Question

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

• A. $\sqrt{2ax}$
• B. $2ax$
• C. $\sqrt{ax}$
• D. $ax$

Answer 1

The correct option is A $\sqrt{2ax}$

Let the passenger moving with constant velocity $v$ catches the bus at point C in time $t$.

For bus :

Initial speed of the bus $u=0$

$\therefore$ $S=0+\frac{1}{2}a{t}^2$ $⟹S=\frac{1}{2}a{t}^2$

For passenger : $x+S=vt$

$\therefore$ $x+\frac{a{t}^2}{2}=vt$

$⟹$ $a{t}^2-2vt+2x=0$

For the above equation to have real roots, then $(-2v{)}^2-4a(2x)\ge 0$

$\therefore$ $v^2\ge 2ax$ $⟹v\ge \sqrt{2ax}$

Thus the minimum velocity of the passenger $v_{min}=\sqrt{2ax}$