Boat across the river

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Published on 2012. december 21. péntek, 11:31

In which direction should we paddle across the river to keep our height as much as it is possible?

We give a geometric solution of the next problem shown on a dynamic figure implemented in JavaScript.

Problem. Given a straight river with parallel banks. On one of the banks there is boat of zero extension at the point $A$. The speed of the river is $w$, and the speed of the boat in a lake is $v$, both are measured in $\frac{m}{s}$. The opposite points of $A$ on other side of the river is $B$. How to set the steering of the boat, to port in the nearest possible point to $B$?

Solution. Let the point $C$ be on the same bank of the river as $A$, so that a slide arrives to it 1 second later than it was droped in the river at $A$. The point $C$ is then in a distance of $1\cdot w=w$ meter from $A$ in the direction the river flows.

The direction of the boat's motion (green on the figure) is the sum $\bf u$ of the vector (red on the figure) determined by the speed of paddling and the direction of the boat's steering and the velocity vector (darkblue on the figure) of the river.

Clearly, putting the vector $\bf u$ to the point $A$ its endpoint $V$ is placed on the circle of radius $v$ centered at $C$. Our task is to choose the steering so that the angle of the straight line $AV$ to $AB$ be minimized.

If the segment $\overline{AB}$ does not intersect the circle centered at $C$, then smallest angle belongs to the straight line thhrough $A$ that touches the circle.