---
title: "Riverboat Navigation Interactive Lab | Sandbox Physics"
description: "Does the quickest crossing land straight opposite?"
canonical_url: "https://sandboxphysics.com/labs/riverboat-navigation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/riverboat-navigation/"
---

M005 · Moving water

# Riverboat Navigation

**Does the quickest crossing land straight opposite?** The quickest crossing points the boat straight across the water, leaving the full water-relative speed available for crossing. The current carries it downstream. Landing opposite needs an upstream heading and takes longer, when a constant-heading solution exists.

Steer a boat through uniform, sheared or faster central currents. Compare the quickest crossing with a constant heading that lands opposite, and watch water-relative and ground velocities add.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The boat keeps a prescribed speed and heading relative to a steady downstream current. Three profiles share the same channel-average speed. The model has no acceleration, drag, waves or hull dynamics. The opposite-bank solution is restricted to constant headings.

Physics tutorial

## Crossing quickly and landing opposite are different tasks

**Background** Point a boat straight across a river. Its own motion carries it toward the far bank, while the water adds downstream motion. The two velocities act at the same time.

**Why it matters** A heading that removes downstream drift spends some of the boat speed on fighting the current. A shaped current makes the path bend even when the boat keeps one heading.

Start with the essentials

- **Focus question**: What changes when the destination is the opposite point rather than anywhere on the far bank?
- **One-sentence intuition**: The boat crosses fastest when all its water-relative speed points across. A constant upstream heading can cancel the total downstream drift when the mean current is small enough.

### Core mathematical model

#### Water-relative and ground velocity

$$
\begin{aligned}\mathbf v_g&=\mathbf v_b+\mathbf u\\v_{g,x}&=u(y)-v_b\sin\theta\\v_{g,y}&=v_b\cos\theta\end{aligned}
$$

Downstream is positive horizontal distance. The heading is measured upstream from the across-river direction.

#### Crossing time and landing drift

$$
\begin{aligned}T&=\frac{W}{v_b\cos\theta}\\X&=\frac{W(\bar u-v_b\sin\theta)}{v_b\cos\theta}\\\bar u&=\frac1W\int_0^W u(y)\,\mathrm dy\end{aligned}
$$

The heading stays fixed during a trial. The across-river speed is constant; downstream drift integrates the flow through the full channel width.

#### The quickest arrival anywhere on the far bank

$$
T_{\min}=\frac{W}{v_b},\qquad\theta=0
$$

The water has no across-river component. No heading can provide more across-river speed than the boat full water-relative speed.

#### A constant heading that lands opposite

$$
\begin{aligned}\sin\theta_{\mathrm{opp}}&=\frac{\bar u}{v_b}\\T_{\mathrm{opp}}&=\frac{W}{\sqrt{v_b^2-\bar u^2}}\end{aligned}
$$

This finite-time solution requires the mean current to be smaller than the boat speed. In a variable current, other time-varying routes lie outside this constant-heading test.

### Common difficulties

#### Aimed across does not mean moving straight across

**Typical misconception** The boat follows the direction its bow points.

**Better mental model** The bow sets its velocity through water. Water motion adds another velocity relative to the bank.

#### Zero landing drift can have a curved path

**Typical misconception** Landing opposite means the boat stayed on the opposite-bank line throughout.

**Better mental model** In a shaped current, upstream and downstream drift can cancel over the whole trip. The path can bend and still end opposite.

#### One heading is one constraint

**Typical misconception** If the constant-heading formula fails, every possible navigation plan fails.

**Better mental model** The test fixes one heading. A route that varies its heading or waits in a slower part of the stream is a different control problem.

### Run the experiment

1. 01

   #### Watch the speed change across the stream

   Choose Curved crossing and watch the gold water markers.

   **What to observe:** The center flows faster than the edges. The blue boat keeps one water-relative heading while its ground path bends.

2. 02

   #### Cross as quickly as possible

   Choose Quickest crossing.

   **What to observe:** The full boat speed carries it across. It lands downstream, with a shorter crossing time.

3. 03

   #### Land opposite

   Choose Land opposite or press Set opposite-bank course.

   **What to observe:** The upstream heading cancels total drift. The crossing takes longer.

4. 04

   #### Keep the average, change the profile

   Compare uniform, shear and central-jet profiles at the same mean current and heading.

   **What to observe:** The final drift and crossing time agree, although the shapes of the paths differ.

5. 05

   #### Make the current too strong for one heading

   Choose Current outruns boat.

   **What to observe:** No finite constant-heading solution lands opposite in this uniform current. The simulator still shows a valid crossing downstream.

6. 06

   #### Read the velocity triangle

   Pause partway across and inspect the vector diagram below the controls.

   **What to observe:** The blue water-relative vector followed by the gold current vector ends at the teal ground-velocity tip.

Physical conclusion

For a constant heading, the current profile changes the path, while its channel average sets the landing point. Local cancellation and zero total drift are different conditions.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)