$A$ passenger train of length $60\, m$ travels at a speed of $80\, km/hr$. Another freight train of length $120\, m$ travels at a speed of $30\, km/hr$. The ratio of times taken by the passenger train to completely cross the freight train when:
$(i)$ they are moving in the same direction,and
$(ii)$ in the opposite directions is

Rain is falling vertically with a speed of $30 \,m/s$. A man rides a bicycle with a speed of $10 \,m/s$ from East to West direction. At what angle with the vertical does he see the rain falling?

Two persons $A$ and $B$ are located in the $X-Y$ plane at the points $(0,0)$ and $(0,10)$ respectively. (The distances are measured in $MKS$ units). At time $t=0$, they start moving simultaneously with velocities $\overrightarrow{v}_A = 2\hat{j} \text{ m/s}$ and $\overrightarrow{v}_B = 2\hat{i} \text{ m/s}$ respectively. The time after which $A$ and $B$ are at their closest distance is:

$A$ 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: (in $; s$)

$A$ river is flowing from west to east at a speed of $8\,m/min$. $A$ man on the south bank of the river,capable of swimming at $20\,m/min$ in still water,wants to swim across the river in the shortest time. He should swim in a direction

$A$ ship is travelling due east at $10 \, km/hr$. $A$ second ship,heading $30^o$ east of north,is always due north of the first ship. The speed of the second ship in $km/hr$ is: