$A$ boat running downstream covers a distance of $16 \, km$ in $2 \, hours$, while for covering the same distance upstream, it takes $4 \, hours$. What is the speed of the boat in still water?

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

Two boys are standing at points $A$ and $B$ on the ground where distance $AB = a$. The boy at point $B$ starts running perpendicular to line $AB$ with velocity $V_1$. The boy at point $A$ starts running simultaneously with velocity $V$ and catches the other boy in time $t$. The value of $t$ is

The width of the river is $1 \; km$. The velocity of the boat is $5 \; km/hr$. The boat covers the width of the river along the shortest possible path in $15 \; min$. The velocity of the river stream is:

Ship $A$ is sailing towards north-east with velocity $\vec{v}_A = 30\hat{i} + 50\hat{j}\,\text{km/hr}$,where $\hat{i}$ points east and $\hat{j}$ points north. Ship $B$ is at a distance of $80\,\text{km}$ east and $150\,\text{km}$ north of Ship $A$ and is sailing towards west at $10\,\text{km/hr}$. After how many hours will Ship $A$ be at the minimum distance from Ship $B$?

$A$ 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 $\,m$)?