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Two buses P and Q starts from a point at the same time and move in a straight line and their positions ar represented by $X_{P} (t) = α t + β t^{2}$ and $X_{Q} (t) = f t - t^{2} .$ At what time both the buses have same velocity?

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$\frac{α - f}{1 + β}$

$\frac{α + f}{2 (β - 1)}$

$\frac{α + f}{2 (1 + β)}$

$\frac{f - α}{2 (1 + β)}$

answer is D.

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Detailed Solution

$V_{P} = \frac{{dx}_{P}}{dt} = α + 2 βt$

$V_{Q} = \frac{{dx}_{Q}}{dt} = f - 2 t$

$Given V_{P} = V_{Q}$

$α + 2 βt = f - 2 t$

$2 (β + 1) t = f - α$

$\Rightarrow t = \frac{f - α}{2 (1 + β)}$

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