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Edexcel M2 2018 June Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88731f1c-5177-4096-841b-cd9c3f87782b-06_314_1118_219_427} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A , B\) and \(C\) lie on a smooth horizontal plane. A small ball of mass 0.2 kg is moving along the line \(A B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the ball is at \(B\), the ball is given an impulse. Immediately after the impulse is given, the ball moves along the line \(B C\) with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The line \(B C\) makes an angle of \(35 ^ { \circ }\) with the line \(A B\), as shown in Figure 1.
  1. Find the magnitude of the impulse given to the ball.
  2. Find the size of the angle between the direction of the impulse and the original direction of motion of the ball.
Edexcel M2 2018 June Q3
8 marks Standard +0.3
3. [The centre of mass of a semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88731f1c-5177-4096-841b-cd9c3f87782b-08_581_460_374_740} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the uniform lamina \(A B C D E\), such that \(A B D E\) is a square with sides of length \(2 a\) and \(B C D\) is a semicircle with diameter \(B D\).
  1. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 20 a } { 3 ( 8 + \pi ) }\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
  2. Find, to the nearest degree, the angle that \(D E\) makes with the downward vertical.
Edexcel M2 2018 June Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88731f1c-5177-4096-841b-cd9c3f87782b-12_510_1082_269_438} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), rests with its end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a light string attached to the rod at \(B\), as shown in Figure 3. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The coefficient of friction between the ground and the rod is \(\mu\).
Show that \(\mu = \frac { \cos \theta \sin \theta } { 2 - \cos ^ { 2 } \theta }\)
Edexcel M2 2018 June Q5
13 marks Standard +0.3
5. A particle \(A\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal floor. Particle \(A\) collides directly with another particle \(B\) of mass \(2 m\) which is moving along the same straight line with speed \(u\) but in the opposite direction to \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).
    1. Show that the speed of \(B\) immediately after the collision is \(\frac { 7 } { 5 } u\)
    2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). The first collision between \(A\) and \(B\) occurred at a distance \(x\) from the wall. The particles collide again at a distance \(y\) from the wall.
  2. Find \(y\) in terms of \(x\).
Edexcel M2 2018 June Q6
14 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where
$$\mathbf { v } = \left( 4 t - 3 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 8 t - 40 \right) \mathbf { j }$$
  1. Find
    1. the magnitude of \(\mathbf { F }\) when \(t = 3\)
    2. the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(- \mathbf { i } - \mathbf { j }\). When \(t = 1 , P\) is at the point \(A\). When \(t = 2 , P\) is at the point \(B\).
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the vector \(\overrightarrow { A B }\).
Edexcel M2 2018 June Q7
15 marks Challenging +1.2
7. A particle, of mass 0.3 kg , is projected from a point \(O\) on horizontal ground with speed \(u\). The particle is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = 2\), and moves freely under gravity. When the particle has moved a horizontal distance \(x\) from \(O\), its height above the ground is \(y\).
  1. Show that $$y = 2 x - \frac { 5 g } { 2 u ^ { 2 } } x ^ { 2 }$$ The particle hits the ground at the point \(A\), where \(O A = 36 \mathrm {~m}\).
  2. Find \(u\), the speed of projection.
  3. Find the minimum kinetic energy of the particle as it moves between \(O\) and \(A\). The point \(B\) lies on the path of the particle. The direction of motion of the particle at \(B\) is perpendicular to the initial direction of motion of the particle.
  4. Find the horizontal distance between \(O\) and \(B\).
Edexcel M2 Q1
5 marks Moderate -0.8
  1. A smooth sphere is moving with speed \(U\) in a straight line on a smooth horizontal plane. It strikes a fixed smooth vertical wall at right angles. The coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\).
Find the fraction of the kinetic energy of the sphere that is lost as a result of the impact.
(5 marks)
Edexcel M2 Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9126ebb1-eaa7-4a40-953f-5dc819c9f479-3_631_581_744_769} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform ladder \(A B\) has one end \(A\) on smooth horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is modelled as a uniform rod of mass \(m\) and length 4a. The ladder is kept in equilibrium by a horizontal force \(F\) acting at a point \(C\) of the ladder where \(A C = a\). The force \(F\) and the ladder lie in a vertical plane perpendicular to the wall. The ladder is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = 2\), as shown in Fig. 1. Find \(F\) in terms of \(m\) and \(g\).
