Questions — AQA M2 (163 questions)

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AQA M2 2012 January Q2
2 A particle, of mass 50 kg , moves on a smooth horizontal plane. A single horizontal force $$\left[ \left( 300 t - 60 t ^ { 2 } \right) \mathbf { i } + 100 \mathrm { e } ^ { - 2 t } \mathbf { j } \right] \text { newtons }$$ acts on the particle at time \(t\) seconds.
The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the particle at time \(t\).
  2. When \(t = 0\), the velocity of the particle is \(( 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
  3. Calculate the speed of the particle when \(t = 1\).
AQA M2 2012 January Q3
3 A uniform ladder \(P Q\), of length 8 metres and mass 28 kg , rests in equilibrium with its foot, \(P\), on a rough horizontal floor and its top, \(Q\), leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the angle between the ladder and the floor is \(69 ^ { \circ }\). A man, of mass 72 kg , is standing at the point \(C\) on the ladder so that the distance \(P C\) is 6 metres. The man may be modelled as a particle at \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-3_679_679_685_678}
  1. Draw a diagram to show the forces acting on the ladder.
  2. With the man standing at the point \(C\), the ladder is on the point of slipping.
    1. Show that the magnitude of the reaction between the ladder and the vertical wall is 256 N , correct to three significant figures.
    2. Find the coefficient of friction between the ladder and the horizontal floor.
AQA M2 2012 January Q4
4 A car travels along a straight horizontal road. When its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car experiences a resistance force of magnitude \(25 v\) newtons.
  1. The car has a maximum constant speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on this road. Show that the power being used to propel the car at this speed is 44100 watts.
  2. The car has mass 1500 kg . Find the acceleration of the car when it is travelling at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on this road under a power of 44100 watts.
AQA M2 2012 January Q5
5 A parcel is placed on a flat rough horizontal surface in a van. The van is travelling along a horizontal road. It travels around a bend of radius 34 m at a constant speed. The coefficient of friction between the parcel and the horizontal surface in the van is 0.85 . Model the parcel as a particle travelling around part of a circle of radius 34 m and centre \(O\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-4_348_700_687_667} Find the greatest speed at which the van can travel around the bend without causing the parcel to slide.
AQA M2 2012 January Q6
6 Alice places a toy, of mass 0.4 kg , on a slope. The toy is set in motion with an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) down the slope. The resultant force acting on the toy is \(( 2 - 4 v )\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the toy's velocity at time \(t\) seconds after it is set in motion.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 10 ( v - 0.5 )\).
  2. By using \(\int \frac { 1 } { v - 0.5 } \mathrm {~d} v = - \int 10 \mathrm {~d} t\), find \(v\) in terms of \(t\).
  3. Find the time taken for the toy's velocity to reduce to \(0.55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    \(7 \quad\) A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). With the string taut, the bead is at the point \(B\), vertically below \(O\), when it is set into vertical circular motion with an initial horizontal velocity \(u\), as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-5_616_613_520_733} The string does not become slack in the subsequent motion. The velocity of the bead at the point \(A\), where \(A\) is vertically above \(O\), is \(v\).
AQA M2 2012 January Q8
8 An elastic string has one end attached to a point \(O\) fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point \(A\), on the surface, 3 metres from the point \(O\).
  1. Calculate the elastic potential energy when the particle is at the point \(A\).
  2. The particle is released from rest at the point \(A\) and moves in a straight line towards \(O\). The particle is next at rest at the point \(B\). The distance \(A B\) is 5 metres.
    \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
  3. Show that the particle does not remain at rest at the point \(B\).
  4. The particle next comes to rest at a point \(C\) with the string slack. Find the distance \(B C\).
  5. Hence, or otherwise, find the total distance travelled by the particle after it is released from the point \(A\).
AQA M2 2013 January Q1
1 Tim is playing cricket. He hits a ball at a point \(A\). The speed of the ball immediately after being hit is \(11 \mathrm {~ms} ^ { - 1 }\). The ball strikes a tree at a point \(B\). The height of \(B\) is 5 metres above the height of \(A\).
The ball is to be modelled as a particle of mass 0.16 kg being acted upon only by gravity.
  1. Calculate the initial kinetic energy of the ball.
  2. Calculate the potential energy gained by the ball as it moves from the point \(A\) to the point \(B\).
    1. Find the kinetic energy of the ball immediately before it strikes the tree.
    2. Hence find the speed of the ball immediately before it strikes the tree.
AQA M2 2013 January Q2
2 A particle moves in a horizontal plane. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 12 \cos \left( \frac { \pi } { 3 } t \right) \mathbf { i } - 9 t ^ { 2 } \mathbf { j }$$
  1. Find an expression for the acceleration of the particle at time \(t\).
  2. The particle, which has mass 4 kg , moves under the action of a single force, \(\mathbf { F }\) newtons.
    1. Find an expression for the force \(\mathbf { F }\) in terms of \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
  3. When \(t = 3\), the particle is at the point with position vector \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
AQA M2 2013 January Q3
3 A van, of mass 1500 kg , travels at a constant speed of \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 25 }\). The van experiences a resistance force of 8000 N .
