Two trains, each $30 \,m$ long, are travelling in opposite directions with velocities $5 \,m/s$ and $10 \,m/s$. They will cross each other after: (in $\,s$)

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?

$A$ 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?

An aeroplane is to go along a straight line from $A$ to $B$,and back again. The relative speed with respect to wind is $V$. The wind blows perpendicular to the line $AB$ with speed $v$. The distance between $A$ and $B$ is $\ell$. The total time for the round trip is:

Two cars $A$ and $B$ are moving with a velocity of $30 \ km/h$ in the same direction. They are separated by $10 \ km$. The speed of another car $C$ moving in the opposite direction,if it meets these two cars at an interval of $8 \ minutes$,is: (in $km/h$)

Two trains are moving with equal speed in opposite directions along two parallel railway tracks. If the wind is blowing with speed $u$ along the track so that the relative velocities of the trains with respect to the wind are in the ratio of $1:2$,then the speed of each train must be