Questions — Edexcel M2 (551 questions)

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Edexcel M2 2016 October Q4
  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
$$\mathbf { r } = \left( t ^ { 3 } - \frac { 9 } { 2 } t ^ { 2 } - 24 t \right) \mathbf { i } + \left( - t ^ { 3 } + 3 t ^ { 2 } + 12 t \right) \mathbf { j }$$ At time \(T\) seconds, \(P\) is moving in a direction parallel to the vector \(\mathbf { - i } - \mathbf { j }\)
Find
  1. the value of \(T\),
  2. the magnitude of the acceleration of \(P\) at the instant when \(t = T\).
Edexcel M2 2016 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be16c17a-c4db-4f0c-9f32-8d5614b4f2f3-12_440_1047_246_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) of length 8 m and weight \(W\) newtons rests in equilibrium against a rough horizontal peg \(P\). The end \(A\) is on rough horizontal ground. The friction is limiting at both \(A\) and \(P\). The distance \(A P\) is 5 m , as shown in Figure 1. The rod rests at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The rod is in a vertical plane which is perpendicular to \(P\). The coefficient of friction between the rod and \(P\) is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the ground is \(\mu\).
  1. Show that the magnitude of the normal reaction between the rod and \(P\) is \(0.48 W\) newtons.
  2. Find the value of \(\mu\).
Edexcel M2 2016 October Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be16c17a-c4db-4f0c-9f32-8d5614b4f2f3-16_1031_915_116_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(L\) shown shaded in Figure 2 is formed by removing two circular discs, \(C _ { 1 }\) and \(C _ { 2 }\), from a circular disc with centre \(O\) and radius \(8 a\). Disc \(C _ { 1 }\) has centre \(A\) and radius \(a\). Disc \(C _ { 2 }\) has centre \(B\) and radius \(2 a\). The diameters \(P R\) and \(Q S\) are perpendicular. The midpoint of \(P O\) is \(A\) and the midpoint of \(O R\) is \(B\).
  1. Show that the centre of mass of \(L\) is \(\frac { 484 } { 59 } a\) from \(R\). The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(S\). The lamina with the attached particle is suspended from \(R\) and hangs freely in equilibrium with the diameter \(P R\) at an angle of arctan \(\left( \frac { 1 } { 4 } \right)\) to the downward vertical through \(R\).
  2. Find the value of \(k\).
Edexcel M2 2016 October Q7
7. [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. Position vectors are given relative to a fixed origin O.] At time \(t = 0\) seconds, the particle \(P\) is projected from \(O\) with velocity ( \(3 \mathbf { i } + \lambda \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a positive constant. The particle moves freely under gravity. As \(P\) passes through the fixed point \(A\) it has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The kinetic energy of \(P\) at the instant it passes through \(A\) is half the initial kinetic energy of \(P\). Find the position vector of \(A\), giving the components to 2 significant figures.
(10)
Edexcel M2 2016 October Q8
8. Particles \(A , B\) and \(C\), of masses \(4 m , k m\) and \(2 m\) respectively, lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(3 u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\) Find
  1. the speed of \(A\) immediately after the collision with \(B\), giving your answer in terms of \(u\) and \(k\),
  2. the range of values of \(k\) for which \(A\) and \(B\) will both be moving in the same direction immediately after they collide. After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\). Given that \(k = 4\),
  3. show that there will not be a second collision between \(A\) and \(B\).
    DO NOT WRITEIN THIS AREA
Edexcel M2 2017 October Q1
  1. A small ball \(B\) of mass 0.2 kg is hit by a bat. Immediately before being hit, \(B\) has velocity \(( 10 \mathbf { i } - 17 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being hit, \(B\) has velocity \(( 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the magnitude of the impulse exerted on \(B\) by the bat.
    (4)
Edexcel M2 2017 October Q2
2. A van of mass 1200 kg is travelling along a straight horizontal road. The resistance to the motion of the van has a constant magnitude of 650 N and the van's engine is working at a rate of 30 kW .
  1. Find the acceleration of the van when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The van now travels up a straight road which is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The resistance to the motion of the van from non-gravitational forces has a constant magnitude of 650 N . The van moves up the road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find, in kW , the rate at which the van's engine is now working.
    "
Edexcel M2 2017 October Q3
3. A particle \(P\) of mass 4 kg moves from point \(A\) to point \(B\) down a line of greatest slope of a fixed rough plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal and \(A B = 12 \mathrm {~m}\). The coefficient of friction between \(P\) and the plane is 0.5
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that the speed of \(P\) at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. use the work-energy principle to find the speed of \(P\) at \(A\).
Edexcel M2 2017 October Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef8231d-5b95-4bbb-a8e2-788c708fa078-12_518_696_319_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The end \(A\) rests against a rough vertical wall. One end of a light inextensible string is attached to the rod at the point \(C\), where \(A C = 2 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is vertically above \(A\), with the string perpendicular to the rod. A particle of mass \(m\) is attached to the rod at the end \(B\). The rod is in equilibrium in a vertical plane which is perpendicular to the wall. The rod is inclined at \(60 ^ { \circ }\) to the wall, as shown in Figure 1. Find, in terms of \(m\) and \(g\),
  1. the tension in the string,
  2. the magnitude of the horizontal component of the force exerted by the wall on the rod. The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
  3. find the value of \(\mu\).
Edexcel M2 2017 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef8231d-5b95-4bbb-a8e2-788c708fa078-16_632_734_248_605} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(A B C\) is in the shape of an equilateral triangle with sides of length \(4 a\). The midpoint of \(B C\) is \(D\). The point \(E\) lies on \(A D\) with \(D E = \frac { 3 a } { 2 }\). A circular hole, with centre \(E\) and radius \(a\), is made in the lamina \(A B C\) to form the lamina \(L\), shown shaded in Figure 2.
  1. Find the distance of the centre of mass of \(L\) from \(D\). The lamina \(L\) is freely suspended from the point \(B\) and hangs in equilibrium.
  2. Find, to the nearest degree, the size of the acute angle between \(A D\) and the downward vertical.
Edexcel M2 2017 October Q6
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) is
    \(( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. When \(t = 3 , v = 2\)
    1. Find \(v\) in terms of \(t\).
      (4)
    The particle first comes to instantaneous rest at the point \(A\) and then comes to instantaneous rest again at the point \(B\).
  2. Find the distance \(A B\).
Edexcel M2 2017 October Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef8231d-5b95-4bbb-a8e2-788c708fa078-24_711_1009_251_479} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(P\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 47.5 m above a horizontal beach. The ball moves freely under gravity and hits the beach at the point \(B\), as shown in Figure 3.
  1. By considering energy, find the speed of \(P\) immediately before it hits the beach. The ball was projected from \(A\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\)
  2. Find the greatest height above the beach of \(P\) as it moved from \(A\) to \(B\).
  3. Find the least speed of \(P\) as it moved between \(A\) and \(B\).
  4. Find the horizontal distance from \(A\) to \(B\).
Edexcel M2 2017 October Q8
8. A particle \(A\) of mass \(3 m\) lies at rest on a smooth horizontal floor. A particle \(B\) of mass \(2 m\) is moving in a straight line on the floor with speed \(u\) when it collides directly with \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). As a result of the collision the direction of motion of \(B\) is reversed.
  1. Find an expression, in terms of \(u\) and \(e\), for
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. The particle \(A\) subsequently strikes a smooth vertical wall. The wall is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(A\) and the wall is \(\frac { 1 } { 7 }\) There is a second collision between \(A\) and \(B\).
  2. Show that \(\frac { 2 } { 3 } < e < \frac { 16 } { 19 }\)
Edexcel M2 2018 October Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-02_273_264_223_831} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle, \(P\), of mass 0.8 kg , moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane, receives a horizontal impulse of magnitude 6 N s. The angle between the initial direction of motion of \(P\) and the direction of the impulse is \(50 ^ { \circ }\), as shown in Figure 1. Find the speed of \(P\) immediately after receiving the impulse.
