$A$ swimmer can swim in still water with speed $v$ and the river is flowing with velocity $v/2$. To cross the river in the shortest distance,the swimmer should swim at an angle $\theta$ with the upstream. What is the ratio of the time taken to swim across in the shortest time to the time taken to swim across the shortest distance?

$A$ girl standing on a road holds her umbrella at $45^{\circ}$ with the vertical to keep the rain away. If she starts running without an umbrella with a speed of $15 \sqrt{2} \; kmh^{-1}$,the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is ........... $kmh^{-1}$

$A$ slider block $A$ moves downward at a speed of $v_A = 2 \ m/s$,at an angle of $75^\circ$ with the horizontal as shown in the figure. The velocity of the portion of belt $B$ between ideal pulleys $C$ and $D$ with respect to $A$ is $v_{CD/A} = 2 \ m/s$ at an angle $\theta$ with the horizontal. The magnitude of the velocity of portion $CD$ of the belt when $\theta = 15^\circ$ is .......... $m/s$.

The stream of a river is flowing with a speed of $2\,km/h$. $A$ swimmer can swim at a speed of $4\,km/h$. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight (in $^\circ$)?

Two particles $P_1$ and $P_2$ are moving with velocities $v_1$ and $v_2$ respectively. Which of the following statements about their relative velocity $v_{12}$ is true?

$A$ 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