Rain is falling vertically with a speed of $12 \ ms^{-1}$. $A$ woman rides a bicycle with a speed of $12 \ ms^{-1}$ in the east to west direction. What is the direction in which she should hold her umbrella?

$A$ 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?

$A$ 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 ........ $m/s$.

$A$ motorboat is racing towards north at $25\; km/h$ and the water current in that region is $10\; km/h$ in the direction of $60^{\circ}$ east of south. Find the resultant velocity of the boat.

$A$ man swimming downstream overtakes a float at a point $M$. After travelling a distance $D$,he turns back and passes the float at a distance of $D/2$ from the point $M$. The ratio of the speed of the swimmer with respect to still water $(v_s)$ to the speed of the river $(v_r)$ is:

$A$ train is moving towards the east and a car is moving towards the north,both with the same speed. The observed direction of the car to a passenger in the train is