(6 marks)
Edexcel M2 Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9126ebb1-eaa7-4a40-953f-5dc819c9f479-4_698_1271_296_488} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform plane lamina is in the shape of an isosceles triangle \(A B C\), where \(A B = A C\). The mid-point of \(B C\) is \(M , A M = 30 \mathrm {~cm}\) and \(B M = 40 \mathrm {~cm}\). The mid-points of \(A C\) and \(A B\) are \(D\) and \(E\) respectively. The triangular portion \(A D E\) is removed leaving a uniform plane lamina \(B C D E\) as shown in Fig. 2.
  1. Show that the centre of mass of the lamina \(B C D E\) is \(6 \frac { 2 } { 3 } \mathrm {~cm}\) from \(B C\).
    (6 marks)
    The lamina \(B C D E\) is freely suspended from \(D\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle which \(D E\) makes with the vertical.
    (3 marks)
Edexcel M2 Q4
9 marks Standard +0.3
4. The resistance to the motion of a cyclist is modelled as \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\), at a constant speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 4\). The cyclist ascends a slope inclined at an angle \(\beta\) to the horizontal, where \(\sin \beta = \frac { 1 } { 40 }\), at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate at which the cyclist is working.
    (6 marks)
Edexcel M2 Q5
9 marks Standard +0.3
5. A smooth sphere \(S\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal plane. The sphere \(S\) collides with another smooth sphere \(T\), of equal radius to \(S\) but of mass \(k m\), moving in the same straight line and in the same direction with speed \(\lambda u , 0 < \lambda < \frac { 1 } { 2 }\). The coefficient of restitution between \(S\) and \(T\) is \(e\). Given that \(S\) is brought to rest by the impact,
  1. show that \(e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }\).
  2. Deduce that \(k > 1\).
Edexcel M2 Q6
9 marks Standard +0.3
6. At time \(t\) seconds the acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of a particle \(P\) relative to a fixed origin \(O\), is given by \(\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }\). Initially the velocity of \(P\) is \(( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the velocity of \(P\) at time \(t\) seconds. At time \(t = 2\) seconds the particle \(P\) is given an impulse ( \(3 \mathbf { i } - 1.5 \mathbf { j }\) ) Ns. Given that the particle \(P\) has mass 0.5 kg ,
  2. find the speed of \(P\) immediately after the impulse has been applied.
Edexcel M2 Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9126ebb1-eaa7-4a40-953f-5dc819c9f479-6_675_1243_392_415} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A shot is projected upwards from the top of a cliff with a velocity of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. It strikes the ground 52.5 m vertically below the level of the point of projection, as shown in Fig. 3. The motion of the shot is modelled as that of a particle moving freely under gravity. Find, to 3 significant figures,
  1. the horizontal distance from the point of projection at which the shot strikes the ground,
  2. the speed of the shot as it strikes the ground.
Edexcel M2 Q8
15 marks Standard +0.3
8. A particle \(P\) is projected up a line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is projected from the point \(O\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to instantaneous rest at the point \(A\). By Using the Work-Energy principle, or otherwise,
  1. find, to 3 significant figures, the length \(O A\).
  2. Show that \(P\) will slide back down the plane.
  3. Find, to 3 significant figures, the speed of \(P\) when it returns to \(O\).
Edexcel M2 Specimen Q1
5 marks Moderate -0.3
  1. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A ball of mass 0.5 kg is moving with velocity \(- 20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The bat gives the ball an impulse of \(( 15 \mathbf { i } + 10 \mathbf { j } )\) Ns.
Find, to 3 significant figures, the speed of the ball immediately after it has been struck.
(5)
Edexcel M2 Specimen Q2
5 marks Moderate -0.3
2. A bullet of mass 6 grams passes horizontally through a fixed, vertical board. After the bullet has travelled 2 cm through the board its speed is reduced from \(400 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The board exerts a constant resistive force on the bullet. Find, to 3 significant figures, the magnitude of this resistive force.
(5)
Edexcel M2 Specimen Q3
7 marks Moderate -0.3
3. At time \(t\) seconds, a particle \(P\) has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\), where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find
  1. the velocity of \(P\) at time \(t\) seconds,
  2. the time when \(P\) is moving parallel to the vector \(\mathbf { i } + \mathbf { j }\).
