AQA M3 (Mechanics 3) 2009 June

1 A ball of mass \(m\) is travelling vertically downwards with speed \(u\) when it hits a horizontal floor. The ball bounces vertically upwards to a height \(h\). It is thought that \(h\) depends on \(m , u\), the acceleration due to gravity \(g\), and a dimensionless constant \(k\), such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).

2 A particle is projected from a point \(O\) on a horizontal plane and has initial velocity components of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) parallel to and perpendicular to the plane respectively. At time \(t\) seconds after projection, the horizontal and upward vertical distances of the particle from the point \(O\) are \(x\) metres and \(y\) metres respectively.

1. Show that \(x\) and \(y\) satisfy the equation $$y = - \frac { g } { 8 } x ^ { 2 } + 5 x$$
2. By using the equation in part (a), find the horizontal distance travelled by the particle whilst it is more than 1 metre above the plane.
3. Hence find the time for which the particle is more than 1 metre above the plane.

3 A fishing boat is travelling between two ports, \(A\) and \(B\), on the shore of a lake. The bearing of \(B\) from \(A\) is \(130 ^ { \circ }\). The fishing boat leaves \(A\) and travels directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A patrol boat on the lake is travelling with speed \(4 \mathrm {~ms} ^ { - 1 }\) on a bearing of \(040 ^ { \circ }\).
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}

1. Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
2. When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
   1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
   2. Given that the patrol boat intercepts the fishing boat before it reaches \(B\), find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
   3. State a modelling assumption necessary for answering this question, other than the boats being particles.

4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude \(\left( t ^ { 3 } + t \right) \mathrm { N }\), where \(t\) is the time, in seconds, for which the force has been acting.

1. Find the magnitude of the impulse exerted by the force on the particle between the times \(t = 0\) and \(t = 4\).
2. Hence find the speed of the particle when \(t = 4\).
3. Find the time taken for the particle to reach a speed of \(12 \mathrm {~ms} ^ { - 1 }\).

5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the coefficient of restitution between the spheres.
3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
4. Find the mass of \(B\).

6 A smooth sphere \(A\) of mass \(m\) is moving with speed \(5 u\) in a straight line on a smooth horizontal table. The sphere \(A\) collides directly with a smooth sphere \(B\) of mass \(7 m\), having the same radius as \(A\) and moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
\includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}

1. Show that the speed of \(B\) after the collision is \(\frac { u } { 2 } ( e + 3 )\).
2. Given that the direction of motion of \(A\) is reversed by the collision, show that \(e > \frac { 3 } { 7 }\).
3. Subsequently, \(B\) hits a wall fixed at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). Given that after \(B\) rebounds from the wall both spheres move in the same direction and collide again, show also that \(e < \frac { 9 } { 13 }\).
   (4 marks)

7 A particle is projected from a point \(O\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The particle is projected down the plane with velocity \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the plane and first strikes it 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]{719b82f7-2ab5-48db-9b2a-98284096a78a-6_449_963_488_529}

1. Show that the time taken by the particle to travel from \(O\) to \(A\) is $$\frac { 20 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
2. Find the components of the velocity of the particle parallel to and perpendicular to the slope as it hits the slope at \(A\).
3. The coefficient of restitution between the slope and the particle is 0.5 . Find the speed of the particle as it rebounds from the slope.