Who will win?

Two cars \(A\) and \(B\) are racing along a straight line, car \(A\) is leading and their relative velocity is directly proportional to the distance between the two cars. When the lead of car \(A\) is \(x_{1}=10\text{ m}\), it is running \(10\text{ ms}^{-1}\) faster than car \(B\). If the time taken by car \(A\) to increase its lead to \(x_{2}=20\text{ m}\) from car \(B\) is in the form \(t=\ln n \), find \(n\).

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