In a car race on straight road, car \(A\) takes a time \(t\) less than car \(B\) at the finish and passes finishing point with a speed \(v\) more than that of car \(B.\) Both the cars start from rest and travel with constant acceleration \(a_1\) and \(a_2\) respectively. Then \(v\) is equal to

\(\begin{array}{l}
\sqrt {\frac{{2\ell }}{{{a_2}}}}  - \sqrt {\frac{{2\ell }}{{{a_1}}}}  = t\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\frac{{\sqrt {2\ell } }}{t} = \frac{{\sqrt {{a_1}{a_2}} }}{{{{\sqrt a }_1} - \sqrt {{a_2}} }}\\
\sqrt {2{a_1}\ell }  - \sqrt {2{a_2}\ell }  = v\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\frac{{\sqrt {2\ell } }}{v} = \frac{1}{{\sqrt {{a_1}}  - \sqrt {{a_2}} }}\\
 \Rightarrow \,\,\frac{v}{t} = \sqrt {{a_1}{a_2}} \,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,v = \left( {\sqrt {{a_1}{a_2}} \,\,\,} \right)t
\end{array}\)