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Edexcel M2 2019 January Q7
13 marks Standard +0.3
7. A particle \(P\) of mass \(3 m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. A second particle \(Q\) of mass \(2 m\) is moving with speed \(2 u\) in the opposite direction to \(P\) along the same straight line. Particle \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the direction of motion of \(P\) is reversed as a result of the collision with \(Q\).
  2. Find the range of values of \(e\) for which the direction of motion of \(Q\) is also reversed as a result of the collision. Given that \(e = \frac { 1 } { 2 }\)
  3. find, in terms of \(m\) and \(u\), the kinetic energy lost in the collision between \(P\) and \(Q\).
Edexcel M2 2019 January Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-24_286_1317_251_317} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A rough ramp \(A B\) is fixed to horizontal ground at \(A\). The ramp is inclined at \(20 ^ { \circ }\) to the ground. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 6 \mathrm {~m}\). The point \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 3 kg is projected with speed \(15 \mathrm {~ms} ^ { - 1 }\) from \(A\) towards \(B\). At the instant \(P\) reaches the point \(B\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force due to friction is modelled as a constant force of magnitude \(F\) newtons.
  1. Use the work-energy principle to find the value of \(F\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). The speed of \(P\) as it hits the ground at \(C\) is \(w \mathrm {~ms} ^ { - 1 }\). Find
    1. the value of \(w\),
    2. the direction of motion of \(P\) as it hits the ground at \(C\),
  3. the greatest height of \(P\) above the ground as \(P\) moves from \(A\) to \(C\).
Edexcel M2 2020 January Q1
5 marks Standard +0.3
  1. A cyclist and his bicycle have a total mass of 75 kg . The cyclist is moving down a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\)
The cyclist is working at a constant rate of 56 W . The magnitude of the resistance to motion is modelled as a constant force of magnitude 40 N . At the instant when the speed of the cyclist is \(\mathrm { Vm } \mathrm { s } ^ { - 1 }\), his acceleration is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 2 }\) Find the value of \(V\).
(5)
Edexcel M2 2020 January Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-04_239_796_246_577} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). The points \(A\) and \(B\) are on a line of greatest slope of the ramp with \(A B = 2.5 \mathrm {~m}\) and \(B\) above \(A\), as shown in Figure 1. A package of mass 2 kg is projected up the ramp from \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and first comes to instantaneous rest at \(B\). The coefficient of friction between the package and the ramp is \(\mu\). The package is modelled as a particle. Use the work-energy principle to find the value of \(\mu\).
(6)
Edexcel M2 2020 January Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-06_291_481_255_733} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 0.75 kg is moving along a straight line on a horizontal surface. At the instant when the speed of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it receives an impulse of magnitude \(\sqrt { 24 } \mathrm { Ns }\). The impulse acts in the plane of the horizontal surface. At the instant when \(P\) receives the impulse, the line of action of the impulse makes an angle of \(60 ^ { \circ }\) with the direction of motion of \(P\), as shown in Figure 2. Find
  1. the speed of \(P\) immediately after receiving the impulse,
  2. the size of the angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse. \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-06_2252_51_311_1980} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-07_36_65_2722_109}
Edexcel M2 2020 January Q4
10 marks Standard +0.3
4. [The centre of mass of a uniform semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-08_437_563_347_701} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The uniform rectangular lamina \(A B C D E F\) has sides \(A C = F D = 6 a\) and \(A F = C D = 3 a\). The point \(B\) lies on \(A C\) with \(A B = 2 a\) and the point \(E\) lies on \(F D\) with \(F E = 2 a\). The template, \(T\), shown shaded in Figure 3, is formed by removing the semicircular lamina with diameter \(B C\) from the rectangular lamina and then fixing this semicircular lamina to the opposite side, \(F D\), of the rectangular lamina. The diameter of the semicircular lamina coincides with \(E D\) and the semicircular arc \(E D\) is outside the rectangle \(A B C D E F\). All points of \(T\) lie in the same plane.
  1. Show that the centre of mass of \(T\) is a distance \(\left( \frac { 9 + 2 \pi } { 6 } \right)\) a from \(A C\). The mass of \(T\) is \(M\). A particle of mass \(k M\) is attached to \(T\) at \(C\). The loaded template is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at angle \(\phi\) to the downward vertical through \(A\). Given that \(\tan \phi = \frac { 3 } { 2 }\)
  2. find the value of \(k\).