Find the power output of the van's engine, giving your answer in kilowatts.
AQA M2 2013 January Q4
4 The diagram shows a uniform lamina which is in the shape of two identical rectangles \(A X G H\) and \(Y B C D\) and a square \(X Y E F\), arranged as shown. The length of \(A X\) is 10 cm , the length of \(X Y\) is 10 cm and the length of \(A H\) is 30 cm .
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-3_1183_1278_513_374}
  1. Explain why the centre of mass of the lamina is 15 cm from \(A H\).
  2. Find the distance of the centre of mass of the lamina from \(A B\).
  3. The lamina is freely suspended from the point \(H\). Find, to the nearest degree, the angle between \(H G\) and the horizontal when the lamina is in equilibrium.
AQA M2 2013 January Q5
5 A particle, of mass 12 kg , is moving along a straight horizontal line. At time \(t\) seconds, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the particle moves, it experiences a resistance force of magnitude \(4 v ^ { \frac { 1 } { 3 } }\). No other horizontal force acts on the particle. The initial speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that $$v = \left( 4 - \frac { 2 } { 9 } t \right) ^ { \frac { 3 } { 2 } }$$
  2. Find the value of \(t\) when the particle comes to rest.
AQA M2 2013 January Q6
6 A light inextensible string has one end attached to a particle, \(P\), of mass 2 kg . The other end of the string is attached to the fixed point \(A\). The point \(A\) is vertically above the point \(B\). The particle moves at a constant speed in a horizontal circle of radius 0.8 m and centre \(B\). The tension in the string is 34 N . The string is inclined at an angle \(\theta\) to the vertical, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-4_760_816_1436_596}
  1. Find the angle \(\theta\).
  2. Find the speed of the particle.
  3. Find the time taken for the particle to make one complete revolution.
AQA M2 2013 January Q7
7 A small ball, of mass 3 kg , is suspended from a fixed point \(O\) by a light inextensible string of length 1.2 m . Initially, the string is taut and the ball is at the point \(P\), vertically below \(O\). The ball is then set into motion with an initial horizontal velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball moves in a vertical circle, centre \(O\). The point \(A\), on the circle, is such that angle \(A O P\) is \(25 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-5_663_702_660_701}
  1. Find the speed of the ball at the point \(A\).
  2. Find the tension in the string when the ball is at the point \(A\).
AQA M2 2013 January Q8
8
  1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
  2. A particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(O\). The string has natural length 1.6 m and modulus of elasticity 392 N .
    1. Find the extension of the string when the particle hangs in equilibrium.
    2. The particle is pulled down to a point \(A\), which is 2.2 m below the point \(O\). Calculate the elastic potential energy in the string.
    3. The particle is released when it is at rest at the point \(A\). Calculate the distance of the particle from the point \(A\) when its speed first reaches \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M2 2013 January Q9
9 A smooth hollow hemisphere, of radius \(a\) and centre \(O\), is fixed so that its rim is in a horizontal plane. A smooth uniform \(\operatorname { rod } A B\), of mass \(m\), is in equilibrium, with one end \(A\) resting on the inside of the hemisphere and the point \(C\) on the rod being in contact with the rim of the hemisphere. The rod, of length \(l\), is inclined at an angle \(\theta\) to the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-6_453_828_559_591}
  1. Explain why the reaction between the rod and the hemisphere at point \(A\) acts through \(O\).
  2. Draw a diagram to show the forces acting on the rod.
  3. Show that \(l = \frac { 4 a \cos 2 \theta } { \cos \theta }\).
AQA M2 2006 June Q1
1 A particle moves in a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, its position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = \left( 2 t ^ { 3 } - t ^ { 2 } + 6 \right) \mathbf { i } + \left( 8 - 4 t ^ { 3 } + t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
    1. Find the velocity of the particle when \(t = \frac { 1 } { 3 }\).
    2. State the direction in which the particle is travelling at this time.
  3. Find the acceleration of the particle when \(t = 4\).
  4. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 4\).
AQA M2 2006 June Q2
2 A ball of mass 0.6 kg is thrown vertically upwards from ground level with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the initial kinetic energy of the ball.
  2. Assuming that no resistance forces act on the ball, use an energy method to find the maximum height reached by the ball.
