$A$ swimmer can swim in still water with $5 \,m/s$. If a river is flowing with $4 \,m/s$, then find the time to cross the river of width $300 \,m$ if he has to reach the exactly opposite point.

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?

$A$ boat covers a certain distance between two spots in a river taking $t_1$ hours going downstream and $t_2$ hours going upstream. What time will be taken by the boat to cover the same distance in still water?

$A$ boat crosses a river from one bank $A$ to another bank $B$ which is opposite. The distance between them is $D$. The speed of water is $V_W$ and that of the boat relative to water is $V_B$. If $V_B = 2V_W$, the time taken by the boat to cross the river directly along $AB$ is $(\sin 30^\circ = 1/2, \cos 30^\circ = \sqrt{3}/2)$.

$A$ river is flowing due east with a speed $3\, ms^{-1}$. $A$ swimmer can swim in still water at a speed of $4\, ms^{-1}$ (figure).
$(a)$ If the swimmer starts swimming due north,what will be his resultant velocity (magnitude and direction)?
$(b)$ If he wants to start from point $A$ on the south bank and reach the opposite point $B$ on the north bank,
$(i)$ In which direction should he swim?
$(ii)$ What will be his resultant speed?
$(c)$ From the two different cases as mentioned in $(a)$ and $(b)$ above,in which case will he reach the opposite bank in a shorter time?

Two trains $A$ and $B$ travel on two parallel rail tracks in opposite directions with speeds $v_1$ and $v_2$, respectively. They take $4 \ s$ to pass each other at these speeds. If the speed of train $A$ is increased by $50 \%$, then they take $3 \ s$ to pass each other. The ratio $v_1 / v_2$ is: