612-d2957211124f-24_157_85_2595_1966}
Edexcel M1 2020 January Q1
8 marks Standard +0.3
  1. Two particles, \(P\) and \(Q\), of mass \(m _ { 1 }\) and \(m _ { 2 }\) respectively, are moving on a smooth horizontal plane. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, both particles are moving with speed \(u\).
The direction of motion of each particle is reversed by the collision.
Immediately after the collision, the speed of \(Q\) is \(\frac { 1 } { 3 } u\).
  1. Find, in terms of \(m _ { 2 }\) and \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision.
  2. Find, in terms of \(m _ { 1 } , m _ { 2 }\) and \(u\), the speed of \(P\) immediately after the collision.
  3. Hence show that \(m _ { 2 } > \frac { 3 } { 4 } m _ { 1 }\)
Edexcel M1 2020 January Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{916543cb-14f7-486c-ba3c-eda9be134045-04_473_1254_221_346} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform beam \(A B\) has length 6 m and weight \(W\) newtons. The beam is supported in equilibrium in a horizontal position by two vertical ropes, one attached to the beam at \(A\) and the other attached to the beam at \(C\), where \(C B = 1.5 \mathrm {~m}\), as shown in Figure 1 . The centre of mass of the beam is 2.625 m from \(A\). The ropes are modelled as light strings. The beam is modelled as a non-uniform rod. Given that the tension in the rope attached at \(C\) is 20 N greater than the tension in the rope attached at \(A\),
  1. find the value of \(W\).
  2. State how you have used the fact that the beam is modelled as a rod.
Edexcel M1 2020 January Q3
11 marks Standard +0.3
3. A particle, \(P\), is projected vertically upwards with speed \(U\) from a fixed point \(O\). At the instant when \(P\) reaches its greatest height \(H\) above \(O\), a second particle, \(Q\), is projected with speed \(\frac { 1 } { 2 } U\) vertically upwards from \(O\).
  1. Find \(H\) in terms of \(U\) and \(g\).
  2. Find, in terms of \(U\) and \(g\), the time between the instant when \(Q\) is projected and the instant when the two particles collide.
  3. Find where the two particles collide. DO NOT WRITEIN THIS AREA
    \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-08_2666_99_107_1957}
Edexcel M1 2020 January Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{916543cb-14f7-486c-ba3c-eda9be134045-10_633_1237_258_356} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two identical small rings, \(A\) and \(B\), each of mass \(m\), are threaded onto a rough horizontal wire. The rings are connected by a light inextensible string. A particle \(C\) of mass \(3 m\) is attached to the midpoint of the string. The particle \(C\) hangs in equilibrium below the wire with angle \(B A C = \beta\), as shown in Figure 2. The tension in each of the parts, \(A C\) and \(B C\), of the string is \(T\)
  1. By considering particle \(C\), find \(T\) in terms of \(m , g\) and \(\beta\)
  2. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between the wire and \(A\). The coefficient of friction between each ring and the wire is \(\frac { 4 } { 5 }\)
    The two rings, \(A\) and \(B\), are on the point of sliding along the wire towards each other.
  3. Find the value of \(\tan \beta\)
    \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M1 2020 January Q5
10 marks Standard +0.3
5. A car travels at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) in a straight line along a horizontal racetrack. At time \(t = 0\), the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car. The motorcyclist accelerates at \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 15 s and then accelerates at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a further \(T\) seconds until he catches up with the car.
  1. Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time \(t = 0\) to the instant when the motorcyclist catches up with the car. At the instant when \(t = t _ { 1 }\) seconds, the car and the motorcyclist are moving at the same speed.
  2. Find the value of \(t _ { 1 }\)
  3. Show that \(T ^ { 2 } + k T - 300 = 0\), where \(k\) is a constant to be found. DO NOT WRITEIN THIS AREA
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M1 2020 January Q6
11 marks Moderate -0.8
6. A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 10 \mathbf { i } + \mathbf { j } ) \mathrm { N }\).
  1. Find the exact value of the magnitude of \(\mathbf { F }\).
  2. Find, in degrees, the size of the angle between the direction of \(\mathbf { F }\) and the direction of the vector \(( \mathbf { i } + \mathbf { j } )\). The resultant of the force \(\mathbf { F }\) and the force \(( - 15 \mathbf { i } + a \mathbf { j } ) \mathrm { N }\), where \(a\) is a constant, is parallel to, but in the opposite direction to, the vector \(( 2 \mathbf { i } - 3 \mathbf { j } )\).
