$A$ river $200 \,m$ wide is flowing at a rate of $3.0 \,m/s$. $A$ boat is sailing at a velocity of $15 \,m/s$ with respect to the water in a direction perpendicular to the river. How far from the point directly opposite to the starting point does the boat reach on the opposite bank (in $\,m$)?

$A$ swimmer crosses a river of width $d$ flowing at velocity $v$. While swimming,he heads himself always at an angle of $120^{\circ}$ with the river flow and on reaching the other end he finds a drift of $d/2$ in the direction of flow of river. The speed of the swimmer with respect to the river is

Two parallel rail tracks run north-south. Train $A$ moves north with a speed of $54 \; km \; h^{-1},$ and train $B$ moves south with a speed of $90 \; km \; h^{-1}.$ What is the velocity of the ground in $m \; s^{-1}$ with respect to train $B$?

$A$ moves with $65 \, km/h$ while $B$ is moving in the same direction as $A$ with $80 \, km/h$. The relative velocity of $B$ with respect to $A$ is ......... $km/h$.

$A$ person walks up a stalled escalator in $90 \, s$. When just standing on the same moving escalator,he is carried in $60 \, s$. The time it would take him to walk up the moving escalator will be $....... \, s$.

$A$ 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.