Questions — Edexcel (9685 questions)

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Edexcel M2 2004 June Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{8e694174-b9a9-4018-8896-31a3b4f0d344-5_424_1324_264_383}
\end{figure} In a ski-jump competition, a skier of mass 80 kg moves from rest at a point \(A\) on a ski-slope. The skier's path is an arc \(A B\). The starting point \(A\) of the slope is 32.5 m above horizontal ground. The end \(B\) of the slope is 8.1 m above the ground. When the skier reaches \(B\), she is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and moving upwards at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Fig. 2. The distance along the slope from \(A\) to \(B\) is 60 m . The resistance to motion while she is on the slope is modelled as a force of constant magnitude \(R\) newtons. By using the work-energy principle,
  1. find the value of \(R\). On reaching \(B\), the skier then moves through the air and reaches the ground at the point \(C\). The motion of the skier in moving from \(B\) to \(C\) is modelled as that of a particle moving freely under gravity.
  2. Find the time for the skier to move from \(B\) to \(C\).
  3. Find the horizontal distance from \(B\) to \(C\).
  4. Find the speed of the skier immediately before she reaches \(C\). END
Edexcel M2 2006 June Q1
6 marks Moderate -0.3
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, its acceleration is \(( 5 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the direction of \(x\) increasing. When \(t = 0\), its velocity is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) measured in the direction of \(x\) increasing. Find the time when \(P\) is instantaneously at rest in the subsequent motion.
  2. A car of mass 1200 kg moves along a straight horizontal road with a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion of the car has magnitude 600 N .
    1. Find, in kW , the rate at which the engine of the car is working.
    The car now moves up a hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 28 }\). The resistance to motion of the car from non-gravitational forces remains of magnitude 600 N . The engine of the car now works at a rate of 30 kW .
  3. Find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2006 June Q3
8 marks Moderate -0.8
3. A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \(( - 30 \mathbf { i } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being struck, the velocity of the ball is \(( 16 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the magnitude of the impulse exerted on the ball by the bat. In the subsequent motion, the position vector of the ball is \(\mathbf { r }\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf { r } = \left[ 16 t \mathbf { i } + \left( 20 t - 5 t ^ { 2 } \right) \mathbf { j } \right]\). Using this model,
  2. find the speed of the ball when \(t = 3\).
Edexcel M2 2006 June Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{223b98fa-6e19-41de-85dc-2974d1529af1-05_383_748_296_667}
\end{figure} Figure 1 shows four uniform rods joined to form a rigid rectangular framework \(A B C D\), where \(A B = C D = 2 a\), and \(B C = A D = 3 a\). Each rod has mass \(m\). Particles, of mass \(6 m\) and \(2 m\), are attached to the framework at points \(C\) and \(D\) respectively.
  1. Find the distance of the centre of mass of the loaded framework from
    1. \(A B\),
    2. \(A D\). The loaded framework is freely suspended from \(B\) and hangs in equilibrium.
  2. Find the angle which \(B C\) makes with the vertical.
Edexcel M2 2006 June Q5
8 marks Standard +0.3
5. A vertical cliff is 73.5 m high. Two stones \(A\) and \(B\) are projected simultaneously. Stone \(A\) is projected horizontally from the top of the cliff with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Stone \(B\) is projected from the bottom of the cliff with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The stones move freely under gravity in the same vertical plane and collide in mid-air. By considering the horizontal motion of each stone,
  1. prove that \(\cos \alpha = \frac { 4 } { 5 }\).
  2. Find the time which elapses between the instant when the stones are projected and the instant when they collide.
Edexcel M2 2006 June Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{223b98fa-6e19-41de-85dc-2974d1529af1-08_314_1171_301_397}
\end{figure} A wooden plank \(A B\) has mass \(4 m\) and length \(4 a\). The end \(A\) of the plank lies on rough horizontal ground. A small stone of mass \(m\) is attached to the plank at \(B\). The plank is resting on a small smooth horizontal peg \(C\), where \(B C = a\), as shown in Figure 2. The plank is in equilibrium making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between the plank and the ground is \(\mu\). The plank is modelled as a uniform rod lying in a vertical plane perpendicular to the peg, and the stone as a particle. Show that
  1. the reaction of the peg on the plank has magnitude \(\frac { 16 } { 5 } \mathrm { mg }\),
  2. \(\mu \geqslant \frac { 48 } { 61 }\).
  3. State how you have used the information that the peg is smooth.
    □ \includegraphics[max width=\textwidth, alt={}, center]{223b98fa-6e19-41de-85dc-2974d1529af1-09_156_136_2597_1822}
Edexcel M2 2006 June Q7
12 marks Standard +0.3
7. A particle \(P\) has mass 4 kg . It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 7 }\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(A B = 2.5 \mathrm {~m}\). It then moves back down the plane to \(A\).
