Questions — Edexcel M2 (623 questions)

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Edexcel M2 2004 January Q5
12 marks Moderate -0.3
5. A particle \(P\) is projected with velocity \(( 2 u \mathbf { i } + 3 u \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a point \(O\) on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively. The particle \(P\) strikes the plane at the point \(A\) which is 735 m from \(O\).
  1. Show that \(u = 24.5\).
  2. Find the time of flight from \(O\) to \(A\). The particle \(P\) passes through a point \(B\) with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the height of \(B\) above the horizontal plane.
Edexcel M2 2004 January Q6
14 marks Moderate -0.3
6. A smooth sphere \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal table when it collides directly with another smooth sphere \(B\) of mass \(3 m\), which is at rest on the table. The coefficient of restitution between \(A\) and \(B\) is \(e\). The spheres have the same radius and are modelled as particles.
  1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } ( 1 + e ) u\).
  2. Find the speed of \(A\) immediately after the collision. Immediately after the collision the total kinetic energy of the spheres is \(\frac { 1 } { 6 } m u ^ { 2 }\).
  3. Find the value of \(e\).
  4. Hence show that \(A\) is at rest after the collision.
Edexcel M2 2004 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{fe64e6f1-e36b-465d-a41c-ac834439623b-6_428_947_404_566}
\end{figure} A loaded plate \(L\) is modelled as a uniform rectangular lamina \(A B C D\) and three particles. The sides \(C D\) and \(A D\) of the lamina have lengths \(5 a\) and \(2 a\) respectively and the mass of the lamina is \(3 m\). The three particles have mass \(4 m , m\) and \(2 m\) and are attached at the points \(A , B\) and \(C\) respectively, as shown in Fig. 3.
  1. Show that the distance of the centre of mass of \(L\) from \(A D\) is \(2.25 a\).
  2. Find the distance of the centre of mass of \(L\) from \(A B\). The point \(O\) is the mid-point of \(A B\). The loaded plate \(L\) is freely suspended from \(O\) and hangs at rest under gravity.
  3. Find, to the nearest degree, the size of the angle that \(A B\) makes with the horizontal. A horizontal force of magnitude \(P\) is applied at \(C\) in the direction \(C D\). The loaded plate \(L\) remains suspended from \(O\) and rests in equilibrium with \(A B\) horizontal and \(C\) vertically below \(B\).
  4. Show that \(P = \frac { 5 } { 4 } \mathrm { mg }\).
  5. Find the magnitude of the force on \(L\) at \(O\).
Edexcel M2 2005 January Q4
9 marks Standard +0.3
4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j } .$$ When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\).
  1. Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
  2. Calculate the distance \(O S\).
Edexcel M2 2005 January Q5
13 marks Standard +0.3
5. A car of mass 1000 kg is towing a trailer of mass 1500 kg along a straight horizontal road. The tow-bar joining the car to the trailer is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having constant magnitude 750 N . The total resistance to motion of the trailer is modelled as of magnitude \(R\) newtons, where \(R\) is a constant. When the engine of the car is working at a rate of 50 kW , the car and the trailer travel at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(R = 1250\). When travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the driver of the car disengages the engine and applies the brakes. The brakes provide a constant braking force of magnitude 1500 N to the car. The resisting forces of magnitude 750 N and 1250 N are assumed to remain unchanged. Calculate
  2. the deceleration of the car while braking,
  3. the thrust in the tow-bar while braking,
  4. the work done, in kJ , by the braking force in bringing the car and the trailer to rest.
  5. Suggest how the modelling assumption that the resistances to motion are constant could be refined to be more realistic.
Edexcel M2 2005 January Q6
14 marks Standard +0.3
6. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal table. The particle \(P\) collides with a particle \(Q\) of mass \(2 m\) moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) after the collision is \(\frac { 1 } { 5 } u ( 9 e + 4 )\). As a result of the collision, the direction of motion of \(P\) is reversed.
