- Particles \(A , B\) and \(C\) of masses \(4 m , 3 m\) and \(m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected towards each other with speeds \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, and collide directly.
As a result of the collision, \(A\) is brought to rest and \(B\) rebounds with speed \(k v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).
- Show that \(u = 3 v\).
- Find the value of \(k\).
Immediately after the collision between \(A\) and \(B\), particle \(C\) is projected with speed \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) so that \(B\) and \(C\) collide directly.
- Show that there is no further collision between \(A\) and \(B\).