---
title: "An airplane travels with a constant airspeed \\(v_a\\) relative to the air. A steady wind blows due east at a constant speed \\(v_w\\) relative to the ground, where \\(v_w < v_a\\). The pilot aims the airplane such that its resultant motion relative to the ground is directed due north.  Which of the following correctly pairs the heading of the airplane with its resulting speed relative to the ground?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/120495/"
date_modified: "2026-08-23T04:39:10+00:00"
---

# An airplane travels with a constant airspeed \(v_a\) relative to the air. A steady wind blows due east at a constant speed \(v_w\) relative to the ground, where \(v_w < v_a\). The pilot aims the airplane such that its resultant motion relative to the ground is directed due north.

Which of the following correctly pairs the heading of the airplane with its resulting speed relative to the ground?

An airplane travels with a constant airspeed \(v_a\) relative to the air. A steady wind blows due east at a constant speed \(v_w\) relative to the ground, where \(v_w < v_a\). The pilot aims the airplane such that its resultant motion relative to the ground is directed due north.

Which of the following correctly pairs the heading of the airplane with its resulting speed relative to the ground?

![A grayscale vector reference schematic. A horizontal vector arrow labeled \vec{v}_w originates at the origin and points horizontally to the right along the positive x-axis, labeled East. A vertical dashed line with an open arrow at its top extends along the positive y-axis, labeled North, indicating the desired resultant ground track. In the upper-left quadrant, a small four-point compass rose displays letters N, S, E, and W. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459950-hkUj9p.jpg)

- **A.** Heading: \(\arcsin\left(\dfrac{v_w}{v_a}\right)\) east of north Ground Speed: \(\sqrt{v_a^2 - v_w^2}\)
- **B.** Heading: \(\arctan\left(\dfrac{v_w}{v_a}\right)\) west of north Ground Speed: \(\sqrt{v_a^2 + v_w^2}\)
- **C.** Heading: \(\arcsin\left(\dfrac{v_w}{v_a}\right)\) west of north Ground Speed: \(\sqrt{v_a^2 + v_w^2}\)
- **D.** Heading: \(\arcsin\left(\dfrac{v_w}{v_a}\right)\) west of north Ground Speed: \(\sqrt{v_a^2 - v_w^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120495/*