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\includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_168_1032_292_552} Three uniform spheres \(A , B\) and \(C\) have masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) (see diagram). Air resistance may be ignored.

1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed with which \(B\) starts to move.
2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).
3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\).
4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers.