Questions — Edexcel M2 (551 questions)

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Edexcel M2 2005 June Q1
  1. A car of mass 1200 kg moves along a straight horizontal road. The resistance to motion of the car from non-gravitational forces is of constant magnitude 600 N . The car moves with constant speed and the engine of the car is working at a rate of 21 kW .
    1. Find the speed of the car.
    The car moves up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\).
    The car's engine continues to work at 21 kW , and the resistance to motion from nongravitational forces remains of magnitude 600 N .
  2. Find the constant speed at which the car can move up the hill.
Edexcel M2 2005 June Q2
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{3847deb8-d86e-4254-828f-5d62f20c186f-03_378_652_294_630}
\end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.
  1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 2005 June Q3
3．A particle \(P\) moves in a horizontal plane．At time \(t\) seconds，the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ，and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant．When \(t = 1.5\) ，the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ．Find
（a）the value of \(c\) ， （b）the acceleration of \(P\) when \(t = 1.5\) ． \(\mathbf { r }\) metres relative to a fixed origin \(O\) ，and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,
\text { where } c \text { is a positive constant．When } t = 1.5 \text { ，the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { ．Find } \end{aligned}$$ （a）the value of \(c\) ， 3．A particle \(P\) moves in a horizontal plane．At time \(t\) seconds，the position vector of \(P\) is D啨
（b）the acceleration of \(P\) when \(t = 1.5\) ．
Edexcel M2 2005 June Q4
4. A darts player throws darts at a dart board which hangs vertically. The motion of a dart is modelled as that of a particle moving freely under gravity. The darts move in a vertical plane which is perpendicular to the plane of the dart board. A dart is thrown horizontally with speed \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits the board at a point which is 10 cm below the level from which it was thrown.
  1. Find the horizontal distance from the point where the dart was thrown to the dart board. The darts player moves his position. He now throws a dart from a point which is at a horizontal distance of 2.5 m from the board. He throws the dart at an angle of elevation \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). This dart hits the board at a point which is at the same level as the point from which it was thrown.
  2. Find the speed with which the dart is thrown.
Edexcel M2 2005 June Q5
5. Two small spheres \(A\) and \(B\) have mass \(3 m\) and \(2 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane, both with speed \(2 u\), when they collide directly. As a result of the collision, the direction of motion of \(B\) is reversed and its speed is unchanged.
  1. Find the coefficient of restitution between the spheres. Subsequently, \(B\) collides directly with another small sphere \(C\) of mass \(5 m\) which is at rest. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 3 } { 5 }\).
  2. Show that, after \(B\) collides with \(C\), there will be no further collisions between the spheres.
Edexcel M2 2005 June Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{3847deb8-d86e-4254-828f-5d62f20c186f-09_442_689_292_632}
\end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find
  1. the thrust in the rod \(C D\),
  2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
  3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.
Edexcel M2 2005 June Q7
7. At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The distance travelled down the chute by each brick is 8 m . A brick of mass 3 kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the potential energy lost by the brick in moving down the chute.
  2. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.
  3. Hence find the coefficient of friction between the brick and the chute. Another brick of mass 3 kg slides down the chute. This brick is given an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the chute.
  4. Find the speed of this brick when it reaches the bottom of the chute.
Edexcel M2 2016 June Q1
  1. A particle \(P\) moves along a straight line. The speed of \(P\) at time \(t\) seconds ( \(t \geqslant 0\) ) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \left( p t ^ { 2 } + q t + r \right)\) and \(p , q\) and \(r\) are constants. When \(t = 2\) the speed of \(P\) has its minimum value. When \(t = 0 , v = 11\) and when \(t = 2 , v = 3\)
Find
  1. the acceleration of \(P\) when \(t = 3\)
  2. the distance travelled by \(P\) in the third second of the motion.
Edexcel M2 2016 June Q2
2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(3 P\) watts. When the car is moving down the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(P\) watts.
  1. Find
    1. the value of \(P\),
    2. the value of \(R\).
      (6) When the car is moving up the road at \(12.5 \mathrm {~ms} ^ { - 1 }\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
  2. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2016 June Q3
3. A particle of mass 0.6 kg is moving with constant velocity ( \(c \mathbf { i } + 2 c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(c\) is a positive constant. The particle receives an impulse of magnitude \(2 \sqrt { 10 } \mathrm {~N} \mathrm {~s}\). Immediately after receiving the impulse the particle has velocity ( \(2 c \mathbf { i } - c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). Find the value of \(c\).
(6)
Edexcel M2 2016 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-07_606_883_260_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(O B C\) is one quarter of a circular disc with centre \(O\) and radius 4 m . The points \(A\) and \(D\), on \(O B\) and \(O C\) respectively, are 3 m from \(O\). The uniform lamina \(A B C D\), shown shaded in Figure 1, is formed by removing the triangle \(O A D\) from \(O B C\). Given that the centre of mass of one quarter of a uniform circular disc of radius \(r\) is at a distance \(\frac { 4 \sqrt { 2 } } { 3 \pi } r\) from the centre of the disc,
  1. find the distance of the centre of mass of the lamina \(A B C D\) from \(A D\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
  2. Find, to the nearest degree, the angle between \(D C\) and the downward vertical.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-09_915_1269_118_356} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
Edexcel M2 2016 June Q5
5. A non-uniform rod \(A B\), of mass 5 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The centre of mass of the rod is \(d\) metres from \(A\). The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a force \(\mathbf { P }\), which acts in a direction perpendicular to the rod at \(B\), as shown in Figure 2. The line of action of \(\mathbf { P }\) lies in the same vertical plane as the rod.
  1. Find, in terms of \(d , g\) and \(\theta\),
    1. the magnitude of the vertical component of the force exerted on the rod by the ground,
    2. the magnitude of the friction force acting on the rod at \(A\). Given that \(\tan \theta = \frac { 5 } { 12 }\) and that the coefficient of friction between the rod and the ground is \(\frac { 1 } { 2 }\),
  2. find the value of \(d\).
Edexcel M2 2016 June Q6
6. [In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is an upward vertical unit vector.] A particle \(P\) is projected from a fixed origin \(O\) with velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity and passes through the point \(A\) with position vector \(\lambda ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(\lambda\) is a positive constant.
  1. Find the value of \(\lambda\).
  2. Find
    1. the speed of \(P\) at the instant when it passes through \(A\),
    2. the direction of motion of \(P\) at the instant when it passes through \(A\).
      HMAV SIHI NITIIIUM ION OC
      VILV SIHI NI JAHM ION OC
      VJ4V SIHI NI JIIYM ION OC
Edexcel M2 2016 June Q7
7. Two particles \(A\) and \(B\), of mass \(2 m\) and \(3 m\) respectively, are initially at rest on a smooth horizontal surface. Particle \(A\) is projected with speed \(3 u\) towards \(B\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\)
  1. Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. After the collision \(B\) hits a fixed 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 \(e\). The magnitude of the impulse received by \(B\) when it hits the wall is \(\frac { 27 } { 4 } m u\).
  2. Find the value of \(e\).
  3. Determine whether there is a further collision between \(A\) and \(B\) after \(B\) rebounds from the wall.
Edexcel M2 Q1
  1. A car of mass 1200 kg decelerates from \(30 \mathrm {~ms} ^ { - 1 }\) to \(20 \mathrm {~ms} ^ { - 1 }\) in 6 seconds at a constant rate.
    1. Find the magnitude, in N , of the decelerating force.
    2. Find the loss, in J , in the car's kinetic energy.
    3. A particle moves in a straight line from \(A\) to \(B\) in 5 seconds. At time \(t\) seconds after leaving \(A\), the velocity of the particle is \(\left( 32 t - 3 t ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\).
    4. Calculate the straight-line distance \(A B\).
    5. Find the acceleration of the particle when \(t = 3\).
    6. Eddie, whose mass is 71 kg , rides a bicycle of mass 25 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). When Eddie is working at a rate of 600 W , he is moving at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\).
      Find the magnitude of the non-gravitational resistance to his motion.
    7. A boat leaves the point \(O\) and moves such that, \(t\) seconds later, its position vector relative to \(O\) is \(\left( t ^ { 2 } - 2 \right) \mathbf { i } + 2 t \mathbf { j }\), where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) both have magnitude 1 metre and are directed parallel and perpendicular to the shoreline through \(O\).
    8. Find the speed with which the boat leaves \(O\).
    9. Show that the boat has constant acceleration and state the magnitude of this acceleration.
    10. Find the value of \(t\) when the boat is 40 m from \(O\).
    11. Comment on the limitations of the given model of the boat's motion.
    \includegraphics[max width=\textwidth, alt={}]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-1_446_595_1965_349}
    The diagram shows a body which may be modelled as a uniform lamina. The body is suspended from the point marked \(A\) and rests in equilibrium.
  2. Calculate, to the nearest degree, the angle which the edge \(A B\) then makes with the vertical.
    (8 marks) Frank suggests that the angle between \(A B\) and the vertical would be smaller if the lamina were made from lighter material.
  3. State, with a brief explanation, whether Frank is correct.
    (2 marks) \section*{MECHANICS 2 (A) TEST PAPER 1 Page 2}
Edexcel M2 Q6
  1. A uniform rod \(A B\), of mass 0.8 kg and length \(10 a\), is supported at the end \(A\) by a light inextensible vertical string and rests in limiting equilibrium on a rough fixed peg at \(C\), where \(A C = 7 a\).
    \includegraphics[max width=\textwidth, alt={}, center]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-2_319_638_228_1293}
  2. Two particles \(A\) and \(B\), of mass \(m\) and \(k m\) respectively, are moving in the same direction on a smooth horizontal surface. \(A\) has speed \(4 u\) and \(B\) has speed \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e \quad A\) collides directly with \(B\), and in the collision the direction of \(A\) 's motion is reversed. Immediately after the impact, \(B\) has speed \(2 u\).
    1. Show that the speed of \(A\) immediately after the impact is \(u ( 3 e - 2 )\).
    2. Deduce the range of possible values of \(e\).
    3. Show that \(4 < k \leq 5\).
    4. A ball is projected from ground level with speed \(34 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 8 } { 15 }\).
    5. Find the greatest height reached by the ball above ground level.
    While it is descending, the ball hits a horizontal ledge 6 metres above ground level.
  3. Find the horizontal distance travelled by the ball before it hits the ledge.
  4. Find the speed of the ball at the instant when it hits the ledge.
Edexcel M2 Q1
\begin{enumerate} \item A constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is \(\frac { 1 } { 4 }\).
Calculate, in J , the work done by the force. \item A stone, of mass 0.9 kg , is projected vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\) in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m . Find the magnitude of the non-gravitational resisting force acting on the stone. \item A particle \(P\), of mass 0.4 kg , moves in a straight line such that, at time \(t\) seconds after passing through a fixed point \(O\), its distance from \(O\) is \(x\) metres, where \(x = 3 t ^ { 2 } + 8 t\).
  1. Show that \(P\) never returns to \(O\).
  2. Find the value of \(t\) when \(P\) has velocity \(20 \mathrm {~ms} ^ { - 1 }\).
  3. Show that the force acting on \(P\) is constant, and find its magnitude. \item Two smooth spheres \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are moving on a smooth horizontal table with velocities \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. They collide, after which \(A\) has velocity \(( 5 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Edexcel M2 Q6
  1. A uniform wire \(A B C D\) is bent into the shape shown, where the sections \(A B , B C\) and \(C D\) are straight and of length \(3 a , 10 a\) and \(5 a\) respectively and \(A D\) is parallel to \(B C\).
    1. Show that the cosine of angle \(B C D\) is \(\frac { 4 } { 5 }\).
      \includegraphics[max width=\textwidth, alt={}, center]{70e0f6f0-9016-45a1-9ae2-b908f6b3911e-2_261_479_283_1462}
    2. Find the distances of the centre of mass of the bent wire from (i) \(A B\), (ii) \(B C\).
    The wire is hung over a smooth peg at \(B\) and rests in equilibrium.
  2. Find, to the nearest \(0.1 ^ { \circ }\), the angle between \(B C\) and the vertical in this position.
Edexcel M2 Q7
7. Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed \(4 \mathrm {~ms} ^ { - 1 }\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed \(2 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 3 }\). Find
  1. the speed of \(Q\) before the collision,
  2. the speed of \(P\) after the collision,
  3. the kinetic energy, in J , lost in the impact.
Edexcel M2 Q8
8. In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of \(8 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{70e0f6f0-9016-45a1-9ae2-b908f6b3911e-2_293_641_1604_1254}
  1. Find the time taken by the ball to travel 6 m horizontally.
  2. Showing your method clearly, decide whether or not the ball will hit the coconut.
  3. Find the greatest height reached by the ball above the level from which it was thrown.
  4. Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  5. State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut.
