A man is, $d$ distance behind a bus. The bus moves away from the man with an acceleration $a$. At the same time, the man starts running towards the bus with a constant velocity $v$.

Let us see the relative motion, distance moved by the man before coming to rest 

                           $0=v^2-2as$ or $s=\frac{v^2}{2a}$

Now, man can catch the bus, if 

                            $s⩾d$ or $v⩾\sqrt{2ad}$

If he just catches the bus, then

                             $0=v-at$

or                           $t=\frac{v}{a}$