Questions — Edexcel M2 (623 questions)

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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 \(O\).
Edexcel M2 2009 June Q3
6 marks Moderate -0.3
3. A truck of mass of 300 kg moves along a straight horizontal road with a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion of the truck has magnitude 120 N .
  1. Find the rate at which the engine of the truck is working. On another occasion the truck moves at a constant speed up a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to motion of the truck from non-gravitational forces remains of magnitude 120 N . The rate at which the engine works is the same as in part (a).
  2. Find the speed of the truck.
Edexcel M2 2009 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e220b8a-46f1-4b9b-88a4-f032c7fbda50-05_568_956_205_516} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\), of length 1.5 m and mass 3 kg , is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium in a horizontal position by a light strut \(C D\) as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The end \(C\) of the strut is freely jointed to the wall at a point 0.5 m vertically below \(A\). The end \(D\) is freely joined to the rod so that \(A D\) is 0.5 m .
  1. Find the thrust in \(C D\).
  2. Find the magnitude and direction of the force exerted on the \(\operatorname { rod } A B\) at \(A\).
Edexcel M2 2009 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e220b8a-46f1-4b9b-88a4-f032c7fbda50-07_564_910_207_523} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A shop sign \(A B C D E F G\) is modelled as a uniform lamina, as illustrated in Figure 2. \(A B C D\) is a rectangle with \(B C = 120 \mathrm {~cm}\) and \(D C = 90 \mathrm {~cm}\). The shape \(E F G\) is an isosceles triangle with \(E G = 60 \mathrm {~cm}\) and height 60 cm . The mid-point of \(A D\) and the mid-point of \(E G\) coincide.
  1. Find the distance of the centre of mass of the sign from the side \(A D\). The sign is freely suspended from \(A\) and hangs at rest.
  2. Find the size of the angle between \(A B\) and the vertical.
Edexcel M2 2009 June Q6
12 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e220b8a-46f1-4b9b-88a4-f032c7fbda50-09_323_1018_274_452} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A child playing cricket on horizontal ground hits the ball towards a fence 10 m away. The ball moves in a vertical plane which is perpendicular to the fence. The ball just passes over the top of the fence, which is 2 m above the ground, as shown in Figure 3. The ball is modelled as a particle projected with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from point \(O\) on the ground at an angle \(\alpha\) to the ground.
  1. By writing down expressions for the horizontal and vertical distances, from \(O\) of the ball \(t\) seconds after it was hit, show that $$2 = 10 \tan \alpha - \frac { 50 g } { u ^ { 2 } \cos ^ { 2 } \alpha }$$ Given that \(\alpha = 45 ^ { \circ }\),
  2. find the speed of the ball as it passes over the fence.
Edexcel M2 2009 June Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8e220b8a-46f1-4b9b-88a4-f032c7fbda50-11_501_1018_116_468} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass 2 kg is projected up a rough plane with initial speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point \(X\) on the plane, as shown in Figure 4. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 8 }\).
  1. Use the work-energy principle to show that \(X Y = 25 \mathrm {~m}\). After reaching \(Y\), the particle \(P\) slides back down the plane.
  2. Find the speed of \(P\) as it passes through \(X\).
Edexcel M2 2009 June Q8
12 marks Standard +0.3
  1. Particles \(A , B\) and \(C\) of masses \(4 m , 3 m\) and \(m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected towards each other with speeds \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, and collide directly.
As a result of the collision, \(A\) is brought to rest and \(B\) rebounds with speed \(k v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).
  1. Show that \(u = 3 v\).
  2. Find the value of \(k\). Immediately after the collision between \(A\) and \(B\), particle \(C\) is projected with speed \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) so that \(B\) and \(C\) collide directly.
