Two Swimmers
Problem: Two swimmers leave point $A$ on one bank of the river to reach point $B$ lying right across on the other bank. One of them crosses the river along the straight line $AB$ while the other swims at right angles to the stream and then walks the distance that he has been carried away by the stream to get to point $B$. What was the velocity $u$ of his walking if both swimmers reached the destination simultaneously? The stream velocity $v_0 = 2.0 \text{ km/hour}$ and the velocity if of each swimmer with respect to water equals $2.5 \text{ km per hour}$. Solution: WLOG assume that the flow of the river (aka the stream) is towards the right. Let $v$ be the final vector from $A$ to $B$. First consider the first person. To swim directly from $A$ to $B$, the person must work diagonally for vector $v_p$. Simultaneously, the stream will drag the person along the vector $v_s$. These two result in the final vector $v_p+v_s=v$. So, he takes $$t_1=\frac{AB}{\sqrt{v_p^2-v_s^2}}$$ hours to get