Questions M2 (1391 questions)

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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
\includegraphics[max width=\textwidth, alt={}, center]{46695667-272a-4ba3-9f48-1d21280aa4d2-1_202_492_690_1471}
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 \(( 2 t + 4 ) \mathbf { i } + \left( k t ^ { 2 } - 5 \right) \mathbf { j }\).
Edexcel M2 Q6
  1. A rectangular piece of cardboard \(A B C D\), measuring 30 cm by 12 cm , has a semicircle of radius 5 cm removed from it as shown.
    \includegraphics[max width=\textwidth, alt={}, center]{46695667-272a-4ba3-9f48-1d21280aa4d2-2_279_675_265_1293}
    1. Calculate the distances of the centre of mass of the remaining piece of cardboard from \(A B\) and from \(B C\).
    The remaining cardboard is suspended from \(A\) and hangs in equilibrium.
  2. Find the angle made by \(A B\) with the vertical.
Edexcel M2 Q7
7. A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v \mathrm { ms } ^ { - 1 }\), is given at time \(t\) seconds after firing by the formula $$v = p t ^ { 2 } + q t .$$ 5 seconds after firing, the rocket is travelling at \(500 \mathrm {~ms} ^ { - 1 }\).
30 seconds after firing, the rocket is travelling at \(12000 \mathrm {~ms} ^ { - 1 }\).
  1. Find the constants \(p\) and \(q\).
  2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\).
  3. Find the initial acceleration of the rocket.
  4. Find the distance of the rocket from \(O 30\) seconds after firing. From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12000 \mathrm {~ms} ^ { - 1 }\).
  5. Find the average speed of the rocket during its first 50 seconds of motion.
Edexcel M2 Q8
8. A golf ball is hit with initial velocity \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. The ball passes over a building which is 15 m tall at a distance of 30 m horizontally from the point where the ball was hit.
  1. Find the smallest possible value of \(u\). When \(u\) has this minimum value,
  2. show that the ball does not rise higher than the top of the building.
  3. Deduce the total horizontal distance travelled by the ball before it hits the ground.
  4. Briefly describe two modelling assumptions that you have made.
Edexcel M2 Q1
  1. The acceleration of a particle \(P\) is \(( 8 t - 18 ) \mathrm { ms } ^ { - 2 }\), where \(t\) seconds is the time that has elapsed since \(P\) passed through a fixed point \(O\) on the straight line on which it is moving.
    At time \(t = 3 , P\) has speed \(2 \mathrm {~ms} ^ { - 1 }\). Find
    1. the velocity of \(P\) at time \(t\),
    2. the values of \(t\) when \(P\) is instantaneously at rest.
    3. A pump raises water from a reservoir at a depth of 25 m below ground level. The water is delivered at ground level with speed \(12 \mathrm {~ms} ^ { - 1 }\) through a pipe of radius 4 cm . Find
    4. the potential and kinetic energy given to the water each second,
    5. the rate, in kW , at which the pump is working.
      [0pt] [ \(1 \mathrm {~m} ^ { 3 }\) of water has a mass of 1000 kg .]
    6. A particle \(P\) of mass 3 kg has position vector \(\mathbf { r } = \left( 2 t ^ { 2 } - 4 t \right) \mathbf { i } + \left( 1 - t ^ { 2 } \right) \mathbf { j } \mathbf { m }\) at time \(t\) seconds.
    7. Find the velocity vector of \(P\) when \(t = 3\).
    8. Find the magnitude of the force acting on \(P\), showing that this force is constant.
    \includegraphics[max width=\textwidth, alt={}]{240016bf-26ac-4ef2-bc0e-e9b48dc5b6c5-1_404_565_1409_377}
    The diagram shows a uniform lamina \(A B C D E\) formed by removing a symmetrical triangular section from a rectangular sheet of metal measuring 30 cm by 25 cm .
  2. Find the distance of the centre of mass of the lamina from \(E D\). The lamina has mass \(m\). A particle \(P\) is attached to the lamina at \(B\). The lamina is then suspended freely from \(A\) and hangs in equilibrium with \(A D\) vertical.
  3. Find, in terms of \(m\), the mass of \(P\).
Edexcel M2 Q5
5. A car, of mass 1100 kg , pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at \(1.2 \mathrm {~ms} ^ { - 2 }\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.
