A swimmer wants to cross a river from point $A$ to point $B$. Line $A B$ makes an angle of $30^{\circ}$ with the flow of river. Magnitude of velocity of the swimmer is same as that of the river. The angle $\theta$ with the line ${AB}$ should be $^{\circ}$, so that the swimmer reaches point ${B}$.

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $v$ and other with a uniform acceleration $a$. If $\alpha$ is the angle between the lines of motion of two particles, then the least value of relative velocity will be at time given by

Four persons $K,\,L,\,M$ and $N$ are initially at the corners of a square of side of length $d$. If every person starts moving, such that $K$ is always headed towards $L,\,L$ towards $M,\,M$ is headed directly towards $N$ and $N$ towards $K$, then the four persons will meet after

A train is moving towards east and a car is along north, both with same speed. The observed direction of car to the passenger in the train is

A boy is running on the plane road with velocity $v$ with a long hollow tube in his hand. The water is failing vertically downwards with velocity $u$. At what angle to the vertical, he must incline the tube so that the water drops enter it without touching its sides?

A man is running at a speed of $5\, m/s$, the rain drops appear to be falling at an angle of $45^o$ from the vertical. If the rain drops are actually falling vertically downwards, then velocity of rain drops (in $m/s$) is