Edexcel M2 Q1
  1. A ball, of mass \(m \mathrm {~kg}\), is moving with velocity \(( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( - 2 \mathbf { i } - 4 \mathbf { j } )\) Ns. Immediately after the impulse is applied, the ball has velocity \(( 3 \mathbf { i } + k \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the values of the constants \(k\) and \(m\).
  2. A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \(( 12 t - 15 ) \mathrm { ms } ^ { - 2 }\). Find
    1. the velocity of \(P\) at time \(t\) seconds after it leaves \(O\),
    2. the value of \(t\) when the speed of \(P\) is \(36 \mathrm {~ms} ^ { - 1 }\).
    3. A non-uniform ladder \(A B\), of length \(3 a\), has its centre of mass at \(G\), where \(A G = 2 a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(A B\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac { 14 } { 9 }\). Calculate the coefficient of friction between the ladder and the ground.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6198c550-927b-4554-9ddf-ef166fc9f2dd-1_355_330_1019_1615} \captionsetup{labelformat=empty} \caption{(7 marks)}
    \end{figure}
Edexcel M2 Q4
  1. A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf { r } \mathbf { m }\) relative to \(O\) after \(t\) seconds is given by \(\mathbf { r } = a t ^ { 2 } \mathbf { i } + b t \mathbf { j }\).
    60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \(( 90 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m }\).
    1. Find the values of the constants \(a\) and \(b\).
    2. Find the speed of \(P\) when it is at \(Q\).
    3. Sketch the path followed by \(P\) for \(0 \leq t \leq 60\).
    4. A lorry of mass 4200 kg can develop a maximum power of 84 kW . On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at \(20 \mathrm {~ms} ^ { - 1 }\) the resisting force has magnitude 2400 N . Find the maximum speed of the lorry when it is
    5. travelling on a horizontal road,
    6. climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\).
    \section*{MECHANICS 2 (A)TEST PAPER 3 Page 2}
Edexcel M2 Q6
  1. Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4 u\) and \(5 u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision.
    2. Show that \(e > \frac { 1 } { 9 }\).
      \(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).
    3. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur.
    4. A uniform lamina is in the form of a trapezium \(A B C D\), as shown. \(A B\) and \(D C\) are perpendicular to \(B C . A B = 17 \mathrm {~cm} , B C = 21 \mathrm {~cm}\) and \(C D = 8 \mathrm {~cm}\).
    5. Find the distances of the centre of mass of the lamina from
      \includegraphics[max width=\textwidth, alt={}, center]{6198c550-927b-4554-9ddf-ef166fc9f2dd-2_273_426_948_1537}
      1. \(A B\),
      2. \(B C\).
    The lamina is freely suspended from \(C\) and rests in equilibrium.
  2. Find the angle between \(C D\) and the vertical.
Edexcel M2 Q8
8. A stone, of mass 1.5 kg , is projected horizontally with speed \(4 \mathrm {~ms} ^ { - 1 }\) from a height of 7 m above horizontal ground.
  1. Show that the stone travels about 4.78 m horizontally before it hits the ground.
  2. Find the height of the stone above the ground when it has travelled half of this horizontal distance.
  3. Calculate the potential energy lost by the stone as it moves from its point of projection to the ground.
  4. Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground.
  5. State two modelling assumptions that you have made in answering this question.
Edexcel M2 Q1
\begin{enumerate} \item A small ball \(A\) is moving with velocity \(( 7 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). It collides in mid-air with another ball \(B\), of mass 0.4 kg , moving with velocity \(( - \mathrm { i } + 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\). Immediately after the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( 6 \cdot 5 \mathbf { i } + 13 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Calculate the mass of \(A\). \item A stick of mass 0.75 kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick
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and the floor is 0.6 . Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \item An engine of mass 20000 kg climbs a hill inclined at \(10 ^ { \circ }\) to the horizontal. The total nongravitational resistance to its motion has magnitude 35000 N and the maximum speed of the engine on the hill is \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Find, in kW , the maximum rate at which the engine can work.
  2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. \item Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m }\) and \(( 12 \mathbf { i } + \mathbf { j } )\) m respectively, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \(( 2 t + 4 ) \mathbf { i } + \left( k t ^ { 2 } - 5 \right) \mathbf { j }\).