There are two particles $P$ and $Q$ separated by a distance of $10 \ km$ and moving with velocities of $10 \ kmph$ as shown in the figure. The time elapsed before they reach their minimum separation is.........$hr$.

$A$ man of mass $60 \,kg$ is standing on a boat of mass $140 \,kg$,which is at rest in still water. The man is initially at $20 \,m$ from the shore. He starts walking on the boat for $4 \,s$ with a constant speed of $1.5 \,m/s$ towards the shore. The final distance of the man from the shore is .......... $m$.

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:

$A$ river of width $200 \ m$ is flowing from west to east with a speed of $18 \ km/h$. $A$ boat, moving with a speed of $36 \ km/h$ in still water, is made to travel one round trip (bank to bank of the river). The minimum time taken by the boat for this journey and the displacement along the river bank are . . . . . . and . . . . . . respectively.

$A$ 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?

Rain is falling vertically with a speed of $35 \; m s^{-1}$. Winds start blowing after sometime with a speed of $12 \; m s^{-1}$ in the east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella?