# Let's collide and then integrate!

**Classical Mechanics**Level 4

Three identical discs \(A,B\) and \(C\) are initially kept at rest on a smooth horizontal plane. The disc \(A\) is set in motion with velocity \(v\) after which it experiences an elastic collision simultaneously with both the discs \(B\) and \(C\). The distance between the centers of discs \(B\) and \(C\) prior to the collision is \(\eta\) times greater than the diameter of each disc. If the range of values for \(\eta\) for which the disc \(A\), after collision recoils back with some velocity can be expressed as \(\eta \in [a, \sqrt{b})\), where \(a\) and \(b\) are coprime positive integers and \(b\) is square-free, then evaluate the value of

\[\large \displaystyle \int_0^1 \dfrac{x^b - x^a}{\ln x} \,dx .\]

Submit your answer to 3 decimal places.

##### This problem is a modified version of a similar one from the book: Problems In General Physics : IE Irodov.

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