Questions — Edexcel M2 (551 questions)

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Edexcel M2 2013 June Q3
3 A particle \(P\) of mass 0.25 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = ( 2 - 4 t ) \mathbf { i } + \left( t ^ { 2 } + 2 t \right) \mathbf { j }$$ When \(t = 0 , P\) is at the point with position vector ( \(2 \mathbf { i } - 4 \mathbf { j }\) ) m with respect to a fixed origin \(O\). When \(t = 3 , P\) is at the point \(A\). Find
  1. the momentum of \(P\) when \(t = 3\),
  2. the magnitude of \(\mathbf { F }\) when \(t = 3\),
  3. the position vector of \(A\).
Edexcel M2 2013 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-5_496_1264_316_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(O\) and \(B\) are on horizontal ground. The point \(A\) is \(h\) metres vertically above \(O\). A particle \(P\) is projected from \(A\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha ^ { \circ }\) to the horizontal. The particle moves freely under gravity and hits the ground at \(B\), as shown in Figure 1. The speed of \(P\) immediately before it hits the ground is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, find the value of \(h\). Given that 1.5 s after it is projected from \(A , P\) is at a point 4 m above the level of \(A\), find
  2. the value of \(\alpha\),
  3. the direction of motion of \(P\) immediately before it reaches \(B\).
Edexcel M2 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-6_501_696_316_726} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform L-shaped lamina \(O A B C D E\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from
  1. \(O E\),
  2. \(O A\). The lamina is freely suspended from \(O\) and hangs in equilibrium with \(O E\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac { 4 } { 3 }\).
  3. Find the value of \(a\).
Edexcel M2 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-7_574_1257_214_448} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\) has weight 30 N and length 3 m . The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at \(15 ^ { \circ }\) to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.
  1. Show that the normal reaction at \(P\) has magnitude 25 N .
  2. Find the magnitude of the force on the rod at \(A\). The coefficient of friction between the rod and the peg is \(\mu\).
  3. Find the range of possible values of \(\mu\).
Edexcel M2 2013 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2656d4b4-7f47-48db-9d7e-07db6ecb8606-8_453_839_219_649} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two smooth particles \(P\) and \(Q\) have masses \(m\) and \(2 m\) respectively. The particles are moving in the same direction in the same straight line, on a smooth horizontal plane, with \(Q\) in front of \(P\). The particles are moving towards a fixed smooth vertical wall which is perpendicular to the direction of motion of the particles, as shown in Figure 4. The speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\). Particle \(Q\) strikes the wall, rebounds and then collides directly with \(P\). The direction of motion of each particle is reversed by this collision. Immediately after this collision the speed of \(P\) is \(v\) and the speed of \(Q\) is \(w\).
  1. Show that \(v = 2 w\). The total kinetic energy of \(P\) and \(Q\) immediately after they collide is half the total kinetic energy of \(P\) and \(Q\) immediately before they collide.
  2. Find the coefficient of restitution between \(P\) and \(Q\).
Edexcel M2 2013 June Q1
  1. A particle \(P\) of mass 2 kg is moving with velocity \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 3 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\).
Find the speed of \(P\) immediately after the impulse is applied.
(5)
Edexcel M2 2013 June Q2
2. A particle \(P\) of mass 3 kg moves from point \(A\) to point \(B\) up a line of greatest slope of a fixed rough plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 Given that \(A B = 15 \mathrm {~m}\) and that the speed of \(P\) at \(A\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find
  1. the work done against friction as \(P\) moves from \(A\) to \(B\),
  2. the speed of \(P\) at \(B\).
Edexcel M2 2013 June Q3
3. 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 $$v = 2 t ^ { 2 } - 14 t + 20 , \quad t \geqslant 0$$ Find
  1. the times when \(P\) is instantaneously at rest,
  2. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\)
  3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
Edexcel M2 2013 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-06_736_725_258_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B C D E F\) is a regular hexagon with centre \(O\) and sides of length 2 m , as shown in Figure 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-06_574_723_1288_605} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The triangles \(O A F\) and \(O E F\) are removed to form the uniform lamina \(O A B C D E\), shown in Figure 2.
  1. Find the distance of the centre of mass of \(O A B C D E\) from \(O\). The lamina \(O A B C D E\) is freely suspended from \(E\) and hangs in equilibrium.
