On a long horizontally moving belt,a child runs to and fro with a speed $9 \, km \, h^{-1}$ (with respect to the belt) between his father and mother located $50 \, m$ apart on the moving belt. The belt moves with a speed of $4 \, km \, h^{-1}$. For an observer on a stationary platform,the speed of the child running in the direction of motion of the belt is ..... $km \, h^{-1}$.

$A$ swimmer can swim in still water with speed $v$ and the river is flowing with velocity $\frac{v}{2}$. To cross the river in the shortest time,he should swim making an angle $\theta$ with the upstream. What is the ratio of the time taken to swim across in the shortest time to that in swimming across over the shortest distance?

$A$ person walks up a stalled escalator in $80 \,s$. When standing on the same escalator, now moving, he is carried up in $20 \,s$. The time taken by him to walk up the moving escalator is (in $\,s$)

Train $A$ and train $B$ are running on parallel tracks in opposite directions with speeds of $36 \, km/h$ and $72 \, km/h$,respectively. $A$ person is walking in train $A$ in the direction opposite to its motion with a speed of $1.8 \, km/h$. The speed (in $m/s$) of this person as observed from train $B$ will be close to:
(take the distance between the tracks as negligible) (in $.5$)

Two towns $A$ and $B$ are connected by a regular bus service with a bus leaving in either direction every $T$ min. $A$ man cycling with a speed of $20 \,km/h$ from $A$ to $B$ notices that a bus travelling in the direction of his motion goes past him every $18 \,min$ and every $6 \,min$ he notices a bus travelling in the opposite direction go past him. Assuming that the buses travel with a constant speed,find $T$ and the constant speed of the buses.

Rain is falling at an angle of $30^{\circ}$ from the vertical due to the wind with a speed of $40 \ m/s$. $A$ car is travelling horizontally in the direction opposite to the wind, at a speed of $40 \ m/s$. At what angle from the vertical will it experience the rain falling from (in $^{\circ}$)?