  2. Find the range of possible values of \(e\). Given that the magnitude of the impulse of \(P\) on \(Q\) is \(\frac { 32 } { 5 } m u\),
  3. find the value of \(e\).
    (4)
Edexcel M2 2005 January Q7
15 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{a9e00b5b-3804-4f8d-9cc8-7d1170027726-6_568_1582_360_239}
\end{figure} A particle \(P\) is projected from a point \(A\) with speed \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The point \(O\) is on horizontal ground, with \(O\) vertically below \(A\) and \(O A = 20 \mathrm {~m}\). The particle \(P\) moves freely under gravity and passes through a point \(B\), which is 16 m above ground, before reaching the ground at the point \(C\), as shown in Figure 4. Calculate
  1. the time of the flight from \(A\) to \(C\),
  2. the distance \(O C\),
  3. the speed of \(P\) at \(B\),
  4. the angle that the velocity of \(P\) at \(B\) makes with the horizontal.
Edexcel M2 2009 January Q1
5 marks Standard +0.3
  1. A car of mass 1500 kg is moving up a straight road, which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to the motion of the car from non-gravitational forces is constant and is modelled as a single constant force of magnitude 650 N . The car's engine is working at a rate of 30 kW .
Find the acceleration of the car at the instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2009 January Q2
10 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8ebad3-0ebb-4dfe-8036-54b651deb9cf-03_602_554_205_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a ladder \(A B\), of mass 25 kg and length 4 m , resting in equilibrium with one end \(A\) on rough horizontal ground and the other end \(B\) against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(\frac { 11 } { 25 }\). The ladder makes an angle \(\beta\) with the ground. When Reece, who has mass 75 kg , stands at the point \(C\) on the ladder, where \(A C = 2.8 \mathrm {~m}\), the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle.
  1. Find the magnitude of the frictional force of the ground on the ladder.
  2. Find, to the nearest degree, the value of \(\beta\).
  3. State how you have used the modelling assumption that Reece is a particle.
Edexcel M2 2009 January Q3
8 marks Moderate -0.3
  1. A block of mass 10 kg is pulled along a straight horizontal road by a constant horizontal force of magnitude 70 N in the direction of the road. The block moves in a straight line passing through two points \(A\) and \(B\) on the road, where \(A B = 50 \mathrm {~m}\). The block is modelled as a particle and the road is modelled as a rough plane. The coefficient of friction between the block and the road is \(\frac { 4 } { 7 }\).
    1. Calculate the work done against friction in moving the block from \(A\) to \(B\).
    The block passes through \(A\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the speed of the block at \(B\).
Edexcel M2 2009 January Q4
8 marks Standard +0.3
4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = \begin{cases} 10 t - 2 t ^ { 2 } , & 0 \leqslant t \leqslant 6 \\ \frac { - 432 } { t ^ { 2 } } , & t > 6 \end{cases}$$ At \(t = 0 , P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when
  1. \(t = 6\),
  2. \(t = 10\).
Edexcel M2 2009 January Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8ebad3-0ebb-4dfe-8036-54b651deb9cf-08_781_541_223_687} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform lamina \(A B C D\) is made by joining a uniform triangular lamina \(A B D\) to a uniform semi-circular lamina \(D B C\), of the same material, along the edge \(B D\), as shown in Figure 2. Triangle \(A B D\) is right-angled at \(D\) and \(A D = 18 \mathrm {~cm}\). The semi-circle has diameter \(B D\) and \(B D = 12 \mathrm {~cm}\).
