$A$ 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...........$km/h$.

$A$ girl riding a bicycle with a speed of $5 \, m/s$ towards the north direction observes rain falling vertically down. If she increases her speed to $10 \, m/s$,the rain appears to meet her at $45^o$ to the vertical. What is the speed of the rain? In what direction does the rain fall as observed by a ground-based observer?

$A$ river is flowing from west to east with a speed of $9 \ km \ h^{-1}$. $A$ boat, capable of moving at a maximum speed of $27 \ km \ h^{-1}$ in still water, crosses the river in half a minute. If the boat moves at an angle of $150^{\circ}$ to the direction of the river flow, then the width of the river is:

The figure shows two ships moving in the $x-y$ plane with velocities $V_A$ and $V_B$. The ships move such that $B$ always remains north of $A$. The ratio $\frac{V_A}{V_B}$ is equal to ........

$A$ 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?

$A$ swimmer wants to cross a river which is flowing at a speed $v$. If the swimmer can swim in still water at speed $V$,the direction he should swim to cross the river in the least time is