$A$ swimmer wants to cross a river which is flowing at a speed $v$. If the swimmer can swim in still water at speed $V$,the direction he should swim to cross the river in the least time is

Two cars $A$ and $B$ are moving in the same direction along a straight line with speeds $100 \text{ km/h}$ and $80 \text{ km/h}$, respectively, such that car $A$ is moving ahead of car $B$. $A$ person in car $B$ throws a stone with a speed $v$ so that it hits car $A$ with a speed of $5 \text{ m/s}$. The value of $v$ is . . . . . . $\text{km/h}$.

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

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 particles are moving along two long straight lines,in the same plane,with the same speed $= 20 \, cm/s$. The angle between the two lines is $60^{\circ}$,and their intersection point is $O$. At a certain moment,the two particles are located at distances $3 \, m$ and $4 \, m$ from $O$,and are moving towards $O$. Subsequently,the shortest distance between them will be

Two particles $A$ and $B$ start moving from the origin in the $xy$-plane. Particle $A$ moves with a velocity of $20 \, m/s$ along the $x$-axis,and particle $B$ moves with a velocity of $30\sqrt{2} \, m/s$ at an angle of $45^{\circ}$ with the $x$-axis. The relative velocity of $B$ with respect to $A$ is ........... $m/s$.