AQA M3 (Mechanics 3) 2012 June

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.

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.

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 $$u ^ { 4 } - 20 k u ^ { 2 } - 400 k ^ { 2 } \geqslant 0$$

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.

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)

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.

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.