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Unit Vectors

A and B parti...

Question

A and B participate in a race with acceleration $a_{1}$ and $a_{2}$ respectively. A reaches t times earlier than B at finish line and there velocities at finish line are $v_{1}$ and $v_{2}$ respectively. If difference between their velocities is v then find the value of v

\frac{a_{1} + a_{2}}{2}

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\sqrt{a_{1} a_{2}} t

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\frac{a_{1} a_{2}}{a_{1} + a_{2}} t

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\frac{2 a_{1} a_{2}}{a_{1} + a_{2}} t

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Solution

The correct option is B $\sqrt{a_{1} a_{2}} t$
$s = \frac{1}{2} a_{1} t_{0}^{2} = \frac{1}{2} a_{2} (t_{0} + t)^{2}$

$\sqrt{a_{1}} - \sqrt{a_{2}} \times \sqrt{2 s} = v$

$(\frac{1}{\sqrt{a_{2}}} - \frac{1}{\sqrt{a_{1}}}) \times \sqrt{2 s} = t$

$= \sqrt{2 s} = \frac{\sqrt{a_{1}} \sqrt{a_{2} t}}{\sqrt{a_{1}} - \sqrt{a_{2}}}$

$\frac{v}{\sqrt{a_{1}} - \sqrt{a_{2}}} = \frac{\sqrt{a_{1}} \sqrt{a_{2} t}}{\sqrt{a_{1}} - \sqrt{a_{2}}}$

$\Rightarrow v = \sqrt{a_{1} a_{2}} t$

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