Questions — Edexcel M2 (551 questions)

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Edexcel M2 2014 January Q1
  1. A particle \(P\) of mass 2 kg is moving with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. Find the magnitude of the impulse.
    2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied.
    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
    $$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).
  2. Find the acceleration of \(P\) when \(t = 3\)
  3. Find the total distance travelled by \(P\) in the first 3 seconds of its motion.
  4. Show that \(P\) never returns to \(O\).
Edexcel M2 2014 January Q3
  1. A car has mass 550 kg . When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.
    1. (i) Find the value of \(R\).
      (ii) Find the value of \(P\).
    2. Find the acceleration of the car when it travels along the straight horizontal road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with the engine working at 50 kW .
Edexcel M2 2014 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad09e19e-c4f3-4b93-9e9a-4987def62f26-07_542_700_219_628} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina \(A B C D\) is formed by removing the isosceles triangle \(A D C\) of height \(h\) metres, where \(h < 2 \sqrt { 3 }\), from a uniform lamina \(A B C\) in the shape of an equilateral triangle of side 4 m , as shown in Figure 1. The centre of mass of \(A B C D\) is at \(D\).
  1. Show that \(h = \sqrt { } 3\) The weight of the lamina \(A B C D\) is \(W\) newtons. The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied at \(B\) so that the lamina is in equilibrium with \(A B\) vertical. The horizontal force acts in the vertical plane containing the lamina.
  2. Find \(F\) in terms of \(W\).
Edexcel M2 2014 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad09e19e-c4f3-4b93-9e9a-4987def62f26-09_620_776_219_584} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(A C = \frac { 2 } { 3 } a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(B D = 2 a\).
  1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac { 1 } { 2 } m g \sin \theta\)
  2. Find the magnitude of the normal reaction on the rod at \(B\). The rod is in limiting equilibrium when \(\tan \theta = \frac { 4 } { 3 }\)
  3. Find the coefficient of friction between the rod and the ground.
Edexcel M2 2014 January Q6
  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]{ad09e19e-c4f3-4b93-9e9a-4987def62f26-11_375_1008_354_475} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \(( 3 \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 } , v > 3\). The ball moves freely under gravity and passes through the point A before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac { 15 } { 49 } \mathrm {~s}\) after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac { 1 } { 2 } E\) joules.
  1. Find the value of \(v\). At another point \(B\) on the path of the ball the kinetic energy is also \(\frac { 1 } { 2 } E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.
  2. Find the value of \(T\).
Edexcel M2 2014 January Q7
7. Three particles \(A , B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5 u\) and \(4 u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3 u\). In the subsequent motion, \(A , B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Find (i) the speed of \(A\) immediately after the collision,
    (ii) the speed of \(B\) immediately after the collision. Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),
  2. find the set of possible values of \(e\).
Edexcel M2 2015 January Q1
  1. A particle \(P\) of mass 0.6 kg is moving with velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { I } \mathrm { N }\) s. Immediately after receiving the impulse, \(P\) has velocity ( \(2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find
  1. the magnitude of \(\mathbf { I }\),
  2. the kinetic energy lost by \(P\) as a result of receiving the impulse.
Edexcel M2 2015 January Q2
2. A car of mass 500 kg is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 150 N .
  1. Find the rate of working of the engine of the car. When the car is travelling up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car then comes to instantaneous rest, without braking, having moved a distance \(d\) metres up the road from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 150 N .
  2. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2015 January Q3
  1. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where
$$\mathbf { r } = \left( \frac { 1 } { 8 } t ^ { 4 } - 2 \lambda t ^ { 2 } + 5 \right) \mathbf { i } + \left( 5 t ^ { 2 } - \lambda t \right) \mathbf { j }$$ and \(\lambda\) is a constant. When \(t = 4 , P\) is moving parallel to the vector \(\mathbf { j }\).
  1. Show that \(\lambda = 2\)
  2. Find the speed of \(P\) when \(t = 4\)
  3. Find the acceleration of \(P\) when \(t = 4\) When \(t = 0 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
  4. Find the distance \(A B\).
Edexcel M2 2015 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-07_887_707_269_621} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform plane lamina \(A B C D E F\) shown in Figure 1 is made from two identical rhombuses. Each rhombus has sides of length \(a\) and angle \(B A D =\) angle \(D A F = \theta\). The centre of mass of the lamina is \(0.9 a\) from \(A\).
