6 Three particles \(A , B\) and \(C\) of masses \(5 \mathrm {~kg} , 1 \mathrm {~kg}\) and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(X Y Z\). Initially \(A\) is at \(X , B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v \mathrm {~ms} ^ { - 1 }\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).

1. Show that after the second collision the speed of \(E\) is \(\frac { 15 - v } { 4 } \mathrm {~ms} ^ { - 1 }\).
2. The total loss of kinetic energy of the system due to the two collisions is 63 J . Use the result from (a) to show that \(v = 3\).
3. It is given that the distance \(X Y\) is 36 m and the distance \(Y Z\) is 98 m .
   1. Find the time between the two collisions.
   2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\).