Two buses P and Q starts from a point at the same time and move in a straight line and their positions are 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

For bus P

$x_{p} (t) = α t + β t^{2}$

$V_{P} (t) = α + 2 β t^{2}$

For bus Q

$x_{Q} (t) = f t - t^{2}$

$V_{Q} (t) = f - 2 t$

$\begin{array}{l} As V_{P} (t) = V_{Q} (t) \\ α + 2 β t = f - 2 t \\ (2 β + 2) t = f - α \\ ∴ t = \frac{f - α}{2 β + 2} \end{array}$

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