Questions — Edexcel M2 (551 questions)

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Edexcel M2 2006 June Q8
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
  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
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
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
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
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
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
  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
  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
  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
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
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
  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
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
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
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
  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
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
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
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
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
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
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
  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
  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.