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.

\(\begin{aligned} t _1 & =30 \min =\frac{1}{2} hr \\ t _2 & =10 \min =\frac{1}{6} hr \\ V _{ B } & =\text { Speed of Bus } \\ V _{ S } & =\text { Speed of scooty (girl) }\end{aligned}\)
(Same direction) (Opposite direction)
\(\begin{array}{l}d=\left(V_B-V_S\right) t_1=\left(V_B+V_S\right) t_2 \\ \Rightarrow\left(V_B-60\right) \frac{1}{2}=\left(V_B+60\right) \frac{1}{6} \\ 3 V_B-180=V_B+60 \\ 2 V_B=240 \\ \Rightarrow V_B=120 km / hr \end{array}\)
Distance \(=\left( V _{ B }- V _{ S }\right) t _1\)
\(\begin{array}{l} D =(120-60) \frac{1}{2}=30 km \\ t =\frac{ d }{ V _{ B }}=\frac{30}{120}=\frac{1}{4} hr =15 min\end{array}\)