Collisions after effects!

4 identical spheres are seen above. Spheres C and D fall freely from a height \(h\) before colliding with spheres A and B on the left and right sides as shown.If the acute angle between the line joining the centres of A and C and the vertical is \(\theta\) and that for B and D with vertical is \(\alpha\) .

If the time at which the velocity vectors of A and B are mutually perpendicular to each other is given by

\(\frac{\sqrt{2gh}( Pcosec2\alpha + \sqrt{Hcosec^{2}2\alpha-YsinS\theta sinI\alpha} )}{hsinC\theta sin S\alpha}\)

find \(P+H+Y+S+I+C+S\)

Details and Assumptions

  • All collisions(including with vertical wall) are ELASTIC
  • ALL spheres are identical
  • path KL has a length \(h\)
  • no rotation takes place
  • entire motion is free of non conservative dissipative forces
  • in case of finding two values of \(t\), consider the larger value

this problem is part of the set "innovative problems in mechanics"


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