6 The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(10 \mathrm {~ms} ^ { - 1 }\) before they collide. The kinetic energy lost due to the collision is 121.5 J .

1. Find the speed and direction of motion of each particle after the collision.
2. Find the coefficient of restitution between \(A\) and \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-5_510_1504_653_271} A particle \(P\) is projected with speed \(32 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac { 24 } { 25 }\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
3. Calculate the height of \(C\) above the ground and the distance \(A B\). Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
4. Given that the mass of \(P\) is 3 kg , find the magnitude and direction of the impulse exerted on \(P\) by the ground. The coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\).
5. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25 ^ { \circ }\) below the horizontal.