Edexcel M2 2018 October Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-04_442_810_237_557} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A truck of mass 1200 kg is being driven up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The resistance to the motion of the truck from non-gravitational forces is modelled as a single constant force of magnitude 250 N . Two points, \(A\) and \(B\), lie on the road, with \(A B = 90 \mathrm {~m}\). The speed of the truck at \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of the truck at \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 2. The truck is modelled as a particle and the road is modelled as a straight line.
  1. Find the work done by the engine of the truck as the truck moves from \(A\) to \(B\). On another occasion, the truck is being driven down the same road. The resistance to the motion of the truck is modelled as a single constant force of magnitude 250 N . The engine of the truck is working at a constant rate of 8 kW .
  2. Find the acceleration of the truck at the instant when its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M2 2018 October Q3
3. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where
(b) the magnitude of the acceleration of \(P\) when \(t = 4\) $$\begin{aligned} & \qquad \mathbf { r } = \left( 16 t - 3 t ^ { 3 } \right) \mathbf { i } + \left( t ^ { 3 } - t ^ { 2 } + 2 \right) \mathbf { j }
& \text { Find }
& \text { (a) the velocity of } P \text { at the instant when it is moving parallel to the vector } \mathbf { j } \text {, } \end{aligned}$$ VILIV SIHI NI IIIIIM ION OC
VILV SIHI NI JAHAM ION OC
VJ4V SIHI NI JIIYM ION OC
Edexcel M2 2018 October Q4
4. At time \(t = 0\) a ball is projected from a fixed point \(A\) on horizontal ground to hit a target. The ball is projected from \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) to the horizontal. At time \(t = 2 \mathrm {~s}\) the ball hits the target. At the instant when it hits the target, the ball is travelling downwards at \(30 ^ { \circ }\) below the horizontal with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle moving freely under gravity and the target is modelled as the point \(T\).
  1. Find
    1. the value of \(\theta\),
    2. the value of \(u\). The height of \(T\) above the ground is \(h\) metres.
  2. Find the value of \(h\).
  3. Find the length of time for which the ball is more than \(h\) metres above the ground during the flight from \(A\) to \(T\).
Edexcel M2 2018 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-16_419_531_214_708} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) with sides of length \(3 a\) and \(k a\), where \(k > 3\). The point \(E\) on side \(A D\) is such that \(D E = 3 a\). Rectangle \(A B C D\) is folded along the line \(C E\) to produce the folded lamina \(L\) shown in Figure 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-16_455_536_941_703} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Find, in terms of \(a\) and \(k\),
  1. the distance of the centre of mass of \(L\) from \(A B\),
  2. the distance of the centre of mass of \(L\) from \(A E\). The folded lamina \(L\) is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at \(45 ^ { \circ }\) to the downward vertical.
  3. Find, to 3 significant figures, the value of \(k\).
Edexcel M2 2018 October Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-20_755_579_267_703} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A uniform rod, \(A B\), of mass \(8 m\) and length \(2 a\), has its end \(A\) resting against a rough vertical wall. One end of a light inextensible string is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\). A particle of mass \(k m\) is fixed to the rod at \(C\), where \(A C = 0.5 a\). The rod is in equilibrium in a vertical plane perpendicular to the wall, and is at an angle of \(60 ^ { \circ }\) to the wall, as shown in Figure 5. The tension in the string is \(T\).
  1. Show that \(T = \frac { \sqrt { 3 } } { 4 } ( 16 + k ) m g\) The coefficient of friction between the wall and the rod is \(\frac { 2 } { 3 } \sqrt { 3 }\).
    Given that the rod is in limiting equilibrium,
  2. find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-23_67_65_2656_1886}
Edexcel M2 2018 October Q7
7. A particle, \(P\), of mass \(k m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal surface. Particle \(P\) collides directly with another particle, \(Q\), of mass \(2 m\) which is moving with speed \(u\) in the same direction along the same straight line. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Given that immediately after the collision \(P\) and \(Q\) are moving in opposite directions and the speed of \(Q\) is \(\frac { 3 } { 2 } u\),
  1. find the range of possible values of \(e\). It is now also given that \(e = \frac { 7 } { 8 }\).
