A man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes $10$ $\min$ with a drift of $120 \mathrm{m}$. If he crosses the river taking the shortest route, he takes $12.5 \min$. Find the velocity of the boat with respect to water.

A river is flowing with a speed of $1 \mathrm{km} {\mathrm{h}}^{-1}$. A swimmer wants to go to point $C$ starting from $A$. He swims with a speed of $5 \mathrm{km} {\mathrm{h}}^{-1}$ at an angle $\theta$ w.r.t. the river flow. If $AB=BC=400 \mathrm{m}$, at what angle with the river bank should the swimmer swim?

A man wants to swim in a river from $A$ to $B$ and back from $B$ to $A$ always following line $AB$. The distance between points $A$ and $B$ is $S$. The velocity of the river current $v$ is constant over the entire width of the river. The line $AB$ makes an angle $\alpha$ with the direction of current. The man moves with velocity $u$ at angle $\beta$ to the line $AB$. The man swim to cover distance $AB$ and back, find the time taken to complete the journey.

A man walking due east with $3 \mathrm{km}/\mathrm{hr}$ observes rain falling vertically at the rate of $3 \mathrm{km}/\mathrm{hr}$. Find the actual speed and direction of the rain with respect to ground.