  1. Show that \(\cos \theta = 0.8\) The weight of the lamina is \(W\). A particle of weight \(k W\) is fixed to the lamina at the point \(A\). The lamina is freely suspended from \(B\) and hangs in equilibrium with \(D A\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 2015 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-09_636_1143_251_468} \captionsetup{labelformat=empty} \caption{Figure 2}
\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 \(k m\) is fixed 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 \(A C = \frac { 5 } { 4 } a\), as shown in Figure 2. The line of action of the force at \(C\) is at right angles to \(A B\) and in the vertical plane containing \(A B\). Given that \(\tan \theta = \frac { 3 } { 4 }\)
  1. show that \(F = \frac { 16 } { 25 } m g ( 1 + 2 k )\),
  2. find, in terms of \(m , g\) and \(k\),
    1. the horizontal component of the force exerted by the hinge on the rod at \(A\),
    2. the vertical component of the force exerted by the hinge on the rod at \(A\). Given also that the force acting on the rod at \(A\) acts at \(45 ^ { \circ }\) above the horizontal,
  3. find the value of \(k\).
Edexcel M2 2015 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-11_452_865_264_495} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(P\) is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A 10 \mathrm {~m}\) above horizontal ground. The angle of projection is \(55 ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 3. Find
  1. the speed of \(P\) as it hits the ground at \(B\),
  2. the direction of motion of \(P\) as it hits the ground at \(B\),
  3. the time taken for \(P\) to move from \(A\) to \(B\).
Edexcel M2 2015 January Q7
7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal surface with \(Q\) between \(P\) and \(R\). Particle \(P\) has mass \(m\), particle \(Q\) has mass \(2 m\) and particle \(R\) has mass \(3 m\). The coefficient of restitution between each pair of particles is \(e\). Particle \(P\) is projected towards \(Q\) with speed \(3 u\) and collides directly with \(Q\).
  1. Find, in terms of \(u\) and \(e\),
    1. the speed of \(Q\) immediately after the collision,
    2. the speed of \(P\) immediately after the collision.
  2. Find the range of values of \(e\) for which the direction of motion of \(P\) is reversed as a result of the collision with \(Q\). Immediately after the collision between \(P\) and \(Q\), particle \(R\) is projected towards \(Q\) with speed \(u\) so that \(R\) and \(Q\) collide directly. Given that \(e = \frac { 2 } { 3 }\)
  3. show that there will be a second collision between \(P\) and \(Q\).
Edexcel M2 2017 January Q1
  1. A car of mass 1200 kg moves up a straight road. The road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 15 }\). The car is moving up the road with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine of the car is working at a constant rate of 11760 watts. The non-gravitational resistance to motion has a constant magnitude of \(R\) newtons.
    1. Find the value of \(R\).
    The rate of working of the car is now increased to 50 kW . At the instant when the speed of the car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the non-gravitational resistance to the motion of the car is 700 N and the acceleration of the car is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(V\).
Edexcel M2 2017 January Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{36cced0d-f982-4534-a3fe-13c32fb37f5b-04_538_625_251_657} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina is in the shape of a trapezium \(A B C D\) with \(A B = a , D A = D C = 2 a\) and angle \(B A D =\) angle \(A D C = 90 ^ { \circ }\), as shown in Figure 1. The centre of mass of the lamina is at the point \(G\).
    1. Show that the distance of \(G\) from \(A B\) is \(\frac { 10 a } { 9 }\).
    2. Find the distance of \(G\) from \(A D\). The mass of the lamina is \(3 M\). A particle of mass \(k M\) is now attached to the lamina at \(B\). The lamina is freely suspended from the midpoint of \(A D\) and hangs in equilibrium with \(A D\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 2017 January Q3
  1. A particle \(P\) moves along a straight line. At time \(t = 0 , P\) passes the point \(A\) on the line and at time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where
$$v = ( 2 t - 3 ) ( t - 2 )$$ At \(t = 3 , P\) reaches the point \(B\). Find the total distance moved by \(P\) as it travels from \(A\) to \(B\).
(6)
Edexcel M2 2017 January Q4
4. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 20 \mathbf { i } - 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse ( \(- 6 \mathbf { i } + 8 \mathbf { j }\) ) N s.
