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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 2010 June Q1
6 marks Standard +0.3
  1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds, \(t \geqslant 0\), is \(( 3 t + 5 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. When \(t = 0\), the velocity of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. When \(t = T\), the velocity of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the value of \(T\).
  2. A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. When \(P\) has moved 12 m , its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that friction is the only non-gravitational resistive force acting on \(P\), find
    1. the work done against friction as the speed of \(P\) increases from \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
    2. the coefficient of friction between the particle and the plane.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0e4552e0-7737-439b-a337-789c83c5258c-04_430_624_297_658} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle \(A B C\), where \(A B = A C = 10 \mathrm {~cm}\) and \(B C = 12 \mathrm {~cm}\), as shown in Figure 1.
  3. Find the distance of the centre of mass of the frame from \(B C\). The frame has total mass \(M\). A particle of mass \(M\) is attached to the frame at the mid-point of \(B C\). The frame is then freely suspended from \(B\) and hangs in equilibrium.
  4. Find the size of the angle between \(B C\) and the vertical.
Edexcel M2 2010 June Q4
8 marks Moderate -0.3
  1. A car of mass 750 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\). The resistance to motion of the car from non-gravitational forces has constant magnitude \(R\) newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(R = 260\).
    The power developed by the car's engine is now increased to 18 kW . The magnitude of the resistance to motion from non-gravitational forces remains at 260 N . At the instant when the car is moving up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the car's acceleration is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(a\).
Edexcel M2 2010 June Q5
9 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A ball of mass 0.5 kg is moving with velocity \(( 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the magnitude of the impulse of the bat on the ball,
  2. the size of the angle between the vector \(\mathbf { i }\) and the impulse exerted by the bat on the ball,
  3. the kinetic energy lost by the ball in the impact.
Edexcel M2 2010 June Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e4552e0-7737-439b-a337-789c83c5258c-10_527_966_310_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a uniform rod \(A B\) of mass \(m\) and length \(4 a\). The end \(A\) of the rod is freely hinged to a point on a vertical wall. A particle of mass \(m\) is attached to the rod at \(B\). One end of a light inextensible string is attached to the rod at \(C\), where \(A C = 3 a\). The other end of the string is attached to the wall at \(D\), where \(A D = 2 a\) and \(D\) is vertically above \(A\). The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is \(T\).
  1. Show that \(T = m g \sqrt { } 13\). The particle of mass \(m\) at \(B\) is removed from the rod and replaced by a particle of mass \(M\) which is attached to the rod at \(B\). The string breaks if the tension exceeds \(2 m g \sqrt { } 13\). Given that the string does not break,
  2. show that \(M \leqslant \frac { 5 } { 2 } m\).
Edexcel M2 2010 June Q7
12 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e4552e0-7737-439b-a337-789c83c5258c-12_631_1041_242_447} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A ball is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) from a point \(P\) on a cliff above horizontal ground. The point \(O\) on the ground is vertically below \(P\) and \(O P\) is 36 m . The ball is projected at an angle \(\theta ^ { \circ }\) to the horizontal. The point \(Q\) is the highest point of the path of the ball and is 12 m above the level of \(P\). The ball moves freely under gravity and hits the ground at the point \(R\), as shown in Figure 3. Find
  1. the value of \(\theta\),
  2. the distance \(O R\),
  3. the speed of the ball as it hits the ground at \(R\).
Edexcel M2 2010 June Q8
15 marks Standard +0.3
  1. A small ball \(A\) of mass \(3 m\) is moving with speed \(u\) in a straight line on a smooth horizontal table. The ball collides directly with another small ball \(B\) of mass \(m\) moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\). The balls have the same radius and can be modelled as particles.
    1. Find
      1. the speed of \(A\) immediately after the collision,
      2. the speed of \(B\) 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 \(\frac { 2 } { 5 }\).
  2. Find the speed of \(B\) immediately after hitting the wall. The first collision between \(A\) and \(B\) occurred at a distance 4a from the wall. The balls collide again \(T\) seconds after the first collision.
  3. Show that \(T = \frac { 112 a } { 15 u }\).
Edexcel M2 2011 June Q1
5 marks Standard +0.2
  1. A car of mass 1000 kg moves with constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The engine of the car is working at a rate of 12 kW . The resistance to motion from non-gravitational forces has magnitude 500 N . Find the value of \(V\).
  2. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4 u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3 m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision.
Show that \(e > \frac { 1 } { 3 }\).
Edexcel M2 2011 June Q3
8 marks Moderate -0.3
3. A ball of mass 0.5 kg is moving with velocity \(12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The impulse received by the ball is \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { N } \mathrm { s }\). By modelling the ball as a particle, find
  1. the speed of the ball immediately after the impact,
  2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf { i }\),
  3. the kinetic energy gained by the ball as a result of the impact.
Edexcel M2 2011 June Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8329378-c976-4068-95ff-e2d254546d6d-05_394_846_239_555} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a uniform lamina \(A B C D E\) such that \(A B D E\) is a rectangle, \(B C = C D\), \(A B = 4 a\) and \(A E = 2 a\). The point \(F\) is the midpoint of \(B D\) and \(F C = a\).
  1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(A E\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find the angle between \(A B\) and the downward vertical.
Edexcel M2 2011 June Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8329378-c976-4068-95ff-e2d254546d6d-07_366_771_267_589} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 0.5 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal and \(A B = 2 \mathrm {~m}\) with \(B\) above \(A\), as shown in Figure 2. The particle \(P\) passes through \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The plane is smooth from \(A\) to \(B\).
  1. Find the speed of projection. The particle \(P\) comes to instantaneous rest at the point \(C\) on the plane, where \(C\) is above \(B\) and \(B C = 1.5 \mathrm {~m}\). From \(B\) to \(C\) the plane is rough and the coefficient of friction between \(P\) and the plane is \(\mu\). By using the work-energy principle,
  2. find the value of \(\mu\).
Edexcel M2 2011 June Q6
11 marks Moderate -0.3
  1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \(( t - 4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , v = 6\).
Find
  1. \(v\) in terms of \(t\),
  2. the values of \(t\) when \(P\) is instantaneously at rest,
  3. the distance between the two points at which \(P\) is instantaneously at rest.
Edexcel M2 2011 June Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8329378-c976-4068-95ff-e2d254546d6d-11_609_773_244_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(4 a\), is held in a horizontal position with the end \(A\) against a rough vertical wall. One end of a light inextensible string \(B D\) is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\) vertically above \(A\), where \(A D = 3 a\). A particle of mass \(3 m\) is attached to the rod at \(C\), where \(A C = x\). The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is \(\frac { 25 } { 4 } m g\). Show that
  1. \(x = 3 a\),
  2. the horizontal component of the force exerted by the wall on the rod has magnitude 5 mg . The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is about to slip,
  3. find the value of \(\mu\).
Edexcel M2 2011 June Q8
13 marks Standard +0.3
  1. A particle is projected from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).
    1. Show that
    $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } \cos ^ { 2 } \alpha }$$ A girl throws a ball from a point \(A\) at the top of a cliff. The point \(A\) is 8 m above a horizontal beach. The ball is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(45 ^ { \circ }\). By modelling the ball as a particle moving freely under gravity,
  2. find the horizontal distance of the ball from \(A\) when the ball is 1 m above the beach. A boy is standing on the beach at the point \(B\) vertically below \(A\). He starts to run in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), leaving \(B 0.4\) seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.
  3. Find the value of \(v\).
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.