    \section*{\textbackslash section*\{Question 4 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-11_149_142_2604_1816}
Edexcel M2 2020 January Q5
10 marks Standard +0.3
5. A t time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + ( 2 t - 4 ) \mathbf { j }$$ When \(t = 0 , P\) is at the fixed point \(O\).
  1. Find the acceleration of \(P\) at the instant when \(t = 0\)
  2. Find the exact speed of \(P\) at the instant when \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } + \mathbf { j } )\) for the second time.
  3. Show that \(P\) never returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-14_2658_1938_107_123} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-15_149_140_2604_1818}
Edexcel M2 2020 January Q6
11 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-16_358_967_248_484} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A uniform rod, \(A B\), of weight \(W\) and length \(8 a\), rests in equilibrium with the end \(A\) on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is \(C\), where \(A C = 7 a\), as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle \(\alpha\) with the horizontal .
  1. Show that the normal reaction of the ground on the rod at \(A\) has $$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$ Given that the coefficient of friction between the rod and the ground is \(\mu\) and that \(\cos \alpha = \frac { 3 } { \sqrt { 10 } }\)
  2. find the range of possible values of \(\mu\).
    \section*{\textbackslash section*\{Question 6 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-19_147_142_2606_1816}
Edexcel M2 2020 January Q7
14 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-20_360_1026_246_466} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} At time \(t = 0\) a particle \(P\) is projected from a fixed point \(A\) on horizontal ground. The particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the ground. The particle moves freely under gravity. At time \(t = 3\) seconds, \(P\) is passing through the point \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at an angle \(\beta\) to the horizontal, as shown in Figure 5.
  1. By considering energy, find the height of \(B\) above the ground.
  2. Find the size of angle \(\alpha\).
  3. Find the size of angle \(\beta\).
  4. Find the least speed of \(P\) as \(P\) travels from \(A\) to \(B\). As \(P\) travels from \(A\) to \(B\), the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 15\) for an interval of \(T\) seconds.
  5. Find the value of \(T\).
    \section*{\textbackslash section*\{Question 7 continued\}}
Edexcel M2 2020 January Q8
12 marks Standard +0.3
  1. A particle \(A\) has mass \(4 m\) and a particle \(B\) has mass \(3 m\). The particles are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(3 u\).
The direction of motion of each particle is reversed by the collision.
The total kinetic energy lost in the collision is \(\frac { 473 } { 24 } m u ^ { 2 }\) Find
  1. the coefficient of restitution between \(A\) and \(B\),
  2. the magnitude of the impulse received by \(A\) in the collision.
    \section*{\textbackslash section*\{Question 8 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-28_2642_1833_118_118}
Edexcel M2 2021 January Q1
7 marks Moderate -0.3
  1. A particle \(P\) of mass 1.5 kg is moving with velocity \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of magnitude 15Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(\boldsymbol { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the two possible values of \(v\).
Edexcel M2 2021 January Q2
5 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-04_760_669_118_641} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B C\) has sides \(A B = A C = 13 a\) and \(B C = 10 a\). The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) is applied to the lamina at \(B\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A B\) vertical. The weight of the lamina is \(W\). Find \(F\) in terms of \(W\).
VILM SIHI NI JAIUM ION OC
VANV SIHI NI I III M LON OO
VI4V SIHI NI JAIUM ION OC
Edexcel M2 2021 January Q3
8 marks Standard +0.3
3. A car of mass 600 kg travels along a straight horizontal road with the engine of the car working at a constant rate of \(P\) watts. The resistance to the motion of the car is modelled as a constant force of magnitude \(R\) newtons. At the instant when the speed of the car is \(15 \mathrm {~ms} ^ { - 1 }\), the magnitude of the acceleration of the car is \(0.2 \mathrm {~ms} ^ { - 2 }\). Later the same car travels up a straight road inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the engine of the car is working at a constant rate of \(P\) watts, the car has a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). Find the value of \(P\).