  1. Find the speed of \(P\) immediately after it receives the impulse.
    (5)
  2. Find the size of the angle between the direction of motion of \(P\) before the impulse is received and the direction of motion of \(P\) after the impulse is received.
    (4)
Edexcel M2 2017 January Q5
5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(3 m\) respectively, are moving in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly and, as a result of the collision, the direction of motion of \(P\) is reversed and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(\frac { 3 v } { 2 }\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\).
  1. Find
    1. the speed of \(P\) immediately before the collision,
    2. the speed of \(Q\) immediately before the collision. After the collision with \(P\), the particle \(Q\) moves on the plane and strikes at right angles a fixed smooth vertical wall and rebounds. The coefficient of restitution between \(Q\) and the wall is \(e\). Given that there is a further collision between the particles,
  2. find the range of possible values of \(e\).
Edexcel M2 2017 January Q6
6. A ball of mass 0.6 kg is projected vertically upwards with speed \(22.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 1.5 m above horizontal ground. The ball moves freely under gravity until it reaches the ground. The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The ball is modelled as a particle and the ground is assumed to exert a constant resistive force of magnitude \(R\) newtons on the ball. Using the work-energy principle, find, to 3 significant figures, the value of \(R\).
(5)
Edexcel M2 2017 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{36cced0d-f982-4534-a3fe-13c32fb37f5b-11_513_429_123_762} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is in contact with rough horizontal ground and the end \(B\) is in contact with a smooth vertical wall. The rod rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. The coefficient of friction between the rod and the ground is \(\mu\).
  1. Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the rod by the wall.
  2. Show that \(\mu \geqslant \frac { \sqrt { 3 } } { 6 }\). A particle of mass \(k m\) is now attached to the rod at \(B\). Given that \(\mu = \frac { \sqrt { 3 } } { 5 }\) and that the rod is now in limiting equilibrium,
  3. find the value of \(k\).
Edexcel M2 2017 January Q8
  1. At time \(t = 0\) seconds, a golf ball is hit from a point \(O\) on horizontal ground. The horizontal and vertical components of the initial velocity of the ball are \(3 U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The ball hits the ground at the point \(A\), where \(O A = 120 \mathrm {~m}\). The ball is modelled as a particle moving freely under gravity.
    1. Show that \(U = 14\)
    2. Find the speed of the ball immediately before it hits the ground at \(A\).
    3. Find the values of \(t\) when the ball is moving at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 1 } { 4 }\).
Edexcel M2 2018 January Q1
  1. A ball of mass 0.5 kg is moving with velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\). Find the gain in kinetic energy of the ball due to the impulse.
    (6)
Edexcel M2 2018 January Q2
2. A particle \(P\) moves in a straight line. At time \(t = 0 , P\) passes through a point \(O\) on the line. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where
  1. Find the acceleration of \(P\) when \(t = \frac { 1 } { 2 }\)
  2. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 1\) At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where $$v = ( 2 t - 1 ) ( 1 - t )$$
  3. Find the acceleration of \(P\) when \(t = \frac { 1 } { 2 }\)
Edexcel M2 2018 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-06_479_608_246_667} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(O A B C D\) is shown in Figure 1, with \(O A = 6 a , A B = 3 a , C D = 2 a\) and \(D O = 6 a\) and with right angles at \(O , A\) and \(D\).
  1. Find the distance of the centre of mass of the lamina
    1. from \(O D\),
    2. from \(O A\). The lamina is suspended from \(C\) and hangs freely in equilibrium with \(C B\) inclined at an angle \(\alpha\) to the vertical.
  2. Find, to the nearest degree, the size of the angle \(\alpha\).
Edexcel M2 2018 January Q4
  1. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(u\) on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\), of mass \(m\), which is moving on the plane along the same straight line as \(P\) but in the opposite direction to \(P\). Immediately before the collision the speed of \(Q\) is \(3 u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 1 } { 8 }\)
    1. Find, in terms of \(u\) and \(e\),
      1. the speed of \(P\) immediately after the collision,
      2. the speed of \(Q\) immediately after the collision.
    2. Show that, for all possible values of \(e\), the direction of motion of \(P\) is reversed by the collision.
    After the collision, \(Q\) strikes a smooth fixed vertical wall, which is perpendicular to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(f\). Given that \(e = \frac { 3 } { 4 }\) and that there is a second collision between \(Q\) and \(P\),
  2. find the range of possible values of \(f\).