$A$ man can swim with a speed of $4 \ km/h$ in still water. How long does he take to cross a river $1 \ km$ wide, if the river flows steadily at $3 \ km/h$ and he makes his strokes normal to the river current (in $min$)?

$A$ particle $(A)$ moves due north at $3\,km/h$ and another particle $(B)$ moves due west at $4\,km/h$. The relative velocity of $A$ with respect to $B$ is $(\tan 37^{\circ} = 3/4)$.

$A$ boat crosses a river with a velocity of $8 \, km/h$ relative to the water. If the resultant velocity of the boat with respect to the ground is $10 \, km/h$,then the velocity of the river water is...........$km/h$.

There are two particles $P$ and $Q$ separated by a distance of $10 \ km$ and moving with velocities of $10 \ kmph$ as shown in the figure. The time elapsed before they reach their minimum separation is.........$hr$.

$A$ steamboat goes across a lake and comes back $(a)$ on a quiet day when the water is still and $(b)$ on a rough day when there is a uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was the same,in which case will it complete the journey in lesser time?

Particle $A$ (which was located at the origin at time $t=0$) is moving along the $x$-axis with a constant speed of $1 \,m/s$. The location of particle $B$, which is moving along the $y$-axis, is given by $y=ct^2$, where $c=1 \,m/s^2$. Find the speed of particle $A$ relative to particle $B$ at $t=1 \,s$.