Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of $60$ km/h in the direction X to Y notices that a bus goes past her every $30$ minutes in the direction of her motion, and every $10$ minutes in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.
Answer: (D) $15$ min, $120$ km/h
Concept. Buses in each direction are spaced $vT$ apart. The time between buses passing the girl is that spacing divided by the relative speed.
Step 1. Buses overtaking her. Relative speed $v - 60$:
$$\frac{vT}{v - 60} = \frac12\ \text{h}$$
Step 2. Oncoming buses. Relative speed $v + 60$:
$$\frac{vT}{v + 60} = \frac16\ \text{h}$$
Step 3. Divide. $\dfrac{v + 60}{v - 60} = 3 \Rightarrow v = 120$ km/h.
Step 4. Period. $vT = \dfrac12(120 - 60) = 30$ km, so $T = \dfrac{30}{120}$ h $= 15$ min.
Answer. $15$ min, $120$ km/h (Option D).
Solution by Sreeraj P, M.Sc Physics