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A boatman sets off from one bank of a straight, uniform canal for a mark directly opposite the starting point. The speed of the water flowing in the canal is v everywhere. The boatman rows steadily at such a rate that, were there no current, the boat’s speed would also be v. He always sets the boat’s course in the direction of the mark, but the water carries him downstream. Fortunately, he never tires! How far downstream does the water carry the boat? What trajectory does it follow with respect to the bank?

Step-by-step

Denote the width of the canal by d and draw a straight line perpendicular to its banks a distance d downstream from the boat’s starting point A.
The boat is initially at distance d both from the mark F on the opposite bank and from this straight line. As both the speed of the water and that of the boat with respect to the water are v, the water takes the boat downstream by the same distance as is covered by the boat in the direction of F. This means that the boat is always equally far from point F and the straight line. The path of the boat is therefore a parabola, with F as its focus and the straight line as its directrix. After a very long time, the boat approaches the opposite bank at a point d/2 from F. Because the speed of the current equals that of the boat, the boatman cannot land closer than this.

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