$A$ person running horizontally observes that rain is falling on his head vertically with a speed of $10\,m/s$. When he stops,he observes that the rain is falling at an angle of $30^o$ with the vertical. Find the speed with which the man was running.

$A$ steam boat goes across a lake and comes back: $(i)$ on a quiet day when the water is still and $(ii)$ on a rough day when there is a uniform current so as to help the journey onward and to impede the journey back. If the speed of the launch,on both days,was same,the time required for the complete journey on the rough day as compared to that on the quiet day,will be

$STATEMENT-1$ 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 $\vec{V}_2 - \vec{V}_1$.

Rain is falling vertically with a speed of $35 \; m s^{-1}$. Winds start blowing after sometime with a speed of $12 \; m s^{-1}$ in the east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella?

$A$ train of $150 \; m$ length is going towards the north direction at a speed of $10 \; m/s$. $A$ parrot flies at a speed of $5 \; m/s$ towards the south direction parallel to the railway track. The time taken by the parrot to cross the train is: (in $; s$)

$A$ swimmer wants to cross a river from point $A$ to point $B$. Line $AB$ makes an angle of $30^{\circ}$ with the flow of the river. The magnitude of the velocity of the swimmer is the same as that of the river. The angle $\theta$ with the line $AB$ should be in degrees,so that the swimmer reaches point $B$.