  3. Find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-19_104_59_2613_1886}
Edexcel M1 2020 January Q7
18 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{916543cb-14f7-486c-ba3c-eda9be134045-20_663_1290_260_335} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(A\) of mass 4 kg is held at rest on a rough horizontal table. Particle \(A\) is attached to one end of a string that passes over a pulley \(P\). The pulley is fixed the the of the table. The other end of the string is attached to a particle \(B\), of mass 3 kg , which hangs freely below \(P\). The part of the string from \(A\) to \(P\) is perpendicular to the edge of the table and \(A , P\) and \(B\) all lie in the same vertical plane. The string is modelled as being light and inextensible and the pulley is modelled as being small, smooth and light. The system is released from rest with the string taut. At the instant of release, \(A\) is 2 m from the edge of the table and \(B\) is 1.4 m above a horizontal floor, as shown in Figure 3. After descending with constant acceleration for 2 seconds, \(B\) hits the floor and does not rebound.
  1. Show that the acceleration of \(A\) before \(B\) hits the floor is \(0.7 \mathrm {~ms} ^ { - 2 }\)
  2. State which of the modelling assumptions you have used in order to answer part (a).
  3. Find the magnitude of the resultant force exerted on the pulley by the string. The coefficient of friction between \(A\) and the table is \(\mu\).
  4. Find the value of \(\mu\).
  5. Determine, by calculation, whether or not \(A\) reaches the pulley. DO NOT WRITEIN THIS AREA
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-23_2255_50_314_34}
Edexcel M1 2021 January Q1
6 marks Moderate -0.5
  1. A small stone is projected vertically upwards with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.
Find
  1. the speed of the stone at the instant when it is 2 m above the ground,
  2. the total time between the instant when the stone is projected from \(O\) and the instant when it first strikes the ground.
Edexcel M1 2021 January Q2
6 marks Moderate -0.3
2. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(m\) respectively. The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision, the speed of \(P\) is \(3 u\) and the speed of \(Q\) is \(2 u\). The magnitude of the impulse exerted on \(Q\) by \(P\) in the collision is 5mu. Find
  1. the speed of \(P\) immediately after the collision,
  2. the speed of \(Q\) immediately after the collision.
Edexcel M1 2021 January Q3
9 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-06_156_1009_255_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3 Two forces, both acting in the same vertical plane, of magnitudes 200 N and \(T \mathrm {~N}\) are applied to the parcel. The line of action of the 200 N force makes an angle of \(15 ^ { \circ }\) with the horizontal and the line of action of the \(T \mathrm {~N}\) force makes an angle of \(25 ^ { \circ }\) with the horizontal, as shown in Figure 1. The parcel is modelled as a particle \(P\). Find the smallest value of \(T\) for which \(P\) remains in equilibrium.
Edexcel M1 2021 January Q4
6 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-08_426_1428_118_258} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} \begin{verbatim} A metal girder \(A B\) has weight \(W\) newtons and length 6 m . The girder rests in a horizontal position on two supports \(C\) and \(D\) where \(A C = D B = 1 \mathrm {~m}\), as shown in Figure 2. When a force of magnitude 900 N is applied vertically upwards to the girder at \(A\), the girder is about to tilt about \(D\). When a force of magnitude 1500 N is applied vertically upwards to the girder at \(B\), the girder is about to tilt about \(C\). The girder is modelled as a non-uniform rod whose centre of mass is a distance \(x\) metres from \(A\). Find the value of \(x\). A metal girder AB has weight When a force of magnitude 1500 N is applied vertically upwards to the girder at \(B\), the girder is about to tilt about \(C\). The girder is modelled as a non-uniform rod whose centre of mass is a distance \(x\) metres from \(A\). Find the value of \(x\). \end{verbatim}
Edexcel M1 2021 January Q5
7 marks Moderate -0.3
5. A particle is acted upon by two forces \(\mathbf { F }\) and \(\mathbf { G }\). The force \(\mathbf { F }\) has magnitude 8 N and acts in a direction with a bearing of \(240 ^ { \circ }\). The force \(\mathbf { G }\) has magnitude 10 N and acts due South. Given that \(\mathbf { R } = \mathbf { F } + \mathbf { G }\), find
  1. the magnitude of \(\mathbf { R }\),
  2. the direction of \(\mathbf { R }\), giving your answer as a bearing to the nearest degree. in a direction with a bearing of \(240 ^ { \circ }\). The force \(\mathbf { G }\) has magnitude 10 N and acts due South. Given that \(\mathbf { R } = \mathbf { F } + \mathbf { G }\), find
Edexcel M1 2021 January Q6
12 marks Moderate -0.3
6. Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles. At time \(t = 0\), Agatha is at the point with position vector \(( 11 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m }\) and Brionie is at the point with position vector \(( 7 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m }\). The position vectors are given relative to the door, \(O\), and \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. Agatha skates with constant velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
Brionie skates with constant velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
  1. Find the position vector of Agatha at time \(t\) seconds. At time \(t = 6\) seconds, Agatha passes through the point \(P\).
