Questions — Edexcel (9685 questions)

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Edexcel M2 2005 January Q7
15 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{a9e00b5b-3804-4f8d-9cc8-7d1170027726-6_568_1582_360_239}
\end{figure} A particle \(P\) is projected from a point \(A\) with speed \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The point \(O\) is on horizontal ground, with \(O\) vertically below \(A\) and \(O A = 20 \mathrm {~m}\). The particle \(P\) moves freely under gravity and passes through a point \(B\), which is 16 m above ground, before reaching the ground at the point \(C\), as shown in Figure 4. Calculate
  1. the time of the flight from \(A\) to \(C\),
  2. the distance \(O C\),
  3. the speed of \(P\) at \(B\),
  4. the angle that the velocity of \(P\) at \(B\) makes with the horizontal.
Edexcel M2 2006 January Q1
6 marks Moderate -0.3
  1. A brick of mass 3 kg slides in a straight line on a horizontal floor. The brick is modelled as a particle and the floor as a rough plane. The initial speed of the brick is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brick is brought to rest after moving 12 m by the constant frictional force between the brick and the floor.
    1. Calculate the kinetic energy lost by the brick in coming to rest, stating the units of your answer.
    2. Calculate the coefficient of friction between the brick and the floor.
    3. A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds is given by
    $$\mathbf { r } = \left( t ^ { 2 } + 4 t \right) \mathbf { i } + \left( 3 t - t ^ { 3 } \right) \mathbf { j } .$$
  2. Calculate the speed of \(P\) when \(t = 3\). When \(t = 3\), the particle \(P\) is given an impulse ( \(8 \mathbf { i } - 12 \mathbf { j }\) ) N s.
  3. Find the velocity of \(P\) immediately after the impulse.
Edexcel M2 2006 January Q3
9 marks Moderate -0.3
3. A car of mass 1000 kg is moving along a straight horizontal road. The resistance to motion is modelled as a constant force of magnitude \(R\) newtons. The engine of the car is working at a rate of 12 kW . When the car is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(R = 600\). The car now moves with constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downhill on a straight road inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 40 }\). The engine of the car is now working at a rate of 7 kW . The resistance to motion from non-gravitational forces remains of magnitude \(R\) newtons.
  2. Calculate the value of \(U\).
    (5)
Edexcel M2 2006 January Q4
13 marks Moderate -0.3
4. A particle \(A\) of mass \(2 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(m\) moving with speed \(2 u\) in the opposite direction to \(A\). Immediately after the collision the speed of \(B\) is \(\frac { 8 } { 3 } u\) and the direction of motion of \(B\) is reversed.
  1. Calculate the coefficient of restitution between \(A\) and \(B\).
  2. Show that the kinetic energy lost in the collision is \(7 m u ^ { 2 }\). After the collision \(B\) strikes a fixed vertical wall that is perpendicular to the direction of motion of \(B\). The magnitude of the impulse of the wall on \(B\) is \(\frac { 14 } { 3 } m u\).
  3. Calculate the coefficient of restitution between \(B\) and the wall.
    (4)
Edexcel M2 2006 January Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{97fbfac6-6c1c-4a5c-ab5d-adc3193bfedc-4_805_1265_258_312}
\end{figure} Figure 1 shows a triangular lamina \(A B C\). The coordinates of \(A , B\) and \(C\) are ( 0,4 ), ( 9,0 ) and \(( 0 , - 4 )\) respectively. Particles of mass \(4 m , 6 m\) and \(2 m\) are attached at \(A , B\) and \(C\) respectively.
  1. Calculate the coordinates of the centre of mass of the three particles, without the lamina. The lamina \(A B C\) is uniform and of mass \(k m\). The centre of mass of the combined system consisting of the three particles and the lamina has coordinates \(( 4 , \lambda )\).
  2. Show that \(k = 6\).
  3. Calculate the value of \(\lambda\). The combined system is freely suspended from \(O\) and hangs at rest.
  4. Calculate, in degrees to one decimal place, the angle between \(A C\) and the vertical.
    (3) \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{97fbfac6-6c1c-4a5c-ab5d-adc3193bfedc-5_693_556_338_712}
    \end{figure} A ladder \(A B\), of weight \(W\) and length \(4 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\mu\). The other end \(B\) rests against a smooth vertical wall. The ladder makes an angle \(\theta\) with the horizontal, where \(\tan \theta = 2\). A load of weight \(4 W\) is placed at the point \(C\) on the ladder, where \(A C = 3 a\), as shown in Figure 2. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The load is modelled as a particle. Given that the system is in limiting equilibrium,
Edexcel M2 2006 January Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{97fbfac6-6c1c-4a5c-ab5d-adc3193bfedc-6_501_1284_306_386}
\end{figure} The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m . The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.
