Question

A man running with a uniform speed $u$ on a straight road observes a stationary bus at a distance $d$ ahead of him. At that instant, the bus starts moving with an acceleration $a.$ The condition that he would be able to catch the bus is

• A. $d\le \frac{u^2}{a}$
• B. $d\le \frac{u^2}{2a}$
• C. $d\le \frac{u^2}{3a}$
• D. $d\le \frac{u^2}{4a}$

Answer 1

The correct option is B

$d\le \frac{u^2}{2a}$

Let the man catch the bus at time $t$.
Velocity of the man is $u,$ and acceleration of the bus is $a,$ then

Distance covered by man is $ut$

Distance covered by bus is $\frac{1}{2}a{t}^2$

Difference of the distance covered by them is $d$
$\Rightarrow ut-\frac{1}{2}a{t}^2=d$

$\Rightarrow ut=\frac{1}{2}a{t}^2+d$
For the time to be positive, the discriminant of the above expression should be greater than or equal to zero.
Required condition $d\le \frac{u^2}{2a}$