Man Or Rocket

Let the minimum velocity the man should run be $v$ . The displacement of the man in time $t$ is $s_{man} = vt$ . That of the bus starting from rest is $s_{bus} = \frac 12 at^2 + 45 = 1.25 t^2 + 45$ . For the man to catch up with the bus,

$\begin{aligned} s_{bus} - s_{man} & \le 0 \\ 1.25t^2 +45 - vt & \le 0 \\ 1.25t^2 - vt + 45 & \le 0 \quad \quad \small \color{#3D99F6}{\text{For the quadratic equation to have a root, the discriminant }b^2-4ac \ge 0} \\ \implies v^2 - 4(1.25)(45) & \ge 0 \\ v^2 & \ge 15^2 \\ \implies v & \ge \boxed{15} \quad \quad \small \color{#3D99F6}{v \text{ must be positive, that is, in the direction of the bus travelling.}} \end{aligned}$ .