Collision followed by wall impact

Two particles \(A\) and \(B\) move on a smooth horizontal table. The mass of \(A\) is \(m\), and the mass of \(B\) is \(4m\). Initially \(A\) is moving with speed \(u\) when it collides directly with \(B\), which is at rest on the table. As a result of the collision, the direction of motion of \(A\) is reversed. The coefficient of restitution between the particles is \(e\).

1. Find expressions for the speed of \(A\) and the speed of \(B\) immediately after the collision. [7]

In the subsequent motion, \(B\) strikes a smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{4}{5}\). Given that there is a second collision between \(A\) and \(B\),

1. show that \(\frac{1}{4} < e < \frac{9}{16}\). [5]

Given that \(e = \frac{1}{2}\),

1. find the total kinetic energy lost in the first collision between \(A\) and \(B\). [3]

A small ball \(A\) of mass \(3m\) is moving with speed \(u\) in a straight line on a smooth horizontal table. The ball collides directly with another small ball \(B\) of mass \(m\) moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\). The balls have the same radius and can be modelled as particles.

1. Find
   1. the speed of \(A\) immediately after the collision,
   2. the speed of \(B\) immediately after the collision.
   (7)

After the collision \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\).

1. Find the speed of \(B\) immediately after hitting the wall. (2)

The first collision between \(A\) and \(B\) occurred at a distance \(4a\) from the wall. The balls collide again \(T\) seconds after the first collision.

1. Show that \(T = \frac{112a}{15u}\). (6)

Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4u\) and \(5u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. [6 marks]
2. Show that \(e > \frac{1}{9}\). [2 marks]

\(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).

1. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur. [3 marks]

A particle \(A\) of mass \(2m\) is moving with speed \(u\) on a smooth horizontal surface when it collides with a stationary particle \(B\) of mass \(m\). After the collision the speed of \(A\) is \(v\), the speed of \(B\) is \(3v\) and the particles move in the same direction.

1. Find \(v\) in terms of \(u\). [3]
2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\). [2]

\(B\) subsequently hits a vertical wall which is perpendicular to the direction of motion. As a result of the impact, \(B\) loses \(\frac{3}{4}\) of its kinetic energy.

1. Show that the speed of \(B\) after hitting the wall is \(\frac{3}{4}u\). [4]
2. \(B\) then hits \(A\). Calculate the speeds of \(A\) and \(B\), in terms of \(u\), after this collision and state their directions of motion. [8]