  3. An experiment is conducted to confirm the maximum height for the ball calculated in part (b). In this experiment the ball rises to a height of only 8 metres.
    1. Find the work done against the air resistance force that acts on the ball as it moves.
    2. Assuming that the air resistance force is constant, find its magnitude.
  4. Explain why it is not realistic to model the air resistance as a constant force.
AQA M2 2006 June Q3
3 The diagram shows a uniform rod, \(A B\), of mass 10 kg and length 5 metres. The rod is held in equilibrium in a horizontal position, by a support at \(C\) and a light vertical rope attached to \(A\), where \(A C\) is 2 metres.
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_237_680_479_648}
  1. Draw and label a diagram to show the forces acting on the rod.
  2. Show that the tension in the rope is 24.5 N .
  3. A package of mass \(m \mathrm {~kg}\) is suspended from \(B\). The tension in the rope has to be doubled to maintain equilibrium.
    1. Find \(m\).
    2. Find the magnitude of the force exerted on the rod by the support.
  4. Explain how you have used the fact that the rod is uniform in your solution.
AQA M2 2006 June Q4
4 A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(P\) vertically below \(O\). The particle is then set into motion with a horizontal velocity \(U\) so that it moves in a complete vertical circle with centre \(O\). The point \(Q\) on the circle is such that \(\angle P O Q = 60 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_566_540_1797_751}
  1. Find, in terms of \(g , l\) and \(U\), the speed of the particle at \(Q\).
  2. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle is at \(Q\).
  3. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle returns to \(P\).
    (2 marks)
AQA M2 2006 June Q5
5 The graph shows a model for the resultant horizontal force on a car, which varies as it accelerates from rest for 20 seconds. The mass of the car is 1200 kg .
\includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-4_373_1203_445_390}
  1. The acceleration of the car at time \(t\) seconds is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that $$a = \frac { 2 } { 3 } + \frac { t } { 20 } , \text { for } 0 \leqslant t \leqslant 20$$
  2. Find an expression for the velocity of the car at time \(t\).
  3. Find the distance travelled by the car in the 20 seconds.
  4. An alternative model assumes that the resultant force increases uniformly from 900 to 2100 newtons during the 20 seconds. Which term in your expression for the velocity would change as a result of this modification? Explain why.
AQA M2 2006 June Q6
6 A car of mass 1200 kg travels round a roundabout on a horizontal, circular path at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the circle is 50 metres. Assume that there is no resistance to the motion of the car and that the car can be modelled as a particle.
  1. A friction force, directed towards the centre of the roundabout, acts on the car as it moves. Show that the magnitude of this friction force is 4704 N .
  2. The coefficient of friction between the car and the road is \(\mu\). Show that \(\mu \geqslant 0.4\).
AQA M2 2006 June Q7
7 A particle of mass 20 kg moves along a straight horizontal line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance force of magnitude \(10 \sqrt { v }\) newtons acts on the particle while it is moving. At time \(t = 0\) the velocity of the particle is \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Show that, at time \(t\) $$v = \left( \frac { 20 - t } { 4 } \right) ^ { 2 }$$
  2. State the value of \(t\) when the particle comes to rest.
AQA M2 2007 June Q1
1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).
  1. Find the initial kinetic energy of the box.
  2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
  3. Hence find the speed of the box when it hits the ground.
  4. State two modelling assumptions which you have made.
AQA M2 2007 June Q2
2 A uniform lamina is in the shape of a rectangle \(A B C D\) and a square \(E F G H\), as shown in the diagram. The length \(A B\) is 20 cm , the length \(B C\) is 30 cm , the length \(D E\) is 5 cm and the length \(E F\) is 10 cm . The point \(P\) is the midpoint of \(A B\) and the point \(Q\) is the midpoint of \(H G\).
\includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-2_615_1221_1585_429}
  1. Explain why the centre of mass of the lamina lies on \(P Q\).
  2. Find the distance of the centre of mass of the lamina from \(A B\).
  3. The lamina is freely suspended from \(A\). Find, to the nearest degree, the angle between \(A D\) and the vertical when the lamina is in equilibrium.
AQA M2 2007 June Q3
3 A particle has mass 800 kg . A single force of \(( 2400 \mathbf { i } - 4800 t \mathbf { j } )\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.
  1. Find the acceleration of the particle at time \(t\).
  2. At time \(t = 0\), the velocity of the particle is \(( 6 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The velocity of the particle at time \(t\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Show that $$\mathbf { v } = ( 6 + 3 t ) \mathbf { i } + \left( 30 - 3 t ^ { 2 } \right) \mathbf { j }$$
  3. Initially, the particle is at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).