Questions — AQA M3 (75 questions)

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AQA M3 2010 June Q2
2 A projectile is fired from a point \(O\) on top of a hill with initial velocity \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal and moves in a vertical plane. The horizontal and upward vertical distances of the projectile from \(O\) are \(x\) metres and \(y\) metres respectively.
    1. Show that, during the flight, the equation of the trajectory of the projectile is given by $$y = x \tan \theta - \frac { g x ^ { 2 } } { 12800 } \left( 1 + \tan ^ { 2 } \theta \right)$$
    2. The projectile hits a target \(A\), which is 20 m vertically below \(O\) and 400 m horizontally from \(O\).
      \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-04_392_1031_970_460} Show that $$49 \tan ^ { 2 } \theta - 160 \tan \theta + 41 = 0$$
    1. Find the two possible values of \(\theta\). Give your answers to the nearest \(0.1 ^ { \circ }\).
    2. Hence find the shortest possible time of the flight of the projectile from \(O\) to \(A\).
  3. State a necessary modelling assumption for answering part (a)(i).
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-05_2484_1709_223_153}
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-07_2484_1709_223_153}
AQA M3 2010 June Q3
3 Three smooth spheres, \(A , B\) and \(C\), of equal radii have masses \(1 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(x \mathrm {~kg}\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(3 u\) directly towards \(B\) and collides with it.
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-08_250_835_511_605} The coefficient of restitution between each pair of spheres is \(\frac { 1 } { 3 }\).
  1. Show that \(A\) is brought to rest by the impact and find the speed of \(B\) immediately after the collision in terms of \(u\).
  2. Subsequently, \(B\) collides with \(C\). Show that the speed of \(C\) immediately after the collision is \(\frac { 4 u } { 3 + x }\).
    Find the speed of \(B\) immediately after the collision in terms of \(u\) and \(x\).
  3. Show that \(B\) will collide with \(A\) again if \(x > 9\).
  4. Given that \(x = 5\), find the magnitude of the impulse exerted on \(C\) by \(B\) in terms of \(u\).
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-09_2484_1709_223_153}
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-10_2484_1712_223_153}
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-11_2484_1709_223_153}
AQA M3 2010 June Q4
4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed east, north and vertically upwards respectively. At time \(t = 0\), the position vectors of two small aeroplanes, \(A\) and \(B\), relative to a fixed origin \(O\) are \(( - 60 \mathbf { i } + 30 \mathbf { k } ) \mathrm { km }\) and \(( - 40 \mathbf { i } + 10 \mathbf { j } - 10 \mathbf { k } ) \mathrm { km }\) respectively. The aeroplane \(A\) is flying with constant velocity \(( 250 \mathbf { i } + 50 \mathbf { j } - 100 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the aeroplane \(B\) is flying with constant velocity \(( 200 \mathbf { i } + 25 \mathbf { j } + 50 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Write down the position vectors of \(A\) and \(B\) at time \(t\) hours.
  2. Show that the position vector of \(A\) relative to \(B\) at time \(t\) hours is \(( ( - 20 + 50 t ) \mathbf { i } + ( - 10 + 25 t ) \mathbf { j } + ( 40 - 150 t ) \mathbf { k } ) \mathrm { km }\).
  3. Show that \(A\) and \(B\) do not collide.
  4. Find the value of \(t\) when \(A\) and \(B\) are closest together.
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-13_2484_1709_223_153}
AQA M3 2010 June Q5
5 A smooth sphere is moving on a smooth horizontal surface when it strikes a smooth vertical wall and rebounds. Immediately before the impact, the sphere is moving with speed \(4 \mathrm {~ms} ^ { - 1 }\) and the angle between the sphere's direction of motion and the wall is \(\alpha\). Immediately after the impact, the sphere is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) and the angle between the sphere's direction of motion and the wall is \(40 ^ { \circ }\). The coefficient of restitution between the sphere and the wall is \(\frac { 2 } { 3 }\).
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-14_529_250_831_909}
  1. Show that \(\tan \alpha = \frac { 3 } { 2 } \tan 40 ^ { \circ }\).
  2. Find the value of \(v\).
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-15_2484_1709_223_153}
AQA M3 2010 June Q6
6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 1 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to the unit vector \(\mathbf { i }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-16_456_1052_721_550}
  1. Briefly state why the components of the velocities of \(A\) and \(B\) parallel to the unit vector \(\mathbf { j }\) are not changed by the collision.
