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Two equal spheres A and B lie on a smoot...

Two equal spheres `A` and `B` lie on a smooth horizontal circular groove at opposite ends of a diameter. At time `t = 0`, A is projected along the groove and it first impinges on `B` at time `t = T_(1)` and again at time `t = T_(2)`. If `e` is the coefficient of restitution, the ratio `T_(2)//T_(1)` is

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`T _(1) = (pi R )/( u _(1)) " "...(1)`
`(v _(2) - v _(1))/(u _(1)) = e implies v _(2) - v _(1) = eu _(1)`
.A. makes `2^(nd)` impact with B when B travels a distance `2pi R` as seen by A after `1 ^(st)` impact. Time taken to collide A and B again:
`T _(2) - T _(1) = (2pi R )/( v _(2) - v _(1)) implies T _(2) - T _(1) = (2pi R )/( eu _(1)) ...(2)`
`(2)div(1) implies(T _(2))/( T _(1)) = (2 + e )/( e)`
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