Questions M2 (1537 questions)

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AQA M2 2015 June Q1
10 marks Standard +0.3
1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$
    1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
    2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
  2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}
AQA M2 2015 June Q2
4 marks Moderate -0.8
2 A uniform rod \(A B\), of mass 4 kg and length 6 metres, has three masses attached to it. A 3 kg mass is attached at the end \(A\) and a 5 kg mass is attached at the end \(B\). An 8 kg mass is attached at a point \(C\) on the rod. Find the distance \(A C\) if the centre of mass of the system is 4.3 m from point \(A\).
[0pt] [4 marks]
AQA M2 2015 June Q3
9 marks Standard +0.3
3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.
  1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
    1. Find the kinetic energy of Simon when he reaches the point \(R\).
    2. Hence find the speed of Simon when he reaches the point \(R\).
  2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
    Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}
AQA M2 2015 June Q4
10 marks Standard +0.3
4 A particle, \(P\), of mass 5 kg is attached to two light inextensible strings, \(A P\) and \(B P\). The other ends of the strings are attached to the fixed points \(A\) and \(B\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in a horizontal circle of radius 0.6 metres with centre \(B\). The string \(A P\) is inclined at \(20 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut when the particle is moving. \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-08_835_568_568_719}
  1. Find the tension in the string \(A P\).
  2. The speed of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the tension, \(T _ { B P }\), in the string \(B P\) is given by $$T _ { B P } = \frac { 25 } { 3 } v ^ { 2 } - 5 g \tan 20 ^ { \circ }$$
  3. Find \(v\) when the tensions in the two strings are equal.
AQA M2 2015 June Q5
6 marks Standard +0.3
5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]
\includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-10_1883_1709_824_153}
AQA M2 2015 June Q6
9 marks Standard +0.3
6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]
AQA M2 2015 June Q7
9 marks Standard +0.3
7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.
  1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
  2. Find \(v\) in terms of \(t\).
  3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
    [0pt] [2 marks]
AQA M2 2015 June Q8
10 marks Standard +0.3
8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
  2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
  3. Find the maximum value of \(x\).
    1. Find the distance fallen by Carol when her speed is a maximum.
    2. Hence find Carol's maximum speed.
AQA M2 2015 June Q9
8 marks Challenging +1.8
9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks] \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}
Edexcel M2 Q1
4 marks Standard +0.2
  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 \(\left( 32 t - 3 t ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\).
    1. Calculate the straight-line distance \(A B\).
    2. Find the acceleration of the particle when \(t = 3\).
    3. 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.
    4. 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 \(O\).
    1. Find the speed with which the boat leaves \(O\).
    2. Show that the boat has constant acceleration and state the magnitude of this acceleration.
    3. Find the value of \(t\) when the boat is 40 m from \(O\).
    4. 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.
    1. 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.
    2. State, with a brief explanation, whether Frank is correct.
      (2 marks) \section*{MECHANICS 2 (A) TEST PAPER 1 Page 2}
Edexcel M2 Q6
10 marks Standard +0.3
  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 \(\tan \alpha = \frac { 8 } { 15 }\).
    1. 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
6 marks Moderate -0.8
A ship, of mass 5000 tonnes, is moving through the sea at a constant speed of \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  1. Calculate the momentum of the ship, in the form \(a \times 10 ^ { n }\), where \(0 \leq a < 10\) and \(n\) is an integer. State the units of your answer. Given that there is a constant force of magnitude 4000 N acting against the ship due to air and water resistances,
  2. find the rate, in kW , at which the ship's engines are working.
Edexcel M2 Q2
7 marks Standard +0.8
2. Two small smooth spheres \(P\) and \(Q\) are moving along a straight line in opposite directions, with equal speeds, and collide directly. Immediately after the impact, the direction of \(P\) 's motion has been reversed and its speed has been halved. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Express the speed of \(Q\) after the impact in the form \(a u ( b e + c )\), where \(a , b\) and \(c\) are constants to be found.
  2. Deduce the range of values of \(e\) for which the direction of motion of \(Q\) remains unaltered.
Edexcel M2 Q3
8 marks Standard +0.3
3. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. At a certain instant, a particle \(P\) of mass 1.8 kg is moving with velocity \(( 24 \mathrm { i } - 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Calculate the kinetic energy of \(P\) at this instant. \(P\) is now subjected to a constant retardation. After 10 seconds, the velocity of \(P\) is \(( - 12 \mathbf { i } + 3 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  2. Calculate the work done by the retarding force over the 10 seconds.
Edexcel M2 Q4
9 marks Standard +0.3
4. A small block of wood, of mass 0.5 kg , slides down a line of \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-1_219_501_2042_338}
greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\). The block is given an initial impulse of magnitude 2 Ns , and reaches the bottom of the plane with kinetic energy 19 J.
  1. Find, in J , the change in the potential energy of the block as it moves down the plane.
  2. Hence find the distance travelled by the block down the plane.
  3. State two modelling assumptions that you have made. \section*{MECHANICS 2 (A) TEST PAPER 6 Page 2}
Edexcel M2 Q5
9 marks Standard +0.3
5.
\includegraphics[max width=\textwidth, alt={}]{3c084e42-d304-4b77-afee-7e4bd801a03c-2_278_483_246_386}
A uniform rod \(X Y\), of length \(2 a\) and mass \(m\), is connected to a vertical wall by a smooth hinge at the end \(X\). A horizontal light inelastic string connects the mid-point of \(X Y\) to the wall and the rod is in equilibrium in this position.