Edexcel M2 2021 January Q4
9 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-10_517_371_260_790} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The number "4", shown in Figure 2, is a rigid framework made from three uniform rods, \(A B , B C\) and \(C D\), where $$A B = 6 a , B C = 5 a \text { and } C D = 4 a$$ The point \(E\) is on \(A B\) and \(C D\), where \(B E = 4 a , C E = 3 a\) and angle \(C E B = 90 ^ { \circ }\) The three rods are all made from the same material and they all lie in the same plane. The framework is suspended from \(B\) and hangs in equilibrium with \(B A\) at an angle \(\theta\) to the downward vertical. Find \(\theta\) to the nearest degree.
VILU SIHI NI JAIUM ION OC
VIUV SIHI NI JAHM ION OC
VIIV SIHI NI EIIIM ION OC
VIIV SIHI NI III HM ION OCVIUV SIHI NI JIHM I ON OOVI4V SIHI NI JIIIM ION OO
Edexcel M2 2021 January Q5
11 marks Standard +0.3
5. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = \left( 5 t ^ { 2 } - 12 t + 15 \right) \mathbf { i } + \left( t ^ { 2 } + 8 t - 10 \right) \mathbf { j }$$ When \(t = 0 , P\) is at the origin \(O\).
At time \(T\) seconds, \(P\) is moving in the direction of \(( \mathbf { i } + \mathbf { j } )\).
  1. Find the value of \(T\). When \(t = 3 , P\) is at the point \(A\).
  2. Find the magnitude of the acceleration of \(P\) as it passes through \(A\).
  3. Find the position vector of \(A\).
Edexcel M2 2021 January Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-16_639_561_246_689} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A ladder \(A B\) has length 6 m and mass 30 kg . The ladder rests in equilibrium at \(60 ^ { \circ }\) to the horizontal with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall, as shown in Figure 3. A man of mass 70 kg stands on the ladder at the point \(C\), where \(A C = 2 \mathrm {~m}\), and the ladder remains in equilibrium. The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall. The man is modelled as a particle.
  1. Find the magnitude of the force exerted on the ladder by the ground. The man climbs further up the ladder. When he is at the point \(D\) on the ladder, the ladder is about to slip. Given that the coefficient of friction between the ladder and the ground is 0.4
  2. find the distance \(A D\).
  3. State how you have used the modelling assumption that the ladder is a rod.
Edexcel M2 2021 January Q7
12 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3eb71ecb-fa88-4cca-a2b6-bcf11f1d689b-20_517_947_212_500} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The fixed point \(A\) is 20 m vertically above the point \(O\) which is on horizontal ground. At time \(t = 0\), a particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The particle moves freely under gravity. At time \(t = 5\) seconds, \(P\) strikes the ground at the point \(B\), where \(O B = 40 \mathrm {~m}\), as shown in Figure 4.
  1. By considering energy, find the speed of \(P\) as it hits the ground at \(B\).
  2. Find the least speed of \(P\) as it moves from \(A\) to \(B\).
  3. Find the length of time for which the speed of \(P\) is more than \(10 \mathrm {~ms} ^ { - 1 }\).
Edexcel M2 2021 January Q8
12 marks Standard +0.3
8. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly. Immediately after the collision, \(A\) and \(B\) are moving in the same direction with speeds \(\frac { u } { 3 }\) and \(u\) respectively. In the collision, \(A\) receives an impulse of magnitude 8mu.
  1. Find the coefficient of restitution between \(A\) and \(B\). When \(A\) and \(B\) collide they are at a distance \(d\) from a smooth vertical wall, which is perpendicular to their direction of motion. After the collision with \(A\), particle \(B\) collides directly with the wall and rebounds so that there is a second collision between \(A\) and \(B\). This second collision takes place at distance \(x\) from the wall. Given that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 4 }\)
  2. find \(x\) in terms of \(d\).
    END
Edexcel M2 2022 January Q1
8 marks Standard +0.3
  1. A particle of mass 0.5 kg is moving with velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of ( \(- 4 \mathbf { i } + 6 \mathbf { j }\) )Ns.
    1. Find the speed of the particle immediately after it receives the impulse.
    2. Find the size of the angle between the direction of motion of the particle immediately before it receives the impulse and the direction of motion of the particle immediately after it receives the impulse.
