---
title: "A boat crosses a straight river of width \\(D\\) that flows with a constant, uniform speed \\(u\\) parallel to its banks. The boat maintains a constant speed \\(v\\) relative to the water, where \\(v < u\\). To minimize the total downstream drift distance \\(\\Delta x\\), the boat steers at an angle \\(\\theta\\) upstream relative to the line perpendicular to the bank, resulting in a minimum drift distance of \\[\\Delta x_{\\text{min}} = D\\sqrt{\\left(\\dfrac{u}{v}\\right)^2 – 1}\\] In the limit as the boat's speed approaches the river's speed from below (\\(v \\to u^{-}\\)), which of the following correctly describes the minimum downstream drift distance and the time required to cross the river?"
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url: "https://nerd-notes.com/ubq/124045/"
date_modified: "2026-09-28T13:30:15+00:00"
---

# A boat crosses a straight river of width \(D\) that flows with a constant, uniform speed \(u\) parallel to its banks. The boat maintains a constant speed \(v\) relative to the water, where \(v < u\). To minimize the total downstream drift distance \(\Delta x\), the boat steers at an angle \(\theta\) upstream relative to the line perpendicular to the bank, resulting in a minimum drift distance of
\[\Delta x_{\text{min}} = D\sqrt{\left(\dfrac{u}{v}\right)^2 – 1}\]
In the limit as the boat's speed approaches the river's speed from below (\(v \to u^{-}\)), which of the following correctly describes the minimum downstream drift distance and the time required to cross the river?

A boat crosses a straight river of width \(D\) that flows with a constant, uniform speed \(u\) parallel to its banks. The boat maintains a constant speed \(v\) relative to the water, where \(v < u\). To minimize the total downstream drift distance \(\Delta x\), the boat steers at an angle \(\theta\) upstream relative to the line perpendicular to the bank, resulting in a minimum drift distance of
\[\Delta x_{\text{min}} = D\sqrt{\left(\dfrac{u}{v}\right)^2 - 1}\]
In the limit as the boat's speed approaches the river's speed from below (\(v \to u^{-}\)), which of the following correctly describes the minimum downstream drift distance and the time required to cross the river?

![A schematic showing a river crossing viewed from above. Two horizontal parallel solid lines represent the river banks separated by a vertical distance labeled D, with a double-headed vertical dimension arrow between them. A horizontal arrow pointing to the right is positioned between the banks and labeled u, indicating the river flow. At the bottom bank, a dot marks the starting position of a boat. A vertical dashed line extends upward from the dot across the river, perpendicular to the banks. An arrow labeled v originates at the dot, directed into the upper-left quadrant at an angle \theta to the left of the vertical dashed line. An arc between the vertical dashed line and the arrow labeled v denotes the angle \theta. An axis arrow along the bottom bank points to the right, labeled with a downstream direction. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602215-vHi2Sk.jpg)

- **A.** The minimum downstream drift distance approaches zero, and the crossing time approaches \(\dfrac{D}{u}\).
- **B.** The minimum downstream drift distance approaches zero, and the crossing time approaches infinity.
- **C.** The minimum downstream drift distance approaches infinity, and the crossing time approaches \(\dfrac{D}{u}\).
- **D.** The minimum downstream drift distance approaches infinity, and the crossing time approaches infinity.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124045/*