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. Hencevt=x+12at2or at2−2vt+2x=0∴ t=+2v±4v2−8ax2a=v±v2−2axat is real, when v2 - 2 a x is positive or zero. Hence minimum speed is given byv2 -2ax=0∴v=(2ax)