      (3)
Edexcel M2 2022 January Q2
9 marks Standard +0.3
2. A car of mass 600 kg tows a trailer of mass 200 kg up a hill along a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The trailer is attached to the car by a light inextensible towbar. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude 150 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 300 N . When the engine of the car is working at a constant rate of \(P \mathrm {~kW}\) the car and the trailer have a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the value of \(P\). Later, at the instant when the car and the trailer are travelling up the hill with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. When the towbar breaks the trailer is at the point \(X\). The trailer continues to travel up the hill before coming to instantaneous rest at the point \(Y\). The resistance to the motion of the trailer from non-gravitational forces is again modelled as a constant force of magnitude 300 N .
  2. Use the work-energy principle to find the distance \(X Y\).
    VIIV SIHI NI III M I0N 00 :
Edexcel M2 2022 January Q3
9 marks Standard +0.3
3. A particle \(P\) of mass 0.25 kg is moving on a smooth horizontal surface under the action of a single force, \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 )\), the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = ( 6 \sin 3 t ) \mathbf { i } + ( 1 + 2 \cos t ) \mathbf { j }$$
  1. Find \(\mathbf { F }\) in terms of \(t\). At time \(t = 0\), the position vector of \(P\) relative to a fixed point \(O\) is \(( 4 \mathbf { i } - \sqrt { 3 } \mathbf { j } ) \mathrm { m }\).
  2. Find the position vector of \(P\) relative to \(O\) when \(P\) is first moving parallel to the vector \(\mathbf { i }\).
Edexcel M2 2022 January Q4
10 marks Standard +0.3
4. Two small balls, \(A\) and \(B\), are moving in opposite directions along the same straight line on smooth horizontal ground. The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(3 m\). The balls collide directly. Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\) By modelling the balls as particles,
  1. show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 5 } u ( 1 + 6 e )\).
    (6) After the collision with ball \(A\), ball \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 5 } { 7 }\) Ball \(B\) rebounds from the wall and there is a second direct collision between \(A\) and \(B\).
  2. Find the range of possible values of \(e\).
Edexcel M2 2022 January Q5
12 marks Standard +0.3
5. A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
  1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
  2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
  3. Find
    1. an expression for \(R\) in terms of \(W\)
    2. the value of \(\theta\) (8) 5 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
      \end{figure} . T a and angle \(C A O = \alpha\), as shown in Figure 1.
      Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
  4. Explain why \(A O = 13 a\)
Edexcel M2 2022 January Q6
11 marks Standard +0.3
  1. \hspace{0pt} [The centre of mass of a semicircular arc of radius \(r\) is \(\frac { 2 r } { \pi }\) from the centre.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-20_668_371_358_790} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework,
  • \(A B C\) is straight and has length \(25 a\)
  • \(A D E\) is straight and has length \(24 a\)
  • \(A B D\) is a semicircular arc of radius \(7 a\)
  • \(E C = 7 a\)
  • angle \(A E C = 90 ^ { \circ }\)
  • the points \(A , B , C , D\) and \(E\) all lie in the same plane
The distance of the centre of mass of the framework from \(A E\) is \(d\).
  1. Show that \(d = \frac { 53 } { 2 ( 7 + \pi ) } a\) The framework is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at angle \(\alpha ^ { \circ }\) to the downward vertical.
  2. Find the value of \(\alpha\).
Edexcel M2 2022 January Q7
16 marks Standard +0.3
  1. A particle \(P\) is projected from a fixed point \(O\) on horizontal ground. The particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal. At the instant when the horizontal distance of \(P\) from \(O\) is \(x\), the vertical distance of \(P\) above the ground is \(y\). The motion of \(P\) is modelled as that of a particle moving freely under gravity.
    1. Show that \(y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)\) (6)
    A small ball is projected from the fixed point \(O\) on horizontal ground. The ball is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta ^ { \circ }\) above the horizontal. A vertical pole \(A B\), of height 2 m , stands on the ground with \(O A = 10 \mathrm {~m}\), as shown in Figure 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-24_246_899_840_525} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The ball is modelled as a particle moving freely under gravity and the pole is modelled as a rod.
    The path of the ball lies in the vertical plane containing \(O , A\) and \(B\).
    Using the model,
  2. find the range of values of \(\theta\) for which the ball will pass over the pole. Given that \(\theta = 40\) and that the ball first hits the ground at the point \(C\)
  3. find the speed of the ball at the instant it passes over the pole,
  4. find the distance \(O C\). \includegraphics[max width=\textwidth, alt={}, center]{0762451f-b951-4d66-9e01-61ecb7b30d95-28_2649_1898_109_169}