  1. Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\).
  2. Show that \(C\) is approximately 0.63 m below \(T\).
    (4) The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the angle of elevation remaining \(30 ^ { \circ }\). This time \(B\) hits \(T\).
  3. Calculate the value of \(V\).
    (6)
  4. Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target.
Edexcel M2 2007 January Q1
6 marks Moderate -0.3
  1. A particle of mass 0.8 kg is moving in a straight line on a rough horizontal plane. The speed of the particle is reduced from \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as the particle moves 20 m . Assuming that the only resistance to motion is the friction between the particle and the plane, find
    1. the work done by friction in reducing the speed of the particle from \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
    2. the coefficient of friction between the particle and the plane.
    3. A car of mass 800 kg is moving at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 24 }\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 900 N .
    4. Find, in kW , the rate of working of the engine of the car.
    When the car is travelling down the road at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car comes to rest in time \(T\) seconds after the engine is switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 900 N .
  2. Find the value of \(T\).
Edexcel M2 2007 January Q3
10 marks Standard +0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{7ae16b00-d388-4c1b-a195-c785a3900548-04_648_732_301_612}
\end{figure} Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm , from a uniform circular lamina, of centre \(O\) and radius 24 cm . The point \(X\) lies on the diameter \(A O B\) of the lamina and \(A X = 16 \mathrm {~cm}\). The centre of mass of \(T\) is at the point \(G\).
  1. Find \(A G\). The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(O B\). A small stud of mass \(\frac { 1 } { 4 } m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(A B\) horizontal. Modelling the stud as a particle,
  2. find the mass of \(T\) in terms of \(m\).
Edexcel M2 2007 January Q4
12 marks Standard +0.3
4. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal table. Another particle \(Q\) of mass \(k m\) is at rest on the table. The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. After the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(3 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 2 }\).
  1. Find, in terms of \(v\) only, the speed of \(P\) before the collision.
  2. Find the value of \(k\). After being struck by \(P\), the particle \(Q\) collides directly with a particle \(R\) of mass \(11 m\) which is at rest on the table. After this second collision, \(Q\) and \(R\) have the same speed and are moving in opposite directions. Show that
  3. the coefficient of restitution between \(Q\) and \(R\) is \(\frac { 3 } { 4 }\),
  4. there will be a further collision between \(P\) and \(Q\).
Edexcel M2 2007 January Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{7ae16b00-d388-4c1b-a195-c785a3900548-07_551_636_306_660}
\end{figure} A horizontal uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(4 a\). The end \(A\) rests against a rough vertical wall. A particle of mass \(2 m\) is attached to the rod at the point \(C\), where \(A C = 3 a\). One end of a light inextensible string \(B D\) is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\), where \(D\) is vertically above \(A\). The rod is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\), as shown in Figure 2.
  1. Find the tension in the string.
  2. Show that the horizontal component of the force exerted by the wall on the rod has magnitude \(\frac { 8 } { 3 } m g\). The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
  3. find the value of \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{7ae16b00-d388-4c1b-a195-c785a3900548-08_158_136_2595_1822}
Edexcel M2 2007 January Q6
13 marks Moderate -0.3
6. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(\mathbf { F } = \left( 1.5 t ^ { 2 } - 3 \right) \mathbf { i } + 2 t \mathbf { j }\). When \(t = 2\), the velocity of \(P\) is \(( - 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the acceleration of \(P\) at time \(t\) seconds.
  2. Show that, when \(t = 3\), the velocity of \(P\) is \(( 9 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf { Q }\) Ns. Immediately after the impulse the velocity of \(P\) is \(( - 3 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  3. the magnitude of \(\mathbf { Q }\),
  4. the angle between \(\mathbf { Q }\) and \(\mathbf { i }\).