  1. Show that the driving force produced by the engine of the car is 2680 N .
  2. Find the tension in the tow-bar between the car and the trailer.
  3. Find the rate, in kW , at which the car's engine is working when the car is moving with speed \(18 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 2 (A) TEST PAPER 5 Page 2} 5 continued...
    When the car is moving at \(18 \mathrm {~ms} ^ { - 1 }\) it starts to climb a straight hill which is inclined at \(6 ^ { \circ }\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,
  4. find the acceleration of the car at the instant when it starts to climb the hill.
  5. Show that tension in the tow-bar remains unchanged.
Edexcel M2 Q6
6. Take \(\mathbf { g } = \mathbf { 1 0 } \mathbf { m s } ^ { - \mathbf { 2 } }\) in this question.
\includegraphics[max width=\textwidth, alt={}, center]{240016bf-26ac-4ef2-bc0e-e9b48dc5b6c5-2_276_1628_824_351} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 5 } { 13 }\), giving it an initial speed of \(52 \mathrm {~ms} ^ { - 1 }\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.
  1. Show that the height, \(y \mathrm {~m}\), of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20 t - 5 t ^ { 2 } .$$
  2. Find the time for which the ball is in flight.
  3. Find the horizontal distance travelled by the ball.
  4. Show that, if the ball is \(x \mathrm {~m}\) horizontally from \(T\) at time \(t\) seconds, then $$y = \frac { 5 } { 12 } x - \frac { 5 } { 2304 } x ^ { 2 } .$$
  5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account.
Edexcel M2 Q7
7. Two smooth spheres, \(A\) and \(B\), of equal radius but of masses \(3 m\) and \(4 m\) respectively, are free to move in a straight horizontal groove. The coefficient of restitution between them is \(e\). \(A\) is projected with speed \(u\) to hit \(B\), which is initially at rest.
  1. Show that \(B\) begins to move with speed \(\frac { 3 } { 7 } u ( 1 + e )\).
  2. Given that \(A\) is brought to rest by the collision, show that \(e = 0.75\). Having been brought to rest, \(A\) is now set in motion again by being given an impulse of magnitude \(k m u \mathrm { Ns }\), where \(k > 2 \cdot 25\). A then collides again with \(B\).
  3. Show that the speed of \(A\) after this second impact is independent of \(k\).
Edexcel M2 Q1
  1. 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
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
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
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
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
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
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
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
  1. A particle \(P\) moves in a straight line so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2 t + \frac { 8 } { t ^ { 2 } }\). Find the time when the acceleration of \(P\) is zero. (5 marks)
\includegraphics[max width=\textwidth, alt={}]{baa3525f-6263-4dec-b581-531d8258a2e1-1_350_641_594_367}
A key is modelled as a lamina which consists of a circle of radius 3 cm , with a circle of radius 1 cm removed from its centre, attached to a rectangle of length 8 cm and width 1 cm , with a rectangle measuring 3 cm by 1 cm fixed to its end as shown. Calculate the distance of the centre of mass of the key from the line marked \(A B\).
Edexcel M2 Q3
3. A van of mass 1600 kg is moving with constant speed down a straight road inclined at \(7 ^ { \circ }\) to the horizontal. The non-gravitational resistance to the van's motion has a constant magnitude of 2000 N and the engine of the van is working at a rate of 1.5 kW . Find
  1. the constant speed of the van,
  2. the acceleration of the van if the resistance is suddenly reduced to 1900 N .
Edexcel M2 Q4
4. \(\quad \mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { N }\). The body moves from the point \(P\) with position vector \(( - 3 \mathbf { i } - 15 \mathbf { j } ) \mathbf { m }\) to the point \(Q\) with position vector \(9 \mathbf { i } \mathrm {~m}\).
  1. Find the work done by the force in moving the body from \(P\) to \(Q\).
  2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\).
Edexcel M2 Q5
5. Two railway trucks \(A\) and \(B\), whose masses are \(6 m\) and \(5 m\) respectively, are moving in the same direction along a straight track with speeds \(5 u\) and \(3 u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(k v\) respectively, in the same direction as before. The coefficient of restitution between \(A\) and \(B\) is \(e\).
Modelling the trucks as particles,
  1. show that
    1. \(v = \frac { 45 u } { 5 k + 6 }\),
    2. \(v = \frac { 2 e u } { k - 1 }\).
