$A$ man running at a speed of $5 \text{ km/h}$ finds that the rain falls vertically. When he stops running,he finds that the rain is falling at an angle of $60^{\circ}$ with the horizontal. The velocity of rain with respect to the running man is

$A$ boat crosses a river with a velocity of $8 \, km/h$ relative to the water. If the resultant velocity of the boat with respect to the ground is $10 \, km/h$,then the velocity of the river water is...........$km/h$.

$A$ butterfly is flying with a velocity $4 \sqrt{2} \, m/s$ in the North-East direction. The wind is blowing at $1 \, m/s$ from North to South. The resultant displacement of the butterfly in $3 \, s$ is $.... \, m$.

$A$ river is flowing from west to east at a speed of $8\,m/min$. $A$ man on the south bank of the river,capable of swimming at $20\,m/min$ in still water,wants to swim across the river in the shortest time. He should swim in a direction

Airplanes $A$ and $B$ are flying with constant velocity in the same vertical plane at angles $30^{\circ}$ and $60^{\circ}$ with respect to the horizontal,respectively,as shown in the figure. The speed of $A$ is $100 \sqrt{3} \ m/s$. At time $t=0$,an observer in $A$ finds $B$ at a distance of $500 \ m$. This observer sees $B$ moving with a constant velocity perpendicular to the line of motion of $A$. If at $t = t_0$,$A$ just escapes being hit by $B$,then $t_0$ in seconds is:

$A$ man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. $A$ student derives the following relation between $\theta$ and $v: \tan \theta = v$ and checks that the relation has a correct limit: as $v \rightarrow 0, \theta \rightarrow 0,$ as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not,guess the correct relation.