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
Let the bus speed be $v$. Buses in each direction are spaced $vT$ apart.
Buses overtaking her (same direction), relative speed $v - 60$:
$$\frac{vT}{v - 60} = \frac{1}{2}\ \text{h}$$
Buses coming towards her, relative speed $v + 60$:
$$\frac{vT}{v + 60} = \frac{1}{6}\ \text{h}$$
Dividing: $\dfrac{v + 60}{v - 60} = 3 \;\Rightarrow\; v = 120$ km/h.
Then $vT = \dfrac{1}{2}(120 - 60) = 30$ km, so $T = \dfrac{30}{120}$ h $= 15$ min.
Solution by Sreeraj P, M.Sc Physics