Edexcel M2 2007 January Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{7ae16b00-d388-4c1b-a195-c785a3900548-10_728_1210_303_376}
\end{figure} A particle \(P\) is projected from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(A B = 45 \mathrm {~m}\). After projection, \(P\) moves freely under gravity passing through a point \(C , 30 \mathrm {~m}\) above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  1. using conservation of energy, or otherwise, show that \(u = 17.5\),
  2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\),
  3. find the distance \(B D\).
Edexcel M2 2008 January Q1
5 marks Moderate -0.8
  1. A parcel of mass 2.5 kg is moving in a straight line on a smooth horizontal floor. Initially the parcel is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The parcel is brought to rest in a distance of 20 m by a constant horizontal force of magnitude \(R\) newtons. Modelling the parcel as a particle, find
    1. the kinetic energy lost by the parcel in coming to rest,
    2. the value of \(R\).
    3. At time \(t\) seconds \(( t \geqslant 0 )\), a particle \(P\) has position vector \(\mathbf { p }\) metres, with respect to a fixed origin \(O\), where
    $$\mathbf { p } = \left( 3 t ^ { 2 } - 6 t + 4 \right) \mathbf { i } + \left( 3 t ^ { 3 } - 4 t \right) \mathbf { j } .$$ Find
  2. the velocity of \(P\) at time \(t\) seconds,
  3. the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\). When \(t = 1\), the particle \(P\) receives an impulse of \(( 2 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\). Given that the mass of \(P\) is 0.5 kg ,
  4. find the velocity of \(P\) immediately after the impulse.
Edexcel M2 2008 January Q3
9 marks Standard +0.3
3. A car of mass 1000 kg is moving at a constant speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal. The rate of working of the engine of the car is 20 kW and the resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 550 N .
  1. Show that \(\sin \theta = \frac { 1 } { 14 }\). When the car is travelling up the road at \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car comes to rest, without braking, having moved a distance \(y\) metres from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 550 N .
  2. Find the value of \(y\).
Edexcel M2 2008 January Q4
12 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7a2bf93-a7fc-43c5-b317-109320f633ba-05_783_1231_237_358} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(A B C\), with \(\angle A B C = 90 ^ { \circ } , A B = 12 \mathrm {~cm}\) and \(B C = 21 \mathrm {~cm}\). The point \(O\) is 5 cm from \(A B\) and 5 cm from \(B C\), as shown in Figure 1.
  1. Find the distance of the centre of mass of \(S\) from
    1. \(A B\),
    2. \(B C\). The set square is freely suspended from \(C\) and hangs in equilibrium.
  2. Find, to the nearest degree, the angle between \(C B\) and the vertical.
Edexcel M2 2008 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7a2bf93-a7fc-43c5-b317-109320f633ba-08_678_568_239_703} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A ladder \(A B\), of mass \(m\) and length \(4 a\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. A load of mass \(3 m\) is fixed on the ladder at the point \(C\), where \(A C = a\). The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground.
Edexcel M2 2008 January Q6
13 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7a2bf93-a7fc-43c5-b317-109320f633ba-10_611_748_246_534} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertical.] A particle \(P\) is projected from the point \(A\) which has position vector 47.5j metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \(( 2 u \mathbf { i } + 5 u \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30 \mathbf { i }\) metres, as shown in Figure 3.
  1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s .
  2. Find the value of \(u\).
  3. Find the speed of \(P\) at \(B\).
Edexcel M2 2008 January Q7
17 marks Standard +0.8
  1. A particle \(P\) of mass \(2 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal plane. A particle \(Q\) of mass \(3 m\) is moving with speed \(u\) in the same direction as \(P\). The particles collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 2 }\).
    1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 8 } { 5 } u\).
    2. Find the total kinetic energy lost in the collision.
    After the collision between \(P\) and \(Q\), the particle \(Q\) collides directly with a particle \(R\) of mass \(m\) which is at rest on the plane. The coefficient of restitution between \(Q\) and \(R\) is \(e\).
  2. Calculate the range of values of \(e\) for which there will be a second collision between \(P\) and \(Q\).
Edexcel M2 2009 January Q1
5 marks Standard +0.3
  1. A car of mass 1500 kg is moving up a straight road, which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to the motion of the car from non-gravitational forces is constant and is modelled as a single constant force of magnitude 650 N . The car's engine is working at a rate of 30 kW .
