# Relative Motion - River Crossing, Wind Triangle, Rain Angle, Closest Approach

> Relative velocity simulator: river crossing (min time vs min drift), aircraft wind triangle, rain seen from a moving car, and closest approach of two ships.

- URL: https://qilingtools.com/en/physics/relative-motion
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Four classic relative-velocity problems in one place: a boat crossing a river, a plane navigating in wind, rain seen from a moving car, and the closest approach of two ships or cars. Each scene draws the velocity-addition triangle and lets you switch the animation between the ground frame and the medium’s frame (the water, the car, or object A), so you can see the same motion from two points of view.

## How to use

1. Choose a scene: river crossing, wind triangle, rain and car, or two moving objects
2. Set speeds and angles with the sliders; for the river you can jump to the minimum-time or minimum-drift heading
3. Use the frame buttons to switch between ground and water, roadside and car, or ground and object A
4. Read the results: crossing time and drift, heading and ground speed, rain angle, closest distance and time

## Features

- River crossing: time, drift, minimum-time and minimum-drift headings (including a boat slower than the current)
- Wind triangle: ground speed and track from a heading, or the heading, wind correction angle and flight time for a desired course
- Rain and car: apparent rain angle and speed, animated from inside the car or the roadside
- Two objects: drag start points and velocity arrows, get relative velocity, closest distance and time, with the perpendicular shown in A’s frame
- A vector triangle diagram for every scene

## FAQ

### Why are the minimum-time and minimum-drift headings different?

The crossing time depends only on the velocity component across the river, t = d/(v_b cos θ), so pointing straight across (θ = 0) is fastest but the boat is carried u·d/v_b downstream. To reduce drift the bow must angle upstream so part of the boat’s speed cancels the current, which makes the across component smaller and the trip longer. If the boat is faster than the current, sin θ = u/v_b lands it exactly opposite the start.

### If the boat is slower than the current, how small can the drift be?

It will be carried downstream whatever it does. The drift is smallest when the ground velocity makes the smallest angle with the straight-across direction, which happens when the boat’s velocity relative to the water is perpendicular to its ground velocity: sin θ = v_b/u, giving a minimum drift of d·√(u² − v_b²)/v_b. The "minimum-drift heading" button picks this angle.

### How are heading, track and wind correction angle related?

The heading is where the nose points; times the airspeed it gives the velocity through the air. Adding the wind vector gives the velocity over the ground, whose direction is the track. To hold a course the nose must point into the wind by a wind correction angle so the sideways part of the airspeed cancels the crosswind: sin WCA = crosswind / TAS. Ground speed is then √(TAS² − crosswind²) plus the tailwind component.

### Why does rain seem to come at a slant from the front when driving?

The rain’s velocity relative to the car is its velocity over the ground minus the car’s: vertical rain plus a backward component −v_c. From inside the car it falls at an angle α = arctan(v_c/v_r) from the vertical, coming from the front, and the faster you drive the more it slants. A walker should tilt an umbrella forward by the same angle.

### How do I tell whether two ships will collide?

In A’s frame, B moves in a straight line at the relative velocity v_B − v_A. The distance from A to that line is the closest approach, reached at t* = −(r·v)/|v|². If it is near zero, B’s bearing seen from A stays constant while the range shrinks, which mariners call "constant bearing, decreasing range", the classic sign of a collision course.

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