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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
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
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
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
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 \(\mu\).
Edexcel M2 2013 January Q4
10 marks Moderate -0.3
4. At time \(t\) seconds the velocity of a particle \(P\) is \([ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(P\) is \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\), relative to a fixed origin \(O\).
  1. Find the value of \(t\) when the velocity of \(P\) is parallel to the vector \(\mathbf { j }\).
  2. Find an expression for the position vector of \(P\) at time \(t\) seconds. A second particle \(Q\) moves with constant velocity \(( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\). The particles \(P\) and \(Q\) collide at the point with position vector ( \(d \mathbf { i } + 14 \mathbf { j }\) ) m.
  3. Find
    1. the value of \(c\),
    2. the value of \(d\).
Edexcel M2 2013 January Q5
11 marks Standard +0.3
5. The point \(A\) lies on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 24 } { 25 }\). A particle \(P\) is projected from \(A\), up a line of greatest slope of the plane, with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg and the coefficient of friction between \(P\) and the plane is \(\frac { 5 } { 12 }\). The particle comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 1.5 \mathrm {~m}\). It then moves back down the plane to \(A\).
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\).
  3. Find the speed of \(P\) when it returns to \(A\).
Edexcel M2 2013 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-11_531_931_230_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A ball is thrown from a point \(O\), which is 6 m above horizontal ground. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. There is a thin vertical post which is 4 m high and 8 m horizontally away from the vertical through \(O\), as shown in Figure 2. The ball passes just above the top of the post 2 s after projection. The ball is modelled as a particle.
  1. Show that \(\tan \theta = 2.2\)
  2. Find the value of \(u\). The ball hits the ground \(T\) seconds after projection.
  3. Find the value of \(T\). Immediately before the ball hits the ground the direction of motion of the ball makes an angle \(\alpha\) with the horizontal.
  4. Find \(\alpha\).
Edexcel M2 2013 January Q7
16 marks Challenging +1.2
7. A particle \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal floor when it collides directly with another particle \(B\), of mass \(3 m\), which is at rest on the floor. The coefficient of restitution between the particles is \(e\). The direction of motion of \(A\) is reversed by the collision.
  1. Find, in terms of \(e\) and \(u\),
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. After being struck by \(A\) the particle \(B\) collides directly with another particle \(C\), of mass \(4 m\), which is at rest on the floor. The coefficient of restitution between \(B\) and \(C\) is \(2 e\). Given that the direction of motion of \(B\) is reversed by this collision,
  2. find the range of possible values of \(e\),
  3. determine whether there will be a second collision between \(A\) and \(B\).
Edexcel M2 2004 June Q1
7 marks Moderate -0.3
A lorry of mass 1500 kg moves along a straight horizontal road. The resistance to the motion of the lorry has magnitude 750 N and the lorry's engine is working at a rate of 36 kW .
  1. Find the acceleration of the lorry when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The lorry comes to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 10 }\). The magnitude of the resistance to motion from non-gravitational forces remains 750 N . The lorry moves up the hill at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate at which the lorry's engine is now working.
    (3)