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 particles $P_1$ and $P_2$ are moving with velocities $v_1$ and $v_2$ respectively. Which of the following statements about their relative velocity $v_{12}$ is true?

Explain this common observation clearly: If you look out of the window of a fast-moving train,the nearby trees,houses,etc.,seem to move rapidly in a direction opposite to the train's motion,but the distant objects (hilltops,the Moon,the stars,etc.) seem to be stationary. (In fact,since you are aware that you are moving,these distant objects seem to move with you).

Airplanes $A$ and $B$ are flying with constant velocity in the same vertical plane at angles $30^{\circ}$ and $60^{\circ}$ with respect to the horizontal,respectively,as shown in the figure. The speed of $A$ is $100 \sqrt{3} \ m/s$. At time $t=0$,an observer in $A$ finds $B$ at a distance of $500 \ m$. This observer sees $B$ moving with a constant velocity perpendicular to the line of motion of $A$. If at $t = t_0$,$A$ just escapes being hit by $B$,then $t_0$ in seconds is:

$A$ man goes $10\,km$ downstream in $2\,hrs$ and $30\,km$ upstream in $10\,hrs$. How much time will he take to swim $40\,km$ in still water?

$A$ 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.