    (5)
Edexcel M2 Specimen Q4
9 marks Standard +0.8
4. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-3_714_565_262_749}
A uniform ladder, of mass \(m\) and length \(2 a\), has one end on rough horizontal ground. The other end rests against a smooth vertical wall. A man of mass \(3 m\) stands at the top of the ladder and the ladder is in equilibrium. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\), and the ladder makes an angle \(\alpha\) with the vertical, as shown in Fig. 1. The ladder is in a vertical plane perpendicular to the wall. Show that \(\tan \alpha \leq \frac { 2 } { 7 }\).
Edexcel M2 Specimen Q5
11 marks Moderate -0.3
5. A straight road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). A lorry of mass 4800 kg moves up the road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to the motion of the lorry is constant and has magnitude 2000 N .
  1. Find, in kW to 3 significant figures, the rate of working of the lorry's engine.
    (5) The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion remains the same. Find
  2. the acceleration of the lorry immediately after the road becomes horizontal,
    (3)
  3. the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, at which the lorry will go along the horizontal road.
    (3)
Edexcel M2 Specimen Q6
12 marks Moderate -0.3
6. A cricket ball is hit from a height of 0.8 m above horizontal ground with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The motion of the ball is modelled as that of a particle moving freely under gravity.
  1. Find, to 2 significant figures, the greatest height above the ground reached by the ball. When the ball has travelled a horizontal distance of 36 m , it hits a window.
  2. Find, to 2 significant figures, the height above the ground at which the ball hits the window.
  3. State one physical factor which could be taken into account in any refinement of the model which would make it more realistic. Figure 2
Edexcel M2 Specimen Q7
13 marks Standard +0.3
7. \includegraphics[max width=\textwidth, alt={}, center]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-4_360_472_1105_815} A uniform plane lamina \(A B C D E\) is formed by joining a uniform square \(A B D E\) with a uniform triangular lamina \(B C D\), of the same material, along the side \(B D\), as shown in Fig. 2. The lengths \(A B , B C\) and \(C D\) are \(18 \mathrm {~cm} , 15 \mathrm {~cm}\) and 15 cm respectively.
  1. Find the distance of the centre of mass of the lamina from \(A E\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle which \(B D\) makes with the vertical.
Edexcel M2 Specimen Q8
13 marks Standard +0.3
8. A particle \(A\) of mass \(m\) is moving with speed \(3 u\) on a smooth horizontal table when it collides directly with a particle \(B\) of mass \(2 m\) which is moving in the opposite direction with speed \(u\). The direction of motion of \(A\) is reversed by the collision. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 3 } ( 1 + 4 e ) u\).
    (6)
  2. Show that \(e > \frac { 1 } { 8 }\).
    (3) Subsequently \(B\) hits a wall fixed at right angles to the line of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). After \(B\) rebounds from the wall, there is a further collision between \(A\) and \(B\).
  3. Show that \(e < \frac { 1 } { 4 }\).
    (4) END
Edexcel M3 2014 January Q1
5 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force of magnitude \(F\) newtons. The force acts along the \(x\)-axis in the direction of \(x\) increasing. When \(P\) is \(x\) metres from the origin \(O\), it is moving away from \(O\) with speed \(\sqrt { \left( 8 x ^ { \frac { 3 } { 2 } } - 4 \right) } \mathrm { ms } ^ { - 1 }\).
Find \(F\) when \(P\) is 4 m from \(O\).
Edexcel M3 2014 January Q2
9 marks Standard +0.8
2. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring, of natural length \(l\) and modulus of elasticity \(2 m g\). The other end of the spring is attached to a fixed point \(A\) on a rough horizontal plane. The particle is held at rest on the plane at a point \(B\), where \(A B = \frac { 1 } { 2 } l\), and released from rest. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\) Find the distance of \(P\) from \(B\) when \(P\) first comes to rest.
Edexcel M3 2014 January Q3
8 marks Challenging +1.2
3. A light rod \(A B\) of length \(2 a\) has a particle \(P\) of mass \(m\) attached to \(B\). The rod is rotating in a vertical plane about a fixed smooth horizontal axis through \(A\). Given that the greatest tension in the rod is \(\frac { 9 m g } { 8 }\), find, to the nearest degree, the angle between the rod and the downward vertical when the speed of \(P\) is \(\sqrt { \left( \frac { a g } { 20 } \right) }\).