  2. The coefficient of restitution between the spheres is 0.5 . Find the velocities of \(A\) and \(B\) immediately after the collision. \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-17_2484_1709_223_153}
    \(7 \quad\) A ball is projected from a point \(O\) on a smooth plane which is inclined at an angle of \(35 ^ { \circ }\) above the horizontal. The ball is projected with velocity \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the plane, as shown in the diagram. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane at the point \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-18_321_838_605_577}
  3. Find the components of the velocity of the ball, parallel and perpendicular to the plane, as it strikes the inclined plane at \(A\).
  4. On striking the plane at \(A\), the ball rebounds. The coefficient of restitution between the plane and the ball is \(\frac { 4 } { 5 }\). Show that the ball next strikes the plane at a point lower down than \(A\).
    \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-19_2484_1709_223_153}
AQA M3 2011 June Q1
1 A ball of mass 0.2 kg is hit directly by a bat. Just before the impact, the ball is travelling horizontally with speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Just after the impact, the ball is travelling horizontally with speed \(32 \mathrm {~ms} ^ { - 1 }\) in the opposite direction.
  1. Find the magnitude of the impulse exerted on the ball.
  2. At time \(t\) seconds after the ball first comes into contact with the bat, the force exerted by the bat on the ball is \(k \left( 0.9 t - 10 t ^ { 2 } \right)\) newtons, where \(k\) is a constant and \(0 \leqslant t \leqslant 0.09\). The bat stays in contact with the ball for 0.09 seconds. Find the value of \(k\).
AQA M3 2011 June Q2
2 The time, \(t\), for a single vibration of a piece of taut string is believed to depend on
the length of the taut string, \(l\),
the tension in the string, \(F\),
the mass per unit length of the string, \(q\), and
a dimensionless constant, \(k\),
such that $$t = k l ^ { \alpha } F ^ { \beta } q ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).
AQA M3 2011 June Q3
3 (In this question, use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).)
A golf ball is hit from a point \(O\) on a horizontal golf course with a velocity of \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\). The golf ball travels in a vertical plane through \(O\). During its flight, the horizontal and upward vertical distances of the golf ball from \(O\) are \(x\) and \(y\) metres respectively.
  1. Show that the equation of the trajectory of the golf ball during its flight is given by $$x ^ { 2 } \tan ^ { 2 } \theta - 320 x \tan \theta + \left( x ^ { 2 } + 320 y \right) = 0$$
    1. The golf ball hits the top of a tree, which has a vertical height of 8 m and is at a horizontal distance of 150 m from \(O\). Find the two possible values of \(\theta\).
    2. Which value of \(\theta\) gives the shortest possible time for the golf ball to travel from \(O\) to the top of the tree? Give a reason for your choice of \(\theta\).
AQA M3 2011 June Q4
4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. A helicopter, \(A\), is travelling in the direction of the vector \(- 2 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\) with constant speed \(140 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Another helicopter, \(B\), is travelling in the direction of the vector \(2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\) with constant speed \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  1. Find the velocity of \(A\) relative to \(B\).
  2. Initially, the position vectors of \(A\) and \(B\) are \(( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) and \(( - 3 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) respectively, relative to a fixed origin. Write down the position vector of \(A\) relative to \(B , t\) hours after they leave their initial positions.
  3. Find the distance between \(A\) and \(B\) when they are closest together.
    \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-10_2486_1714_221_153}
    \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-11_2486_1714_221_153}
AQA M3 2011 June Q5
5 A ball is dropped from a height of 2.5 m above a horizontal floor. The ball bounces repeatedly on the floor.
  1. Find the speed of the ball when it first hits the floor.
  2. The coefficient of restitution between the ball and the floor is \(e\).
    1. Show that the time taken between the first contact of the ball with the floor and the second contact of the ball with the floor is \(\frac { 10 e } { 7 }\) seconds.
    2. Find, in terms of \(e\), the time taken between the second contact and the third contact of the ball with the floor.
  3. Find, in terms of \(e\), the total vertical distance travelled by the ball from when it is dropped until its third contact with the floor.
  4. State a modelling assumption for answering this question, other than the ball being a particle.
AQA M3 2011 June Q6
6 A projectile is fired from a point \(O\) on a plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The projectile is fired up the plane with velocity \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the inclined plane. The projectile travels in a vertical plane containing a line of greatest slope of the inclined plane. The projectile hits a target \(T\) on the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-16_481_922_664_593}
  1. Given that \(O T = 200 \mathrm {~m}\), determine the value of \(u\).
  2. Find the greatest perpendicular distance of the projectile from the inclined plane.
    (4 marks)
    \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-18_2486_1714_221_153}
    \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-19_2486_1714_221_153}
AQA M3 2011 June Q7
7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(4 m\) and \(3 m\) respectively. The sphere \(A\) is moving on a smooth horizontal surface and collides with the sphere \(B\), which is stationary on the same surface. Just before the collision, \(A\) is moving with speed \(u\) at an angle of \(30 ^ { \circ }\) to the line of centres, as shown in the diagram below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Before collision} \includegraphics[alt={},max width=\textwidth]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_362_933_664_450}
\end{figure} Immediately after the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0590950d-145c-4ae2-bc3c-f61a9433d158-20_449_927_1244_456} The coefficient of restitution between the spheres is \(\frac { 5 } { 9 }\).
