$A$ man standing on a road holds his umbrella at $30^\circ$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10 \ km/hr$. He finds that raindrops are hitting his head vertically. The speed of raindrops with respect to the road will be ......... $km/hr$.

$STATEMENT-1$ For an observer looking out through the window of a fast-moving train,the nearby objects appear to move in the opposite direction to the train,while the distant objects appear to be stationary.
$STATEMENT-2$ If the observer and the object are moving at velocities $\vec{V}_1$ and $\vec{V}_2$ respectively with reference to a laboratory frame,the velocity of the object with respect to the observer is $\vec{V}_2 - \vec{V}_1$.

The speed of a swimmer in still water is $20 \; m/s$. The speed of river water is $10 \; m/s$ and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path,the angle at which he should make his strokes with respect to north is given by ......$^o$ west.

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 outside,what is the:
$(a)$ speed of the child running in the direction of motion of the belt?
$(b)$ speed of the child running opposite to the direction of motion of the belt?
$(c)$ time taken by the child in $(a)$ and $(b)$?
Which of the answers alter if motion is viewed by one of the parents?

$A$ river of width $200 \ m$ is flowing from west to east with a speed of $18 \ km/h$. $A$ boat, moving with a speed of $36 \ km/h$ in still water, is made to travel one round trip (bank to bank of the river). The minimum time taken by the boat for this journey and the displacement along the river bank are . . . . . . and . . . . . . respectively.

Two trains are moving with equal speed in opposite directions along two parallel railway tracks. If the wind is blowing with speed $u$ along the track so that the relative velocities of the trains with respect to the wind are in the ratio of $1:2$,then the speed of each train must be