  1. Find the work done by friction as \(P\) moves from \(A\) to \(B\).
  2. Using the work-energy principle, find the speed with which \(P\) is projected from \(A\).
  3. Find the speed of \(P\) when it returns to \(A\).
Edexcel M2 2006 June Q8
15 marks Standard +0.8
8. Two particles \(A\) and \(B\) move on a smooth horizontal table. The mass of \(A\) is \(m\), and the mass of \(B\) is \(4 m\). Initially \(A\) is moving with speed \(u\) when it collides directly with \(B\), which is at rest on the table. As a result of the collision, the direction of motion of \(A\) is reversed. The coefficient of restitution between the particles is \(e\).
  1. Find expressions for the speed of \(A\) and the speed of \(B\) immediately after the collision. In the subsequent motion, \(B\) strikes a smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 4 } { 5 }\). Given that there is a second collision between \(A\) and \(B\),
  2. show that \(\frac { 1 } { 4 } < e < \frac { 9 } { 16 }\). Given that \(e = \frac { 1 } { 2 }\),
  3. find the total kinetic energy lost in the first collision between \(A\) and \(B\).
Edexcel M2 2007 June Q1
4 marks Moderate -0.3
  1. A cyclist and his bicycle have a combined mass of 90 kg . He rides on a straight road up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\). He works at a constant rate of 444 W and cycles up the hill at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the magnitude of the resistance to motion from non-gravitational forces as he cycles up the hill.
Edexcel M2 2007 June Q2
6 marks Moderate -0.8
2. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } + ( 1 - 4 t ) \mathbf { j }$$ Find
  1. the acceleration of \(P\) at time \(t\) seconds,
  2. the magnitude of \(\mathbf { F }\) when \(t = 2\).
Edexcel M2 2007 June Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-04_568_568_205_685} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina \(A B C D E F\) is formed by taking a uniform sheet of card in the form of a square \(A X E F\), of side \(2 a\), and removing the square \(B X D C\) of side \(a\), where \(B\) and \(D\) are the mid-points of \(A X\) and \(X E\) respectively, as shown in Figure 1.
  1. Find the distance of the centre of mass of the lamina from \(A F\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle which \(A F\) makes with the vertical.
Edexcel M2 2007 June Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-06_330_1118_203_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).
  1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
  2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.
Edexcel M2 2007 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-08_376_874_205_525} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform beam \(A B\) of mass 2 kg is freely hinged at one end \(A\) to a vertical wall. The beam is held in equilibrium in a horizontal position by a rope which is attached to a point \(C\) on the beam, where \(A C = 0.14 \mathrm {~m}\). The rope is attached to the point \(D\) on the wall vertically above \(A\), where \(\angle A C D = 30 ^ { \circ }\), as shown in Figure 3. The beam is modelled as a uniform rod and the rope as a light inextensible string. The tension in the rope is 63 N . Find
  1. the length of \(A B\),
  2. the magnitude of the resultant reaction of the hinge on the beam at \(A\).
Edexcel M2 2007 June Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-10_447_908_205_516} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A golf ball \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on a cliff above horizontal ground. The angle of projection is \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 4.
  1. Find the greatest height of \(P\) above the level of \(A\). The horizontal distance from \(A\) to \(B\) is 168 m .
  2. Find the height of \(A\) above the ground. By considering energy, or otherwise,
  3. find the speed of \(P\) as it hits the ground at \(B\).
Edexcel M2 2007 June Q7
13 marks Standard +0.8
  1. Two small spheres \(P\) and \(Q\) of equal radius have masses \(m\) and \(5 m\) respectively. They lie on a smooth horizontal table. Sphere \(P\) is moving with speed \(u\) when it collides directly with sphere \(Q\) which is at rest. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 1 } { 5 }\).
    1. (i) Show that the speed of \(P\) immediately after the collision is \(\frac { u } { 6 } ( 5 e - 1 )\).
      (ii) Find an expression for the speed of \(Q\) immediately after the collision, giving your answer in the form \(\lambda u\), where \(\lambda\) is in terms of \(e\).
      (6)
    Three small spheres \(A , B\) and \(C\) of equal radius lie at rest in a straight line on a smooth horizontal table, with \(B\) between \(A\) and \(C\). The spheres \(A\) and \(C\) each have mass \(5 m\), and the mass of \(B\) is \(m\). Sphere \(B\) is projected towards \(C\) with speed \(u\). The coefficient of restitution between each pair of spheres is \(\frac { 4 } { 5 }\).
  2. Show that, after \(B\) and \(C\) have collided, there is a collision between \(B\) and \(A\).
  3. Determine whether, after \(B\) and \(A\) have collided, there is a further collision between \(B\) and \(C\).
Edexcel M2 2007 June Q8
16 marks Standard +0.3
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v\) is given by
$$v = \left\{ \begin{array} { l c } 8 t - \frac { 3 } { 2 } t ^ { 2 } , & 0 \leqslant t \leqslant 4 , \\ 16 - 2 t , & t > 4 . \end{array} \right.$$ When \(t = 0 , P\) is at the origin \(O\).
