$A$ river $200 \,m$ wide is flowing at a rate of $3.0 \,m/s$. $A$ boat is sailing at a velocity of $15 \,m/s$ with respect to the water in a direction perpendicular to the river. How far from the point directly opposite to the starting point does the boat reach on the opposite bank (in $\,m$)?

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 ........... $m/s$.

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

$A$ bird is flying to and fro between two cars moving towards each other on a straight road. One car has a speed of $18 \,km/h$ while the other has a speed of $27 \,km/h$. The bird starts moving from the first car towards the second car with a speed of $36 \,km/h$ when the two cars are separated by $36 \,km$. What is the total distance covered by the bird?

$A$ swimmer can swim in still water with speed $v$ and the river is flowing with velocity $v/2$. To cross the river in the shortest distance,the swimmer should swim at an angle $\theta$ with the upstream. What is the ratio of the time taken to swim across in the shortest time to the time taken to swim across the shortest distance?

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).