  2. Find the size of the angle between \(E O\) and the downward vertical.
Edexcel M2 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-09_522_997_276_477} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point \(A\). A particle of mass \(m\) is attached to the rod at \(B\). The rod is held in equilibrium at an angle \(\theta\) to the horizontal by a force of magnitude \(F\) acting at the point \(C\) on the rod, where \(A C = b\), as shown in Figure 3. The force at \(C\) acts at right angles to \(A B\) and in the vertical plane containing \(A B\).
  1. Show that \(F = \frac { 3 a m g \cos \theta } { b }\).
  2. Find, in terms of \(a , b , g , m\) and \(\theta\),
    1. the horizontal component of the force acting on the rod at \(A\),
    2. the vertical component of the force acting on the rod at \(A\). Given that the force acting on the rod at \(A\) acts along the rod,
  3. find the value of \(\frac { a } { b }\).
Edexcel M2 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf960066-46b8-42a3-8a8b-d8deb76e7c70-11_694_1004_264_529} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A ball is projected from a point \(A\) which is 8 m above horizontal ground as shown in Figure 4. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\). The speed of the ball immediately before it hits the ground is \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, find the value of \(u\). The time taken for the ball to move from \(A\) to \(B\) is 2 seconds. Find
  2. the value of \(\theta\),
  3. the minimum speed of the ball on its path from \(A\) to \(B\).
Edexcel M2 2013 June Q7
7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). The particles \(P , Q\) and \(R\) have masses \(2 m , 3 m\) and \(4 m\) respectively. Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with it. The coefficient of restitution between each pair of particles is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision with \(P\) is \(\frac { 2 } { 5 } ( 1 + e ) u\). After the collision between \(P\) and \(Q\) there is a direct collision between \(Q\) and \(R\).
    Given that \(e = \frac { 3 } { 4 }\), find
    1. the speed of \(Q\) after this collision,
    2. the speed of \(R\) after this collision. Immediately after the collision between \(Q\) and \(R\), the rate of increase of the distance between \(P\) and \(R\) is \(V\).
  3. Find \(V\) in terms of \(u\).
Edexcel M2 2014 June Q1
  1. A van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 16 }\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the van’s engine is working at a constant rate of 25 kW .
    1. Find the value of \(R\).
    The power developed by the van’s engine is now increased to 30 kW . The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(a\).
Edexcel M2 2014 June Q2
  1. A ball of mass 0.4 kg is moving in a horizontal plane when it is struck by a bat. The bat exerts an impulse \(( - 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) s on the ball. Immediately after receiving the impulse the ball has velocity \(( 12 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find
  1. the speed of the ball immediately before the impact,
  2. the size of the angle through which the direction of motion of the ball is deflected by the impact.
Edexcel M2 2014 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-05_617_604_226_678} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform rod, \(A B\), of mass \(m\) and length 2l, rests in equilibrium with one end \(A\) on a rough horizontal floor and the other end \(B\) against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of \(60 ^ { \circ }\) with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the wall is \(\frac { 2 } { 3 }\). The rod is on the point of slipping at both ends.
  1. Find the magnitude of the vertical component of the force exerted on the rod by the floor. The centre of mass of the rod is at \(G\).
  2. Find the distance \(A G\).
Edexcel M2 2014 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-07_737_823_223_532} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a lamina \(L\). It is formed by removing a square \(P Q R S\) from a uniform triangle \(A B C\). The triangle \(A B C\) is isosceles with \(A C = B C\) and \(A B = 12 \mathrm {~cm}\). The midpoint of \(A B\) is \(D\) and \(D C = 8 \mathrm {~cm}\). The vertices \(P\) and \(Q\) of the square lie on \(A B\) and \(P Q = 4 \mathrm {~cm}\). The centre of the square is \(O\). The centre of mass of \(L\) is at \(G\).
  1. Find the distance of \(G\) from \(A B\). When \(L\) is freely suspended from \(A\) and hangs in equilibrium, the line \(A B\) is inclined at \(25 ^ { \circ }\) to the vertical.
  2. Find the distance of \(O\) from \(D C\).
Edexcel M2 2014 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-09_460_974_242_484} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 2 kg is released from rest at a point \(A\) on a rough inclined plane and slides down a line of greatest slope. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) is 5 m from \(A\) on the line of greatest slope through \(A\), as shown in Figure 3.