      (8 marks)
  2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). \section*{MECHANICS 2 (A) TEST PAPER 7 Page 2}
Edexcel M2 Q6
  1. A piece of lead and a table tennis ball are dropped together from a point \(P\) near the top of the Leaning Tower of Pisa. The lead hits the ground after \(3 \cdot 3\) seconds.
    1. Calculate the height above ground from which the lead was dropped.
    According to a simple model, the ball hits the ground at the same time as the lead.
  2. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution. The piece of lead is now thrown again from \(P\), with speed \(7 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal, as shown.
    \includegraphics[max width=\textwidth, alt={}, center]{baa3525f-6263-4dec-b581-531d8258a2e1-2_582_848_854_658}
  3. Find expressions in terms of \(t\) for \(x\) and \(y\), the horizontal and vertical displacements respectively of the piece of lead from \(P\) at time \(t\) seconds after it is thrown.
  4. Deduce that \(y = \frac { \sqrt { } 3 } { 3 } x - \frac { 2 } { 15 } x ^ { 2 }\).
  5. Find the speed of the piece of lead when it has travelled 10 m horizontally from \(P\).
Edexcel M2 Q7
7.
\includegraphics[max width=\textwidth, alt={}]{baa3525f-6263-4dec-b581-531d8258a2e1-2_332_410_1837_402} A uniform ladder \(A B\), of mass \(m \mathrm {~kg}\) and length \(2 a \mathrm {~m}\), rests with its upper end \(A\) in contact with a smooth vertical wall and its lower end \(B\) in contact with a fixed peg on horizontal ground. The ladder makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac { 3 } { 4 }\).
  1. Show that the magnitude of the resultant force acting on the ladder at \(B\) is \(\frac { \sqrt { } 13 } { 3 } \mathrm { mg }\).
  2. Find, to the nearest degree, the direction of this resultant force at \(B\). The peg will break when the horizontal force acting on it exceeds \(2 m g \mathrm {~N}\). A painter of mass \(6 m \mathrm {~kg}\) starts to climb the ladder from \(B\).
  3. Find, in terms of \(a\), the greatest distance up the ladder that the painter can safely climb.
Edexcel M2 Q1
\begin{enumerate} \item A heavy ball, of mass 2 kg , rolls along a horizontal surface. It strikes a vertical wall at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and rebounds. The coefficient of restitution between the ball and the wall is \(0 \cdot 4\). Find the kinetic energy lost in the impact. \item The velocity, \(v \mathrm {~ms} ^ { - 1 }\), of a particle at time \(t \mathrm {~s}\) is given by \(v = 4 t ^ { 2 } - 9\).
  1. Find the acceleration of the particle when it is instantaneously at rest.
  2. Find the distance travelled by the particle from time \(t = 0\) until it comes to rest. \item A particle \(P\) moves in a plane such that its position vector \(\mathbf { r }\) metres at time \(t\) seconds, relative to a fixed origin \(O\), is \(\mathbf { r } = \mathrm { e } ^ { t } \mathbf { i } - 2 t \mathbf { j }\).
Edexcel M2 Q6
  1. \(P Q R\) is a triangular lamina with \(P Q = 18 \mathrm {~cm} , Q R = 24 \mathrm {~cm}\) and \(P R = 30 \mathrm {~cm}\).
    1. Verify that angle \(P Q R\) is a right angle and find the distances of the centre of mass of the lamina from (i) \(P Q\), (ii) \(Q R\).
      \includegraphics[max width=\textwidth, alt={}, center]{632b3cd2-a610-4fb1-90be-cb4efbd988d3-2_207_527_454_383}
    The lamina is held in a vertical plane and placed on a line of greatest slope of a rough plane inclined at an angle \(\theta\) to the horizontal, as shown.
  2. Find the largest value of \(\theta\) for which equilibrium will not be broken by toppling.
Edexcel M2 Q7
7. Two smooth spheres \(A\) and \(B\), of equal radius and masses \(9 m\) and \(4 m\) respectively, are moving towards each other along a straight line with speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(6 \mathrm {~ms} ^ { - 1 }\) respectively. They collide, after which the direction of motion of \(A\) remains unchanged.
  1. Show that the speed of \(B\) after the impact cannot be more than \(3 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
  2. Show that \(e < \frac { 3 } { 10 }\).
  3. Find the speeds of \(A\) and \(B\) after the impact in the case when \(e = 0\).