Question

Two towns A and B are connected by a regular bus service with a bus leaving in every T min. A man cycling with a speed of 20 km/hr in the direction A to B notices that a bus goes past him every 18 min in the direction of his motion, and every 6 min in the opposite direction. What is the period T of the bus service?
A. $\dfrac{1}{20}$hr
B. $\dfrac{2}{20}$hr
C. $\dfrac{3}{20}$hr
D. $\dfrac{4}{20}$hr

Answer 1

Use the concept of relative velocity to determine the time required for the bus to cover the distance for each trip. The time required by the bus to cover the distance is the same as the separation between two successive buses T.

Formula used:
If an observer moves with a velocity $v_A$ in the same direction as $v_B$ the resultant velocity observed by the observer moving with velocity $v_A$ is given as:
$v = v_B - v_A$

Complete answer:
Consider following two situations:
1. As a cyclist moves in the same direction as bus, he must observe the bus to be moving with velocity:
(v - 20) km/hr.
For the bus, v is given as:
$v = \dfrac{d}{T}$ ,
where T is the time required by the bus to travel the distance between the two towns A and B given by d.
But, the cyclist observes the bus in every 18 minutes so, in his frame, the passing bus takes 18 minutes to cover d distance, we can write:
$\dfrac{d}{18} = \dfrac{d}{T} - 20$
from the relative velocity expression.

2. In the second situation when the bus is coming towards the cyclist, he observes the velocity of the bus to be (v + 20) km/hr.
Now, he observes a bus coming towards him every 6 minutes so we may write:
$\dfrac{d}{6} = \dfrac{d}{T} + 20$
like we did before.
This way we have formed two equations with two unknowns.
We attempt to substitute d, we get:
$d = \dfrac{20}{\left( \dfrac{1}{T} - \dfrac{1}{18} \right)}$
from the first equation and;
$d = \dfrac{20}{\left( \dfrac{1}{6} - \dfrac{1}{T} \right)}$
from the second equation.
Using these two, we may write:
$\dfrac{1}{6} - \dfrac{1}{T} = \dfrac{1}{T} - \dfrac{1}{18}$
Solving this, we get:
T = 9 minutes.
In hours we may write:
$T = \dfrac{9}{60} = \dfrac{3}{20}$hr.

Therefore the correct answer is option (C).

Note:
It can be quite tricky to understand that the bus separation T is the same as the time it requires for the bus to cover the distance between A and B. Also, in the frame of the cyclist, if a bus is succeeded by another bus in a time of 18 minutes then the first bus has covered the distance d in 18 minutes.