  1. Show that, to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(A D\) is 4.69 cm . Given that the centre of mass of a uniform semicircular lamina, radius \(r\), is at a distance \(\frac { 4 r } { 3 \pi }\) from the centre of the bounding diameter,
  2. find, in cm to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(B D\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
  3. Find, to the nearest degree, the angle which \(B D\) makes with the vertical.
Edexcel M2 2009 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8ebad3-0ebb-4dfe-8036-54b651deb9cf-10_506_1361_205_299} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A cricket ball is hit from a point \(A\) with velocity of \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), at an angle \(\alpha\) above the horizontal. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertically upwards. The point \(A\) is 0.9 m vertically above the point \(O\), which is on horizontal ground. The ball takes 3 seconds to travel from \(A\) to \(B\), where \(B\) is on the ground and \(O B = 57.6 \mathrm {~m}\), as shown in Figure 3. By modelling the motion of the cricket ball as that of a particle moving freely under gravity,
  1. find the value of \(p\),
  2. show that \(q = 14.4\),
  3. find the initial speed of the cricket ball,
  4. find the exact value of \(\tan \alpha\).
  5. Find the length of time for which the cricket ball is at least 4 m above the ground.
  6. State an additional physical factor which may be taken into account in a refinement of the above model to make it more realistic.
Edexcel M2 2009 January Q7
17 marks Standard +0.3
  1. A particle \(P\) of mass \(3 m\) is moving in a straight line with speed \(2 u\) on a smooth horizontal table. It collides directly with another particle \(Q\) of mass \(2 m\) which is moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 1 } { 5 } ( 9 e + 4 ) u\).
    The speed of \(P\) immediately after the collision is \(\frac { 1 } { 2 } u\).
  2. Show that \(e = \frac { 1 } { 4 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision \(Q\) hits a smooth fixed vertical wall which is at right-angles to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\).
  3. Show that \(P\) is a distance \(\frac { 3 } { 5 } d\) from the wall at the instant when \(Q\) hits the wall. Particle \(Q\) rebounds from the wall and moves so as to collide directly with particle \(P\) at the point \(B\). Given that the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 5 }\),
  4. find, in terms of \(d\), the distance of the point \(B\) from the wall.
Edexcel M2 2011 January Q1
6 marks Moderate -0.8
  1. A cyclist starts from rest and moves along a straight horizontal road. The combined mass of the cyclist and his cycle is 120 kg . The resistance to motion is modelled as a constant force of magnitude 32 N . The rate at which the cyclist works is 384 W . The cyclist accelerates until he reaches a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find
  1. the value of \(v\),
  2. the acceleration of the cyclist at the instant when the speed is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2011 January Q2
5 marks Moderate -0.3
2. A particle of mass 2 kg is moving with velocity \(( 5 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( - 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { N }\) s. Find the kinetic energy of the particle immediately after receiving the impulse.
(5) \includegraphics[max width=\textwidth, alt={}, center]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-03_41_1571_504_185}
Edexcel M2 2011 January Q3
8 marks Moderate -0.3
3. A particle moves along the \(x\)-axis. At time \(t = 0\) the particle passes through the origin with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The acceleration of the particle at time \(t\) seconds, \(t \geqslant 0\), is \(\left( 4 t ^ { 3 } - 12 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. Find
  1. the velocity of the particle at time \(t\) seconds,
  2. the displacement of the particle from the origin at time \(t\) seconds,
  3. the values of \(t\) at which the particle is instantaneously at rest.
Edexcel M2 2011 January Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-06_365_776_264_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A box of mass 30 kg is held at rest at point \(A\) on a rough inclined plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. Point \(B\) is 50 m from \(A\) up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from \(A\) to \(B\) by a force acting parallel to \(A B\) and then held at rest at \(B\). The coefficient of friction between the box and the plane is \(\frac { 1 } { 4 }\). Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,
  1. find the work done in dragging the box from \(A\) to \(B\). The box is released from rest at the point \(B\) and slides down the slope. Using the workenergy principle, or otherwise,
  2. find the speed of the box as it reaches \(A\).