  2. Show that Brionie also passes through \(P\) and find the value of \(t\) when this occurs. At time \(t\) seconds, Agatha is at the point \(A\) and Brionie is at the point \(B\).
  3. Show that \(\overrightarrow { A B } = [ ( t - 4 ) \mathbf { i } + ( 5 - t ) \mathbf { j } ] \mathrm { m }\)
  4. Find the distance between the two girls when they are closest together. \includegraphics[max width=\textwidth, alt={}, center]{ca445c1e-078c-4a57-94df-de90f30f8efd-13_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M1 2021 January Q7
12 marks Moderate -0.3
7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s . His parachute then opens and causes him to immediately decelerate at a constant rate of \(3.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds ( \(T < 6\) ), until his speed is reduced to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then moves with this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until he hits the ground. While he is decelerating, he falls a distance of 73.75 m . The total time between the instant when he leaves \(H\) and the instant when he hits the ground is 20 s . The parachutist is modelled as a particle.
  1. Find the speed of the parachutist at the instant when his parachute opens.
  2. Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves \(H\) to the instant when he hits the ground.
  3. Find the value of \(T\).
  4. Find, to the nearest metre, the height of the point \(H\) above the ground.
    7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps
Edexcel M1 2021 January Q8
17 marks Standard +0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-20_369_1264_248_342} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles, \(A\) and \(B\), have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The particle \(A\) is held at rest on the plane at a distance \(d\) metres from the pulley. The particle \(B\) hangs freely at rest, vertically below the pulley, at a distance \(h\) metres above the ground, as shown in Figure 3. The part of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\) The system is released from rest with the string taut and \(B\) descends.
  1. Find the tension in the string as \(B\) descends. On hitting the ground, \(B\) immediately comes to rest. Given that \(A\) comes to rest before reaching the pulley,
  2. find, in terms of \(h\), the range of possible values of \(d\).
  3. State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.
Edexcel M1 2022 January Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-02_486_638_248_653} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of weight 5 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a force of magnitude \(F\) newtons. The direction of this force is perpendicular to the string and \(O P\) makes an angle of \(60 ^ { \circ }\) with the vertical, as shown in Figure 1. Find
  1. the value of \(F\)
  2. the tension in the string.
Edexcel M1 2022 January Q2
7 marks Moderate -0.3
2. A particle \(P\) has mass \(k m\) and a particle \(Q\) has mass \(m\). The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, \(P\) has speed \(3 u\) and \(Q\) has speed \(u\).
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 \(Q\) in the collision.
Edexcel M1 2022 January Q3
7 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-06_328_1356_244_296} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A beam \(A D C B\) has length 5 m . The beam lies on a horizontal step with the end \(A\) on the step and the end \(B\) projecting over the edge of the step. The edge of the step is at the point \(D\) where \(D B = 1.3 \mathrm {~m}\), as shown in Figure 2. When a small boy of mass 30 kg stands on the beam at \(C\), where \(C B = 0.5 \mathrm {~m}\), the beam is on the point of tilting. The boy is modelled as a particle and the beam is modelled as a uniform rod.
  1. Find the mass of the beam. A block of mass \(X \mathrm {~kg}\) is now placed on the beam at \(A\).
    The block is modelled as a particle.
  2. Find the smallest value of \(X\) that will enable the boy to stand on the beam at \(B\) without the beam tilting.
  3. State how you have used the modelling assumption that the block is a particle in your calculations.