  1. Find the value of \(\alpha\).
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) during the collision.
    \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-23_2349_1707_221_153}
AQA M3 2012 June Q1
1 An ice-hockey player has mass 60 kg . He slides in a straight line at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on the horizontal smooth surface of an ice rink towards the vertical perimeter wall of the rink, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-02_476_594_769_715} The player collides directly with the wall, and remains in contact with the wall for 0.5 seconds. At time \(t\) seconds after coming into contact with the wall, the force exerted by the wall on the player is \(4 \times 10 ^ { 4 } t ^ { 2 } ( 1 - 2 t )\) newtons, where \(0 \leqslant t \leqslant 0.5\).
  1. Find the magnitude of the impulse exerted by the wall on the player.
  2. The player rebounds from the wall. Find the player's speed immediately after the collision.
AQA M3 2012 June Q2
2 A pile driver of mass \(m _ { 1 }\) falls from a height \(h\) onto a pile of mass \(m _ { 2 }\), driving the pile a distance \(s\) into the ground. The pile driver remains in contact with the pile after the impact. A resistance force \(R\) opposes the motion of the pile into the ground. Elizabeth finds an expression for \(R\) as $$R = \frac { g } { s } \left[ s \left( m _ { 1 } + m _ { 2 } \right) + \frac { h \left( m _ { 1 } \right) ^ { 2 } } { m _ { 1 } + m _ { 2 } } \right]$$ where \(g\) is the acceleration due to gravity.
Determine whether the expression is dimensionally consistent.
AQA M3 2012 June Q3
3 (In this question, take \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).)
A projectile is fired from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal so as to pass through a point \(P\). The projectile travels in a vertical plane through \(O\) and \(P\). The point \(P\) is at a horizontal distance \(2 k\) from \(O\) and at a vertical distance \(k\) above \(O\).
  1. Show that \(\alpha\) satisfies the equation $$20 k \tan ^ { 2 } \alpha - 2 u ^ { 2 } \tan \alpha + u ^ { 2 } + 20 k = 0$$
  2. Deduce that $$u ^ { 4 } - 20 k u ^ { 2 } - 400 k ^ { 2 } \geqslant 0$$
AQA M3 2012 June Q4
4 The diagram shows part of a horizontal snooker table of width 1.69 m . A player strikes the ball \(B\) directly, and it moves in a straight line. The ball hits the cushion of the table at \(C\) before rebounding and moving to the pocket at \(P\) at the corner of the table, as shown in the diagram. The point \(C\) is 1.20 m from the corner \(A\) of the table. The ball has mass 0.15 kg and, immediately before the collision with the cushion, it has velocity \(u\) in a direction inclined at \(60 ^ { \circ }\) to the cushion. The table and the cushion are modelled as smooth.
\includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-08_517_963_719_511}
  1. Find the coefficient of restitution between the ball and the cushion.
  2. Show that the magnitude of the impulse on the cushion at \(C\) is approximately \(0.236 u\).
  3. Find, in terms of \(u\), the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\).
  4. Explain how you have used the assumption that the cushion is smooth in your answers.
AQA M3 2012 June Q5
5 A particle is projected from a point \(O\) on a smooth plane, which is inclined at \(25 ^ { \circ }\) to the horizontal. The particle is projected up the plane with velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(30 ^ { \circ }\) above the plane. The particle strikes the plane for the first time at a point \(A\). The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane.
\includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-12_518_839_552_630}
  1. Find the time taken by the particle to travel from \(O\) to \(A\).
  2. The coefficient of restitution between the particle and the inclined plane is \(\frac { 2 } { 3 }\). Find the speed of the particle as it rebounds from the inclined plane at \(A\). (8 marks)
AQA M3 2012 June Q6
6 At noon, two ships, \(A\) and \(B\), are a distance of 12 km apart, with \(B\) on a bearing of \(065 ^ { \circ }\) from \(A\). The ship \(B\) travels due north at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The ship \(A\) travels at a constant speed of \(18 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-16_492_585_445_738}
  1. Find the direction in which \(A\) should travel in order to intercept \(B\). Give your answer as a bearing.
  2. In fact, the ship \(A\) actually travels on a bearing of \(065 ^ { \circ }\).
    1. Find the distance between the ships when they are closest together.
    2. Find the time when the ships are closest together.
AQA M3 2012 June Q7
7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it collides with the sphere \(B\), which has velocity \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision, the velocity of the sphere \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the velocity of \(A\) immediately after the collision.
