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Two buses P and Q start from a point at the same time and move in a straight line and their positions are represented by $X_{P} (t) = αt + {βt}^{2}$ and $X_{Q} (t) = ft - t^{2}$ . At what time, both the buses have same velocity?

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$\frac{α + f}{2 (β - 1)}$

$\frac{f - α}{2 (1 + β)}$

$\frac{α + f}{2 (1 + β)}$

$\frac{α - f}{1 + β}$

answer is D.

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Detailed Solution

$\begin{aligned} X_{p} (t) = α t + β t^{2} X_{Q} = ft - t^{2} \\ V_{P} (t) = α + 2 β t V_{Q} = f - 2 t \\ V_{P} = V_{Q} \\ α + 2 β t = f - 2 t \\ t = \frac{f - α}{2 β + 2} \end{aligned}$

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