Find
  1. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\),
  2. the distance of \(P\) from \(O\) when \(t = 4\),
  3. the time at which \(P\) is instantaneously at rest for \(t > 4\),
  4. the total distance travelled by \(P\) in the first 10 s of its motion.
Edexcel M2 2008 June Q1
6 marks Moderate -0.8
  1. A lorry of mass 2000 kg is moving down a straight road inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The resistance to motion is modelled as a constant force of magnitude 1600 N . The lorry is moving at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find, in kW , the rate at which the lorry's engine is working.
Edexcel M2 2008 June Q2
9 marks Standard +0.3
2. A particle \(A\) of mass \(4 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle \(A\) collides directly with a particle \(B\) of mass \(3 m\) moving with speed \(2 u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Immediately after the collision the speed of \(B\) is \(4 e u\).
  1. Show that \(e = \frac { 3 } { 4 }\).
  2. Find the total kinetic energy lost in the collision.
Edexcel M2 2008 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-04_511_922_260_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The package slides down a line of greatest slope of the plane from a point \(A\) to a point \(B\), where \(A B = 14 \mathrm {~m}\). At \(A\) the package has speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the package has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Find
  1. the total energy lost by the package in travelling from \(A\) to \(B\),
  2. the coefficient of friction between the package and the ramp.
Edexcel M2 2008 June Q4
12 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds,
$$\mathbf { F } = ( 6 t - 5 ) \mathbf { i } + \left( t ^ { 2 } - 2 t \right) \mathbf { j }$$ The velocity of \(P\) at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = \mathbf { i } - 4 \mathbf { j }\).
  1. Find \(\mathbf { v }\) at time \(t\) seconds. When \(t = 3\), the particle \(P\) receives an impulse ( \(- 5 \mathbf { i } + 12 \mathbf { j }\) ) N s.
  2. Find the speed of \(P\) immediately after it receives the impulse.
Edexcel M2 2008 June Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-07_501_918_274_502} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod \(A B\) and the pole as a smooth horizontal peg perpendicular to the vertical plane containing \(A B\). The rod has length \(3 a\) and weight \(W\) and rests on the peg at \(C\), where \(A C = 2 a\). The end \(A\) of the rod rests on rough horizontal ground and \(A B\) makes an angle \(\alpha\) with the ground, as shown in Figure 2.
  1. Show that the normal reaction on the rod at \(A\) is \(\frac { 1 } { 4 } \left( 4 - 3 \cos ^ { 2 } \alpha \right) W\). Given that the rod is in limiting equilibrium and that \(\cos \alpha = \frac { 2 } { 3 }\),
  2. find the coefficient of friction between the rod and the ground.
Edexcel M2 2008 June Q6
13 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-09_600_968_292_486} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a rectangular lamina \(O A B C\). The coordinates of \(O , A , B\) and \(C\) are ( 0,0 ), \(( 8,0 ) , ( 8,5 )\) and \(( 0,5 )\) respectively. Particles of mass \(k m , 5 m\) and \(3 m\) are attached to the lamina at \(A , B\) and \(C\) respectively. The \(x\)-coordinate of the centre of mass of the three particles without the lamina is 6.4.
  1. Show that \(k = 7\). The lamina \(O A B C\) is uniform and has mass \(12 m\).
  2. Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina. The combined system is freely suspended from \(O\) and hangs at rest.
  3. Find the angle between \(O C\) and the horizontal.
Edexcel M2 2008 June Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-11_755_1073_246_287} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A ball is thrown from a point \(A\) at a target, which is on horizontal ground. The point \(A\) is 12 m above the point \(O\) on the ground. The ball is thrown from \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal. The ball is modelled as a particle and the target as a point \(T\). The distance \(O T\) is 15 m . The ball misses the target and hits the ground at the point \(B\), where \(O T B\) is a straight line, as shown in Figure 4. Find
  1. the time taken by the ball to travel from \(A\) to \(B\),
  2. the distance \(T B\). The point \(X\) is on the path of the ball vertically above \(T\).
  3. Find the speed of the ball at \(X\).
Edexcel M2 2009 June Q1
5 marks Moderate -0.5
  1. A particle of mass 0.25 kg is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives the impulse \(( 5 \mathbf { i } - 3 \mathbf { j } )\) N s.
Find the speed of the particle immediately after the impulse.
Edexcel M2 2009 June Q2
9 marks Moderate -0.8
2. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 8 t - t ^ { 2 }$$
  1. Find the maximum value of \(v\).
  2. Find the time taken for \(P\) to return to \(O\).