A passenger is at a distance x from a bus, when the bus begins to move with a constant acceleration a. What is the minimum velocity with which the passenger should run towards the bus so as to catch it ?

Let v be the velocity of the passenger and the passenger catches the bus after a time t. Then, the distance covered by both is same. Hence
$\mathrm{vt}=\mathrm{x}+\frac{1}{2}{\mathrm{at}}^2$
or ${\mathrm{at}}^2−2\mathrm{vt}+2\mathrm{x}=0$
$∴$ $\mathrm{t}=\frac{+2\mathrm{v}Â\pm \sqrt{\left(4{\mathrm{v}}^2−8\mathrm{ax}\right)}}{2\mathrm{a}}$
$=\frac{\mathrm{v}Â\pm \sqrt{\left({\mathrm{v}}^2−2\mathrm{ax}\right)}}{\mathrm{a}}$
t is real, when v² - 2 a x is positive or zero. Hence minimum speed is given by
v² -2ax=0
$∴\mathrm{v}=\sqrt{\left(2\mathrm{ax}\right)}$