Two buses P and Q start from a point at the same time and move in a...

As we know, $v = \dfrac{dx}{dt}$.

Given, ${x}_{P}\left(t\right) = \alphat + \beta{t}^{2}$

Velocity of bus $P$ is ${v}_{P} = \dfrac{d{x}_{P}}{dt} = \alpha + 2\betat$ and velocity of bus $Q$ is ${v}_{Q} = f - 2t$

When ${v}_{P} = {v}_{Q}$

$⇒ \alpha + 2\betat = f - 2t$

$⇒ f - \alpha = 2\left(\beta + 1\right)t ⇒ t = \dfrac{f - \alpha}{2\left(\beta + 1\right)}$