  1. Find the potential energy lost by \(P\) as it moves from \(A\) to \(B\). The speed of \(P\) as it reaches \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Use the work-energy principle to find the magnitude of the constant frictional force acting on \(P\) as it moves from \(A\) to \(B\).
    2. Find the coefficient of friction between \(P\) and the plane. The particle \(P\) is now placed at \(A\) and projected down the plane towards \(B\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the frictional force remains constant,
  3. find the speed of \(P\) as it reaches \(B\).
Edexcel M2 2014 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82cadc37-4cb0-455e-9531-e09ec0c19533-11_711_917_219_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) is projected from a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 4 } { 5 }\). The point \(A\) is 10 m vertically above the point \(O\) which is on horizontal ground, as shown in Figure 4. The particle \(P\) moves freely under gravity and reaches the ground at the point \(B\). Calculate
  1. the greatest height above the ground of \(P\), as it moves from \(A\) to \(B\),
  2. the distance \(O B\). The point \(C\) lies on the path of \(P\). The direction of motion of \(P\) at \(C\) is perpendicular to the direction of motion of \(P\) at \(A\).
  3. Find the time taken by \(P\) to move from \(A\) to \(C\).
Edexcel M2 2014 June Q7
7. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal table. A second particle \(Q\) of mass \(3 m\) is moving in the opposite direction to \(P\) along the same straight line with speed \(u\). The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 5 } ( 8 e + 3 )\)
  2. Find the range of possible values of \(e\). The total kinetic energy of the particles before the collision is \(T\). The total kinetic energy of the particles after the collision is \(k T\). Given that \(e = \frac { 1 } { 2 }\)
  3. find the value of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{82cadc37-4cb0-455e-9531-e09ec0c19533-14_104_61_2407_1836}
Edexcel M2 2014 June Q1
  1. Three particles of mass \(3 m , 2 m\) and \(k m\) are placed at the points whose coordinates are \(( 1,5 ) , ( 6,4 )\) and \(( a , 1 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( 3,3 )\).
Find
  1. the value of \(k\),
  2. the value of \(a\).
Edexcel M2 2014 June Q2
2. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) is moving on a horizontal plane with acceleration \(\left[ \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + ( 6 t - 5 ) \mathbf { j } \right] \mathrm { m } \mathrm { s } ^ { - 2 }\). When \(t = 3\) the velocity of \(P\) is \(( 11 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the velocity of \(P\) at time \(t\) seconds,
  2. the speed of \(P\) when it is moving parallel to the vector \(\mathbf { i }\).
Edexcel M2 2014 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{47420c50-c232-41e9-8c4d-a890d86ea933-04_814_1127_219_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B C D E F\), shown shaded in Figure 1, is symmetrical about the line through \(B\) and \(E\). It is formed by removing the isosceles triangle \(F E D\), of height \(6 a\) and base \(8 a\), from the isosceles triangle \(A B C\) of height \(9 a\) and base \(12 a\).
  1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(A C\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find, to the nearest degree, the size of the angle between \(A B\) and the downward vertical.
Edexcel M2 2014 June Q4
  1. A truck of mass 1800 kg is towing a trailer of mass 800 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 20 }\). The truck is connected to the trailer by a light inextensible rope which is parallel to the direction of motion of the truck. The resistances to motion of the truck and the trailer from non-gravitational forces are modelled as constant forces of magnitudes 300 N and 200 N respectively. The truck is moving at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine of the truck is working at a rate of 40 kW .
    1. Find the value of \(v\).
    As the truck is moving up the road the rope breaks.
  2. Find the acceleration of the truck immediately after the rope breaks.
Edexcel M2 2014 June Q5
5. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface. Initially the particle is at rest at a point \(O\) midway between a pair of fixed parallel vertical walls. The walls are 2 m apart. At time \(t = 0\) the particle is projected from \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\). The magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u \mathrm {~N} \mathrm {~s}\).
  1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 3\) seconds.
  2. Find the value of \(u\).
Edexcel M2 2014 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{47420c50-c232-41e9-8c4d-a890d86ea933-10_645_1196_125_351} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small ball is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on horizontal ground. The angle of projection is \(\alpha\) above the horizontal. A horizontal platform is at height \(h\) metres above the ground. The ball moves freely under gravity until it hits the platform at the point B, as shown in Figure 2. The speed of the ball immediately before it hits the platform at \(B\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(h\). Given that \(\sin \alpha = 0.85\),
  2. find the horizontal distance from \(A\) to \(B\).