$A$ man sitting in a bus travelling from west to east with a speed of $40 \, km/h$ observes that the rain-drops are falling vertically down. To another man standing on the ground,the rain will appear:

$A$ river is flowing from $W$ to $E$ with a speed of $5 \, m/min$. $A$ man can swim in still water with a velocity of $10 \, m/min$. In which direction should the man swim to take the shortest possible path to reach the opposite bank (South)?

$STATEMENT-1$ For an observer looking out through the window of a fast-moving train,the nearby objects appear to move in the opposite direction to the train,while the distant objects appear to be stationary.
$STATEMENT-2$ If the observer and the object are moving at velocities $\vec{V}_1$ and $\vec{V}_2$ respectively with reference to a laboratory frame,the velocity of the object with respect to the observer is $\vec{V}_2 - \vec{V}_1$.

$A$ boat crosses a river from one bank $A$ to another bank $B$ which is opposite. The distance between them is $D$. The speed of water is $V_W$ and that of the boat relative to water is $V_B$. If $V_B = 2V_W$, the time taken by the boat to cross the river directly along $AB$ is $(\sin 30^\circ = 1/2, \cos 30^\circ = \sqrt{3}/2)$.

$A$ man can swim with a speed of $4.0 \; km/h$ in still water. How long does he take to cross a river $1.0 \; km$ wide if the river flows steadily at $3.0 \; km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?

Out of the two cars $A$ and $B$,car $A$ is moving towards the east with a velocity of $10 \, m/s$,whereas car $B$ is moving towards the north with a velocity of $20 \, m/s$. The velocity of $A$ with respect to $B$ is ......... $m/s$ (nearly).