Find the acceleration of the car at the instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2009 January Q2
10 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8ebad3-0ebb-4dfe-8036-54b651deb9cf-03_602_554_205_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a ladder \(A B\), of mass 25 kg and length 4 m , resting in equilibrium with one end \(A\) on rough horizontal ground and the other end \(B\) against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(\frac { 11 } { 25 }\). The ladder makes an angle \(\beta\) with the ground. When Reece, who has mass 75 kg , stands at the point \(C\) on the ladder, where \(A C = 2.8 \mathrm {~m}\), the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle.
  1. Find the magnitude of the frictional force of the ground on the ladder.
  2. Find, to the nearest degree, the value of \(\beta\).
  3. State how you have used the modelling assumption that Reece is a particle.
Edexcel M2 2009 January Q3
8 marks Moderate -0.3
  1. A block of mass 10 kg is pulled along a straight horizontal road by a constant horizontal force of magnitude 70 N in the direction of the road. The block moves in a straight line passing through two points \(A\) and \(B\) on the road, where \(A B = 50 \mathrm {~m}\). The block is modelled as a particle and the road is modelled as a rough plane. The coefficient of friction between the block and the road is \(\frac { 4 } { 7 }\).
    1. Calculate the work done against friction in moving the block from \(A\) to \(B\).
    The block passes through \(A\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the speed of the block at \(B\).
Edexcel M2 2009 January Q4
8 marks Standard +0.3
4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = \begin{cases} 10 t - 2 t ^ { 2 } , & 0 \leqslant t \leqslant 6 \\ \frac { - 432 } { t ^ { 2 } } , & t > 6 \end{cases}$$ At \(t = 0 , P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when
  1. \(t = 6\),
  2. \(t = 10\).
Edexcel M2 2009 January Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8ebad3-0ebb-4dfe-8036-54b651deb9cf-08_781_541_223_687} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform lamina \(A B C D\) is made by joining a uniform triangular lamina \(A B D\) to a uniform semi-circular lamina \(D B C\), of the same material, along the edge \(B D\), as shown in Figure 2. Triangle \(A B D\) is right-angled at \(D\) and \(A D = 18 \mathrm {~cm}\). The semi-circle has diameter \(B D\) and \(B D = 12 \mathrm {~cm}\).
  1. Show that, to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(A D\) is 4.69 cm . Given that the centre of mass of a uniform semicircular lamina, radius \(r\), is at a distance \(\frac { 4 r } { 3 \pi }\) from the centre of the bounding diameter,
  2. find, in cm to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(B D\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
  3. Find, to the nearest degree, the angle which \(B D\) makes with the vertical.
Edexcel M2 2009 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8ebad3-0ebb-4dfe-8036-54b651deb9cf-10_506_1361_205_299} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A cricket ball is hit from a point \(A\) with velocity of \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), at an angle \(\alpha\) above the horizontal. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertically upwards. The point \(A\) is 0.9 m vertically above the point \(O\), which is on horizontal ground. The ball takes 3 seconds to travel from \(A\) to \(B\), where \(B\) is on the ground and \(O B = 57.6 \mathrm {~m}\), as shown in Figure 3. By modelling the motion of the cricket ball as that of a particle moving freely under gravity,
  1. find the value of \(p\),
  2. show that \(q = 14.4\),
  3. find the initial speed of the cricket ball,
  4. find the exact value of \(\tan \alpha\).
  5. Find the length of time for which the cricket ball is at least 4 m above the ground.
  6. State an additional physical factor which may be taken into account in a refinement of the above model to make it more realistic.
Edexcel M2 2009 January Q7
17 marks Standard +0.3
  1. A particle \(P\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal table. It collides directly with another particle \(Q\) of mass \(2 m\) which is moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 1 } { 5 } ( 9 e + 4 ) u\).
    The speed of \(P\) immediately after the collision is \(\frac { 1 } { 2 } u\).
  2. Show that \(e = \frac { 1 } { 4 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision \(Q\) hits a smooth fixed vertical wall which is at right-angles to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\).
  3. Show that \(P\) is a distance \(\frac { 3 } { 5 } d\) from the wall at the instant when \(Q\) hits the wall. Particle \(Q\) rebounds from the wall and moves so as to collide directly with particle \(P\) at the point \(B\). Given that the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 5 }\),
  4. find, in terms of \(d\), the distance of the point \(B\) from the wall.