  3. Show that there is no further collision between \(A\) and \(B\).
Edexcel M2 2012 June Q1
10 marks Moderate -0.5
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A particle \(P\) moves in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 1 \right) \mathbf { i } + \left( 4 t - t ^ { 2 } \right) \mathbf { j }$$
  1. Find the magnitude of the acceleration of \(P\) when \(t = 1\) Given that, when \(t = 0\), the position vector of \(P\) is i metres,
  2. find the position vector of \(P\) when \(t = 3\)
Edexcel M2 2012 June Q2
11 marks Moderate -0.3
2. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4 m\) moving on the plane with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Find the speed of \(Q\) immediately after the collision. Given that the direction of motion of \(P\) is reversed by the collision,
  2. find the range of possible values of \(e\).
Edexcel M2 2012 June Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-04_374_798_255_559} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\), of mass 5 kg and length 4 m , has its end \(A\) smoothly hinged at a fixed point. The rod is held in equilibrium at an angle of \(25 ^ { \circ }\) above the horizontal by a force of magnitude \(F\) newtons applied to its end \(B\). The force acts in the vertical plane containing the rod and in a direction which makes an angle of \(40 ^ { \circ }\) with the rod, as shown in Figure 1.
  1. Find the value of \(F\).
  2. Find the magnitude and direction of the vertical component of the force acting on the rod at \(A\).
Edexcel M2 2012 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-06_796_789_276_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform circular disc has centre \(O\) and radius 4a. The lines \(P Q\) and \(S T\) are perpendicular diameters of the disc. A circular hole of radius \(2 a\) is made in the disc, with the centre of the hole at the point \(R\) on \(O P\) where \(O R = 2 a\), to form the lamina \(L\), shown shaded in Figure 2.
  1. Show that the distance of the centre of mass of \(L\) from \(P\) is \(\frac { 14 a } { 3 }\). The mass of \(L\) is \(m\) and a particle of mass \(k m\) is now fixed to \(L\) at the point \(P\). The system is now suspended from the point \(S\) and hangs freely in equilibrium. The diameter \(S T\) makes an angle \(\alpha\) with the downward vertical through \(S\), where \(\tan \alpha = \frac { 5 } { 6 }\).
  2. Find the value of \(k\).
Edexcel M2 2012 June Q5
6 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-08_330_570_242_657} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(B\) of mass 0.25 kg is moving in a straight line with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it is given an impulse. The impulse has magnitude 12.5 N s and is applied in a horizontal direction making an angle of \(\left( 90 ^ { \circ } + \alpha \right)\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the initial direction of motion of the ball, as shown in Figure 3.
  1. Find the speed of \(B\) immediately after the impulse is applied.
  2. Find the direction of motion of \(B\) immediately after the impulse is applied.
Edexcel M2 2012 June Q6
14 marks Standard +0.3
6. A car of mass 1200 kg pulls a trailer of mass 400 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 14 }\). The trailer is attached to the car by a light inextensible towbar which is parallel to the road. The car's engine works at a constant rate of 60 kW . The non-gravitational resistances to motion are constant and of magnitude 1000 N on the car and 200 N on the trailer. At a given instant, the car is moving at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the acceleration of the car at this instant,
  2. the tension in the towbar at this instant. The towbar breaks when the car is moving at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find, using the work-energy principle, the further distance that the trailer travels before coming instantaneously to rest.
Edexcel M2 2012 June Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-12_602_1175_237_386} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small stone is projected from a point \(O\) at the top of a vertical cliff \(O A\). The point \(O\) is 52.5 m above the sea. The stone rises to a maximum height of 10 m above the level of \(O\) before hitting the sea at the point \(B\), where \(A B = 50 \mathrm {~m}\), as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.
  1. Show that the vertical component of the velocity of projection of the stone is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the speed of projection.
  3. Find the time after projection when the stone is moving parallel to \(O B\).
Edexcel M2 2013 June Q1
5 marks Moderate -0.8
  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
7 marks Standard +0.3
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
13 marks Moderate -0.8
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
11 marks Standard +0.3
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
13 marks Standard +0.3
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
11 marks Standard +0.3
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
15 marks Standard +0.3
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
8 marks Moderate -0.3
  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
7 marks Moderate -0.3
  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.