$A$ river is flowing from west to east with a speed of $9 \ km \ h^{-1}$. $A$ boat, capable of moving at a maximum speed of $27 \ km \ h^{-1}$ in still water, crosses the river in half a minute. If the boat moves at an angle of $150^{\circ}$ to the direction of the river flow, then the width of the river is:

$A$ boat is moving with a velocity $3i + 4j$ with respect to the ground. The water in the river is moving with a velocity $-3i - 4j$ with respect to the ground. The relative velocity of the boat with respect to the water is:

Two persons $A$ and $B$ are located in the $X-Y$ plane at the points $(0,0)$ and $(0,10)$ respectively. (The distances are measured in $MKS$ units). At time $t=0$, they start moving simultaneously with velocities $\overrightarrow{v}_A = 2\hat{j} \text{ m/s}$ and $\overrightarrow{v}_B = 2\hat{i} \text{ m/s}$ respectively. The time after which $A$ and $B$ are at their closest distance is:

$A$ river is flowing with a velocity of $5 \ km/hr$ as shown in the figure. $A$ boat starts from $A$ and reaches the other bank. The width of the river is $b = 300 \ m$. The velocity of the boat in still water is $v_{br} = 3 \ km/hr$. If the boat aims to reach the point directly opposite to $A$ (point $B$),but due to the river flow,it reaches point $C$,what is the distance $AC$ covered by the boat? (Note: The boat is steered such that it maintains a straight path relative to the ground).

$A$ river has a steady speed of $v$. $A$ man swims upstream for a distance of $d$ and swims back to the starting point in a total time $t$. The man can swim at a speed of $2v$ in still water. If the time taken by the man in still water is $t_0$ to complete the same total distance of $2d$, then $\frac{t}{t_0}$ is

If the velocities of particles $A$ and $B$ with respect to the ground are ${\overrightarrow v _A}$ and ${\overrightarrow v _B}$ respectively,then:
$(a)$ Write the formula for the velocity of $A$ with respect to $B$.
$(b)$ Write the formula for the velocity of $B$ with respect to $A$.
$(c)$ Is the relation ${\overrightarrow v _{AB}} = -{\overrightarrow v _{BA}}$ correct?