Question

A man running with a uniform speed u on a straight road observes a stationary bus at a distance d ahead on him. At that instant, the bus starts with an acceleration a . The condition that he would be able to catch the bus is
(a) d≤u²/a
(b) d≤u²/2a
(c) d≤u²/3a
(d) d≤u²/4a

Answer 1

→ (b)

» Let they will meet at time t. At that time, velocity of man must be greater or equal to as that of bus.

The distances travelled by the man and the bus in time t

S1 = ut and S2 = 1/2at²

For catching the bus by the man the final separation must be smaller than or equal to the initial separation d.
As, the bus is stationary with respect to man (initially). So,

tmin = u/a

This implies d≤u(u / a) - 1/2 a u²/a² ( °.° t = u/a)

»› d ≤ u²/2a

Answer 2

Option (b)

At t = 0 sec,
   speed of the man = u 
   distance of the bus ahead of the man = d
   acceleration of the bus = a
   initial speed of the bus = 0

Suppose the man meets the bus at time t sec.
Distance traveled by the man in t sec = u * t
Displacement of the bus during time t = 0 * t + 1/2 * a t²

The condition for the man to catch the bus: 
          u * t ≥ d + 1/2 * a t²
          a t² - 2 u t  +  2 d ≤ 0

The value of t is real if and only if discriminant is ≥ 0.
=>     4 u² ≥ 8 a d
=>      u² ≥ 2 a d
=>      d ≤ u² / (2 a)