Ground Speed Calculator

True Airspeed (vₐ):

Wind Speed (v_w):

Course (δ):

Wind Direction (ω):

Wind Correction Angle (α):

Ground Speed (v_g):

1. What is a Ground Speed Calculator?

Definition: This calculator determines the ground speed of an aircraft and the wind correction angle, accounting for wind effects.

Purpose: It is used in aviation to calculate the actual speed of an aircraft over the ground and the necessary heading adjustment, aiding in navigation and flight planning.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Wind Correction Angle: \[ \alpha = \sin^{-1}\left[\frac{v_w}{v_a} \sin(\omega - \delta)\right] \] Ground Speed: \[ v_g = \sqrt{v_a^2 + v_w^2 - 2 v_a v_w \cos(\delta - \omega + \alpha)} \] Where:

\(v_g\): Ground speed (m/s, km/h, mph, kn)
\(v_a\): True airspeed (m/s, km/h, mph, kn)
\(v_w\): Wind speed (m/s, km/h, mph, kn)
\(\delta\): Course (degrees or radians)
\(\omega\): Wind direction (degrees or radians)
\(\alpha\): Wind correction angle (degrees or radians)

Unit Conversions:

Speed Units (v_a, v_w, v_g): m/s, km/h, mph, kn
Angle Units (δ, ω, α): degrees, radians

Steps:

Enter the true airspeed (vₐ), selecting the unit (m/s, km/h, mph, kn)
Enter the wind speed (v_w), selecting the unit (m/s, km/h, mph, kn)
Enter the course (δ), selecting the unit (degrees or radians)
Enter the wind direction (ω), selecting the unit (degrees or radians)
Convert all speeds to m/s and angles to radians for calculation
Calculate the wind correction angle (α) using the formula
Calculate the ground speed using the formula
Select the desired units for the results and view the converted values

3. Importance of Ground Speed and Wind Correction Angle Calculation

Calculating ground speed and wind correction angle is crucial for:

Navigation: Determining the actual speed over the ground and the correct heading helps pilots plan flight routes and estimate arrival times.
Fuel Efficiency: Adjusting for wind effects allows for more efficient flight paths.
Safety: Accurate calculations ensure safe takeoffs, landings, and in-flight adjustments.

4. Using the Calculator

Examples:

Example 1: For a true airspeed \(v_a = 100 \, \text{m/s}\), wind speed \(v_w = 20 \, \text{m/s}\), course \(\delta = 90^\circ\), wind direction \(\omega = 0^\circ\):
- Angle difference = \(\omega - \delta = 0 - 90 = -90^\circ = -\frac{\pi}{2} \, \text{radians}\)
- Wind correction angle = \(\sin^{-1}\left[\frac{20}{100} \sin(-\frac{\pi}{2})\right] = \sin^{-1}(-0.2) = -0.201 \, \text{radians} = -11.537^\circ\)
- Angle for ground speed = \(90 - 0 + (-11.537) = 78.463^\circ = 1.369 \, \text{radians}\)
- Ground speed = \(\sqrt{100^2 + 20^2 - 2 \times 100 \times 20 \times \cos(1.369)} = 101.980 \, \text{m/s}\)
Example 2: For a true airspeed \(v_a = 200 \, \text{kn}\), wind speed \(v_w = 30 \, \text{kn}\), course \(\delta = \pi \, \text{radians}\), wind direction \(\omega = \pi \, \text{radians}\):
- True airspeed in m/s = \(200 \times 0.514444 = 102.889 \, \text{m/s}\)
- Wind speed in m/s = \(30 \times 0.514444 = 15.433 \, \text{m/s}\)
- Angle difference = \(\pi - \pi = 0 \, \text{radians}\)
- Wind correction angle = \(\sin^{-1}\left[\frac{15.433}{102.889} \sin(0)\right] = \sin^{-1}(0) = 0.000 \, \text{radians} = 0.000^\circ\)
- Angle for ground speed = \(\pi - \pi + 0 = 0 \, \text{radians}\)
- Ground speed = \(\sqrt{102.889^2 + 15.433^2 - 2 \times 102.889 \times 15.433 \times \cos(0)} = 87.456 \, \text{m/s}\)
- In knots = \(87.456 \times 1.94384 = 169.999 \, \text{kn}\)

5. Frequently Asked Questions (FAQ)

Q: What is ground speed?
A: Ground speed is the speed of an aircraft relative to the ground, accounting for wind effects, calculated as \(v_g = \sqrt{v_a^2 + v_w^2 - 2 v_a v_w \cos(\delta - \omega + \alpha)}\).

Q: What is true airspeed?
A: True airspeed (\(v_a\)) is the speed of the aircraft relative to the surrounding air, typically measured by the aircraft's instruments and corrected for atmospheric conditions.

Q: What is the wind correction angle?
A: The wind correction angle (\(\alpha\)) is the angle between the aircraft's course and its heading, calculated as \(\alpha = \sin^{-1}\left[\frac{v_w}{v_a} \sin(\omega - \delta)\right]\), to compensate for wind drift.