    January 2011
Edexcel M2 2011 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-10_823_908_269_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform L-shaped lamina \(A B C D E F\), shown in Figure 2, has sides \(A B\) and \(F E\) parallel, and sides \(B C\) and \(E D\) parallel. The pairs of parallel sides are 9 cm apart. The points \(A , F\), \(D\) and \(C\) lie on a straight line. \(A B = B C = 36 \mathrm {~cm} , F E = E D = 18 \mathrm {~cm} . \angle A B C = \angle F E D = 90 ^ { \circ }\), and \(\angle B C D = \angle E D F = \angle E F D = \angle B A C = 45 ^ { \circ }\).
  1. Find the distance of the centre of mass of the lamina from
    1. side \(A B\),
    2. side \(B C\). 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 vertical.
Edexcel M2 2011 January Q6
12 marks Moderate -0.3
  1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-12_689_1042_360_459} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} At time \(t = 0\), a particle \(P\) is projected from the point \(A\) which has position vector 10j metres with respect to a fixed origin \(O\) at ground level. The ground is horizontal. The velocity of projection of \(P\) is \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), as shown in Figure 3. The particle moves freely under gravity and reaches the ground after \(T\) seconds.
  1. For \(0 \leqslant t \leqslant T\), show that, with respect to \(O\), the position vector, \(\mathbf { r }\) metres, of \(P\) at time \(t\) seconds is given by $$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$
  2. Find the value of \(T\).
  3. Find the velocity of \(P\) at time \(t\) seconds \(( 0 \leqslant t \leqslant T )\). When \(P\) is at the point \(B\), the direction of motion of \(P\) is \(45 ^ { \circ }\) below the horizontal.
  4. Find the time taken for \(P\) to move from \(A\) to \(B\).
  5. Find the speed of \(P\) as it passes through \(B\).
Edexcel M2 2011 January Q7
10 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-14_442_986_264_479} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A uniform plank \(A B\), of weight 100 N and length 4 m , rests in equilibrium with the end \(A\) on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 3 }\). The coefficient of friction between the plank and the ground is \(\mu\). Modelling the plank as a rod, find the least possible value of \(\mu\).
Edexcel M2 2011 January Q8
13 marks Standard +0.3
  1. A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J . The coefficient of restitution between \(P\) and the wall is \(\frac { 1 } { 3 }\).
    1. Show that \(m = 4\).
      (6)
    After rebounding from the wall, \(P\) collides directly with a particle \(Q\) which is moving towards \(P\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(Q\) is 2 kg and the coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 3 }\).
  2. Show that there will be a second collision between \(P\) and the wall.
Edexcel M2 2013 January Q1
5 marks Standard +0.3
  1. Two uniform rods \(A B\) and \(B C\) are rigidly joined at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). Rod \(A B\) has length 0.5 m and mass 2 kg . Rod \(B C\) has length 2 m and mass 3 kg . The centre of mass of the framework of the two rods is at \(G\).
    1. Find the distance of \(G\) from \(B C\).
    The distance of \(G\) from \(A B\) is 0.6 m .
    The framework is suspended from \(A\) and hangs freely in equilibrium.
  2. Find the angle between \(A B\) and the downward vertical at \(A\).
Edexcel M2 2013 January Q2
9 marks Moderate -0.3
2. A lorry of mass 1800 kg travels along a straight horizontal road. The lorry's engine is working at a constant rate of 30 kW . When the lorry's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The magnitude of the resistance to the motion of the lorry is \(R\) newtons.
  1. Find the value of \(R\). The lorry now travels up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The magnitude of the non-gravitational resistance to motion is \(R\) newtons. The lorry travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the new rate of working of the lorry's engine.
Edexcel M2 2013 January Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-05_876_757_125_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A ladder, of length 5 m and mass 18 kg , has one end \(A\) resting on rough horizontal ground and its other end \(B\) resting against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 1. The coefficient of friction between the ladder and the ground is \(\mu\). A woman of mass 60 kg stands on the ladder at the point \(C\), where \(A C = 3 \mathrm {~m}\). The ladder is on the point of slipping. The ladder is modelled as a uniform rod and the woman as a particle. Find the value of \(\mu\).