$A$ river is flowing from east to west at a speed of $5\, m/min$. $A$ man on the south bank of the river,capable of swimming $10\, m/min$ in still water,wants to swim across the river in the shortest time. He should swim:

$A$ 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.

$A$ 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:

$A$ river is flowing from $W$ to $E$ with a speed of $5 \, m/min$. $A$ man can swim in still water with a velocity of $10 \, m/min$. In which direction should the man swim to take the shortest possible path to reach the opposite bank (South)?

$A$ bus appears to go with a speed of $25 \, km/hr$ to a car driver,driving at the rate of $7 \, km/hr$ northwards. If the bus actually travels in the east direction,its speed is ......... $km/h$.

$A$ 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........$s$.