Questions M2 (1391 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 \(A\).
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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.
OCR M2 2007 January Q1
1 A uniform solid cylinder has height 20 cm and diameter 12 cm . It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cylinder topples when the angle of inclination is \(\alpha\). Find \(\alpha\).
OCR M2 2007 January Q2
4 marks
2 Two smooth spheres \(A\) and \(B\), of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with \(B\), which is stationary. The collision is perfectly elastic. Calculate the speed of \(A\) after the impact. [4]
OCR M2 2007 January Q3
3 A small sphere of mass 0.2 kg is projected vertically downwards with speed \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate
  1. the coefficient of restitution between the sphere and the ground,
  2. the magnitude of the impulse which the ground exerts on the sphere.
OCR M2 2007 January Q4
4 A skier of mass 80 kg is pulled up a slope which makes an angle of \(20 ^ { \circ }\) with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(A\) to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\), and the distance \(A B\) is 25 m .
  1. By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from \(A\) to \(B\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-2_451_1019_1425_603} It is given that the pulling force has constant magnitude \(P \mathrm {~N}\), and that it acts at a constant angle of \(30 ^ { \circ }\) above the slope (see diagram). Calculate \(P\).
OCR M2 2007 January Q5
5 A model train has mass 100 kg . When the train is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the resistance to its motion is \(3 v ^ { 2 } \mathrm {~N}\) and the power output of the train is \(\frac { 3000 } { v } \mathrm {~W}\).
  1. Show that the driving force acting on the train is 120 N at an instant when the train is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the train at an instant when it is moving horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train moves with constant speed up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 98 }\).
  3. Calculate the speed of the train.
OCR M2 2007 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-3_540_878_989_632} A uniform lamina \(A B C D E\) of weight 30 N consists of a rectangle and a right-angled triangle. The dimensions are as shown in the diagram.
  1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina. The lamina is freely suspended from a hinge at \(B\).
  2. Calculate the angle that \(A B\) makes with the vertical. The lamina is now held in a position such that \(B D\) is horizontal. This is achieved by means of a string attached to \(D\) and to a fixed point 15 cm directly above the hinge at \(B\).
  3. Calculate the tension in the string.
OCR M2 2007 January Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-4_782_1006_274_571} One end of a light inextensible string of length 0.8 m is attached to a fixed point \(A\) which lies above a smooth horizontal table. The other end of the string is attached to a particle \(P\), of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram).
  1. Calculate the tension in the string.
  2. Calculate the normal contact force between the particle and the table. The particle now moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is on the point of leaving the surface of the table.
  3. Calculate \(v\).
OCR M2 2007 January Q8
8 A missile is projected with initial speed \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate
  1. the maximum height of the missile above the level of the point of projection,
  2. the distance of the missile from the point of projection at the instant when it is moving downwards at an angle of \(10 ^ { \circ }\) to the horizontal. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
OCR M2 2008 January Q1
1 A ball is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(55 ^ { \circ }\) above the horizontal. At the instant when the ball reaches its greatest height, it hits a vertical wall, which is perpendicular to the ball's path. The coefficient of restitution between the ball and the wall is 0.65 . Calculate the speed of the ball
  1. immediately before its impact with the wall,
  2. immediately after its impact with the wall.
OCR M2 2008 January Q2
2 A particle of mass \(m \mathrm {~kg}\) is projected directly up a rough plane with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The plane makes an angle of \(30 ^ { \circ }\) with the horizontal and the coefficient of friction is 0.2 . Calculate the distance the particle travels up the plane before coming instantaneously to rest.
OCR M2 2008 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{982647bd-8514-40cf-b4ee-674f51df32c5-2_412_380_909_884} A uniform rod \(A B\), of weight 25 N and length 1.6 m , rests in equilibrium in a vertical plane with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall which is inclined at \(80 ^ { \circ }\) to the horizontal. The rod is inclined at \(60 ^ { \circ }\) to the horizontal (see diagram). Calculate the magnitude of the force acting on the rod at \(B\).