  2. Show that the kinetic energy lost by \(P\) in the collision with \(Q\) is \(\frac { 11 } { 8 } m u ^ { 2 }\). The collision between \(P\) and \(Q\) takes place at the point \(A\). After the collision, \(Q\) hits a fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The distance from \(A\) to the wall is \(d\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\). Particle \(Q\) rebounds from the wall and moves so that \(P\) and \(Q\) collide directly at the point \(B\).
  3. Find, in terms of \(d\) and \(u\), the time interval between the collision at \(A\) and the collision at \(B\).
    \includegraphics[max width=\textwidth, alt={}]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-28_2639_1833_121_118}
Edexcel M2 2021 October Q1
1. \section*{Figure 1} Figure 1 A particle of mass \(m\) is held at rest at a point \(A\) on a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\)
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 5 }\)
The points \(A\) and \(B\) lie on a line of greatest slope of the plane, with \(B\) above \(A\), and \(A B = d\), as shown in Figure 1. The particle is pushed up the line of greatest slope from \(A\) to \(B\).
  1. Show that the work done against friction as the particle moves from \(A\) to \(B\) is \(\frac { 12 } { 65 } m g d\) The particle is then held at rest at \(B\) and released.
  2. Use the work-energy principle to find, in terms of \(g\) and \(d\), the speed of the particle at the instant it reaches \(A\).
Edexcel M2 2021 October Q2
2. A vehicle of mass 450 kg is moving on a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\) At the instant when the vehicle is moving down the road at \(12 \mathrm {~ms} ^ { - 1 }\)
  • the engine of the vehicle is working at a rate of \(P\) watts
  • the acceleration of the vehicle is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  • the resistance to the motion of the vehicle is modelled as a constant force of magnitude \(R\) newtons
At the instant when the vehicle is moving up the road at \(12 \mathrm {~ms} ^ { - 1 }\)
  • the engine of the vehicle is working at a rate of \(2 P\) watts
  • the deceleration of the vehicle is \(0.5 \mathrm {~ms} ^ { - 2 }\)
  • the resistance to the motion of the vehicle from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons
Find the value of \(P\).
Edexcel M2 2021 October Q3
3. A particle \(P\) moves on the \(x\)-axis. At time \(t = 0 , P\) is instantaneously at rest at \(O\).
At time \(t\) seconds, \(t > 0\), the \(x\) coordinate of \(P\) is given by $$x = 2 t ^ { \frac { 7 } { 2 } } - 14 t ^ { \frac { 5 } { 2 } } + \frac { 56 } { 3 } t ^ { \frac { 3 } { 2 } }$$ Find
  1. the non-zero values of \(t\) for which \(P\) is at instantaneous rest
  2. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
  3. the acceleration of \(P\) when \(t = 4\)
    \(\_\_\_\_\)}
Edexcel M2 2021 October Q4
4. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after \(P\) receives the impulse, the speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Given that \(\mathbf { J } = c ( - \mathbf { i } + 2 \mathbf { j } )\), where \(c\) is a constant, find the two possible values of \(c\).
(6)
Edexcel M2 2021 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-12_597_502_210_721} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A pole \(A B\) has length 2.5 m and weight 70 N .
The pole rests with end \(B\) against a rough vertical wall. One end of a cable of length 4 m is attached to the pole at \(A\). The other end of the cable is attached to the wall at the point \(C\). The point \(C\) is vertically above \(B\) and \(B C = 2.5 \mathrm {~m}\).
The angle between the cable and the wall is \(\alpha\), as shown in Figure 2.
The pole is in a vertical plane perpendicular to the wall.
The cable is modelled as a light inextensible string and the pole is modelled as a uniform rod. Given that \(\tan \alpha = \frac { 3 } { 4 }\)
  1. show that the tension in the cable is 56 N . Given also that the pole is in limiting equilibrium,
  2. find the coefficient of friction between the pole and the wall. \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-15_90_61_2613_1886}