  2. Show that the impulse exerted on \(B\) in the collision is \(( 6 m \mathbf { j } )\) Ns.
  3. Find the coefficient of restitution between the two spheres.
  4. After the collision, each sphere moves in a straight line with constant speed. Given that the radius of each sphere is 0.05 m , find the time taken, from the collision, until the centres of the spheres are 1.10 m apart.
AQA M3 2013 June Q1
1 A stone, of mass 2 kg , is moving in a straight line on a smooth horizontal sheet of ice under the action of a single force which acts in the direction of motion. At time \(t\) seconds, the force has magnitude \(( 3 t + 1 )\) newtons, \(0 \leqslant t \leqslant 3\). When \(t = 0\), the stone has velocity \(1 \mathrm {~ms} ^ { - 1 }\).
When \(t = T\), the stone has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(T\).
(6 marks)
AQA M3 2013 June Q2
2 A car has mass \(m\) and travels up a slope which is inclined at an angle \(\theta\) to the horizontal. The car reaches a maximum speed \(v\) at a height \(h\) above its initial position. A constant resistance force \(R\) opposes the motion of the car, which has a maximum engine power output \(P\). Neda finds a formula for \(P\) as $$P = m g v \sin \theta + R v + \frac { 1 } { 2 } m v ^ { 3 } \frac { \sin \theta } { h }$$ where \(g\) is the acceleration due to gravity.
Given that the engine power output may be measured in newton metres per second, determine whether the formula is dimensionally consistent.
AQA M3 2013 June Q3
3 A player projects a basketball with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The basketball travels in a vertical plane through the point of projection and goes into the basket. During the motion, the horizontal and upward vertical displacements of the basketball from the point of projection are \(x\) metres and \(y\) metres respectively.
\includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-06_737_937_513_550}
  1. Find an expression for \(y\) in terms of \(x , u , g\) and \(\tan \theta\).
  2. The player projects the basketball with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 0.5 metres vertically below and 5 metres horizontally from the basket.
    1. Show that the two possible values of \(\theta\) are approximately \(63.1 ^ { \circ }\) and \(32.6 ^ { \circ }\), correct to three significant figures.
    2. Given that the player projects the basketball at \(63.1 ^ { \circ }\) to the horizontal, find the direction of the motion of the basketball as it enters the basket. Give your answer to the nearest degree.
  3. State a modelling assumption needed for answering parts (a) and (b) of this question.
    (1 mark)
AQA M3 2013 June Q4
4 A smooth sphere \(A\), of mass \(m\), is moving with speed \(4 u\) in a straight line on a smooth horizontal table. A smooth sphere \(B\), of mass \(3 m\), has the same radius as \(A\) and is moving on the table with speed \(2 u\) in the same direction as \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-10_289_780_493_625} The sphere \(A\) collides directly with sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Find, in terms of \(u\) and \(e\), the speeds of \(A\) and \(B\) immediately after the collision.
  2. Show that the speed of \(B\) after the collision cannot be greater than \(3 u\).
  3. Given that \(e = \frac { 2 } { 3 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) in the collision.
AQA M3 2013 June Q5
5 A particle is projected from a point \(O\) on a plane which is inclined at an angle \(\theta\) to the horizontal. The particle is projected down the plane with velocity \(u\) at an angle \(\alpha\) above the plane. The particle first strikes the plane at a point \(P\), as shown in the diagram. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane.
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  1. Given that the time of flight from \(O\) to \(P\) is \(T\), find an expression for \(u\) in terms of \(\theta , \alpha , T\) and \(g\).
  2. Using the identity \(\cos ( X - Y ) = \cos X \cos Y + \sin X \sin Y\), show that the distance \(O P\) is given by \(\frac { 2 u ^ { 2 } \sin \alpha \cos ( \alpha - \theta ) } { g \cos ^ { 2 } \theta }\).
    (6 marks)
AQA M3 2013 June Q6
6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 4 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to unit vector \(\mathbf { i }\). The direction of motion of \(B\) is changed through \(90 ^ { \circ }\) by the collision, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-14_332_1184_566_543}
  1. Show that the velocity of \(B\) immediately after the collision is \(\left( \frac { 9 } { 2 } \mathbf { i } - 3 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Find the coefficient of restitution between the spheres.
  3. Find the impulse exerted on \(B\) during the collision. State the units of your answer.