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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