$A$ 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?

For the two given particles $A$ and $B$ as shown in the figure,how much time will it take for their horizontal separation to become zero?

$A$ boat running downstream covers a distance of $16 \, km$ in $2 \, hours$, while for covering the same distance upstream, it takes $4 \, hours$. What is the speed of the boat in still water?

$A$ man is crossing a river flowing with a velocity of $5 \ m/s$. He reaches a point directly across at a distance of $60 \ m$ in $5 \ s$. His velocity in still water should be: (in $m/s$)

$A$ river is flowing with a velocity of $5 \ km/hr$ as shown in the figure. $A$ boat starts from $A$ and reaches the other bank. The width of the river is $b = 300 \ m$. The velocity of the boat in still water is $v_{br} = 3 \ km/hr$. If the boat aims to reach the point directly opposite to $A$ (point $B$),but due to the river flow,it reaches point $C$,what is the distance $AC$ covered by the boat? (Note: The boat is steered such that it maintains a straight path relative to the ground).

$A$ 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 ......... $km/hr$.