AQA M3 (Mechanics 3) 2016 June

At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of \(400 \text{ m s}^{-1}\). The mass of the gun is \(1.5\) kg and the mass of the bullet is \(30\) grams.

1. Find the speed of recoil of the gun. [2 marks]
2. Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired. [2 marks]

A lunar mapping satellite of mass \(m_1\) measured in kg is in an elliptic orbit around the moon, which has mass \(m_2\) measured in kg. The effective potential, \(E\), of the satellite is given by $$E = \frac{K^2}{2m_1r^2} - \frac{Gm_1m_2}{r}$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G\) Nm\(^2\)kg\(^{-2}\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:

1. \(E\), [3 marks]
2. \(K\). [3 marks]

A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}

1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
2. Find the distance \(PQ\). [7 marks]

A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4m\), has the same radius as \(A\) and is moving on the table with velocity \(u\). \includegraphics{figure_4} The sphere \(A\) collides directly with the sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. 1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision. [6 marks]
   2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined. [2 marks]
2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{5}\). The sphere \(B\) collides with \(A\) again after rebounding from the wall. Show that \(e < b\), where \(b\) is a constant to be determined. [3 marks]
3. Given that \(e = \frac{4}{7}\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall. [3 marks]

A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. 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 for the first time at a point \(A\). \includegraphics{figure_5}

1. 1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
   2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]

In this question use \(\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}\). A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60°\) with the wall, as shown in the diagram. \includegraphics{figure_6} The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{1}{4}u(1 + e)\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision. [7 marks]
2. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac{1 + e}{7 - e}\). [7 marks]

A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \text{ m s}^{-1}\), the truck is moving on a bearing of \(340°\). Relative to the car, which is travelling due east at a speed of \(15 \text{ m s}^{-1}\), the truck is moving on a bearing of \(300°\).

1. Show that the speed of the truck is approximately \(24.7 \text{ m s}^{-1}\) and that it is moving on a bearing of \(318°\), correct to the nearest degree. [8 marks]
2. At the instant when the truck is at a distance of \(400\) metres from the quad-bike, the bearing of the truck from the quad-bike is \(060°\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \text{ m s}^{-1}\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible. [5 marks]