  1. Draw a diagram to show all the forces acting on the rod. Given that the tension in the horizontal string is of magnitude \(2 m g\),
  2. find the angle which \(X Y\) makes with the vertical.
Edexcel M2 Q6
10 marks Standard +0.3
6. \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-2_424_492_813_379} The diagram shows a uniform lamina \(A B C D E F\).
  1. Calculate the distance of the centre of mass of the lamina from (i) \(A F\), (ii) \(A B\). The lamina is hung over a smooth peg at \(D\) and rests in equilibrium in a vertical plane.
  2. Find the angle between \(C D\) and the vertical.
Edexcel M2 Q7
11 marks Moderate -0.3
7. A particle \(P\) moves in a straight line so that its displacement \(s\) metres from a fixed point \(O\) at time \(t\) seconds is given by the formula \(s = t ^ { 3 } - 7 t ^ { 2 } + 8 t\).
  1. Find the values of \(t\) when the velocity of \(P\) equals zero, and briefly describe what is happening to \(P\) at these times.
  2. Find the distance travelled by \(P\) between the times \(t = 3\) and \(t = 5\).
  3. Find the value of \(t\) when the acceleration of \(P\) is \(- 2 \mathrm {~ms} ^ { - 2 }\). Briefly explain the significance of a negative acceleration at this time.
Edexcel M2 Q8
15 marks Standard +0.3
8. A particle \(P\) is projected from a point \(O\) with initial velocity \(( 3 \cdot 5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and moves under gravity. \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the horizontal and vertical directions respectively.
  1. Find the initial speed of \(P\).
  2. Show that the position vector \(\mathbf { r } \mathbf { m }\) of \(P\) at time \(t\) seconds after projection is given by $$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - 4 \cdot 9 t ^ { 2 } \right) \mathbf { j } .$$
  3. Find the horizontal distance of \(P\) from \(O\) at each of the times when it is 4.4 m vertically above the level of \(O\). In a refined model of the motion of \(P\), the position vector of \(P\) at time \(t\) seconds is taken to be $$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - t ^ { 3 } \right) \mathbf { j } \mathbf { ~ m } .$$
  4. Using this model, find the position vector of the highest point reached by \(P\).
Edexcel M2 Q1
5 marks Moderate -0.3
  1. A snooker ball \(A\) is moving on a horizontal table with velocity \(( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
It collides with another ball \(B\), whose mass is twice the mass of \(A\).
After the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Find the velocity of \(B\) before the collision.
Edexcel M2 Q2
6 marks Standard +0.3
2. Charlotte, whose mass is 55 kg , is running up a straight hill inclined at \(6 ^ { \circ }\) to the horizontal. She passes two points \(P\) and \(Q , 80\) metres apart, with speeds \(2 \cdot 5 \mathrm {~ms} ^ { - 1 }\) and \(1 \cdot 5 \mathrm {~ms} ^ { - 1 }\) respectively.
Calculate, in J to the nearest whole number, the total work done by Charlotte as she runs from \(P\) to \(Q\).
Edexcel M2 Q3
7 marks Moderate -0.8
3. A particle \(P\) moves in a horizontal plane such that, at time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - t ^ { \frac { 1 } { 2 } } \mathbf { j }\). When \(t = 0 , P\) is at the point with position vector \(- 10 \mathbf { i } + \mathbf { j }\) relative to a fixed origin \(O\).
  1. Find the position vector \(\mathbf { r }\) of \(P\) at time \(t\) seconds.
  2. Find the distance \(O P\) when \(t = 4\).
Edexcel M2 Q4
7 marks Standard +0.8
4. A small stone, of mass 600 grams, is released from rest a height of 2 metres above ground level and falls under gravity. The time it takes to reach the ground is \(T\) seconds. The stone is then again released from rest at the surface of a tank containing a 2 metre depth of liquid and reaches the bottom after \(2 T\) seconds. It may be assumed that the resisting force acting on the stone is constant.
  1. Find the magnitude of the resisting force exerted on the stone by the liquid.
  2. Find the speed with which the stone hits the bottom of the tank.
Edexcel M2 Q5
7 marks Standard +0.3
5. \includegraphics[max width=\textwidth, alt={}, center]{9e1d8a2f-0c35-4398-98ff-083ec76653ec-1_367_529_2122_383} A sign-board consists of a rectangular sheet of metal, of mass \(M\), which is 3 metres wide and 1 metre high, attached to two thin metal supports, each of mass \(m\) and length 2 metres. The board stands on horizontal ground.
  1. Calculate the height above the ground of the centre of mass of the sign-board, in terms of \(M\) and \(m\). Given now that the centre of mass of the sign-board is \(2 \cdot 2\) metres above the ground, (b) find the ratio \(M : m\), in its simplest form. \section*{MECHANICS 2 (A) TEST PAPER 9 Page 2}
Edexcel M2 Q6
12 marks Standard +0.3
A ball is hit with initial speed \(u \mathrm {~ms} ^ { - 1 }\), at an angle \(\theta\) above the horizontal, from a point at a height of \(h \mathrm {~m}\) above horizontal ground. The ball, which is modelled as a particle moving freely under gravity, hits the ground at a horizontal distance \(d \mathrm {~m}\) from the point of projection.
  1. Prove that \(\frac { g d ^ { 2 } } { 2 u ^ { 2 } } \sec ^ { 2 } \theta - d \tan \theta - h = 0\). Given further that \(u = 14 , h = 7\) and \(d = 14\), and assuming the result \(\sec ^ { 2 } \theta = 1 + \tan ^ { 2 } \theta\),
  2. find the value of \(\theta\).