# Mathematical Solution to the Triangle of Velocities

The motion of an aircraft relative to the surface of the Earth is made up of two velocities. These are the aircraft moving relative to the air mass, and the air mass moving relative to the surface of the Earth. Adding these two vectors together gives us the aircrafts motion over the ground. Together, they form the “triangle of velocities”.

A wind that is blowing from the left will carry an aircraft onto a track that is to the right of the heading. In order to achieve a particular track from A to B the aircraft must be turned into-wind by an amount that corrects for the drift.

Each of the three vectors in the triangle of velocities has two properties - magnitude and direction. This means that there are a total of six components. These are the True Air Speed (TAS) and heading (HDG) of the aircraft, the speed and direction of the wind (W/V), and the Ground Speed (GS) and track (TR) of the path over the ground. The triangle of velocities is shown below. Typical navigation problems involve finding two of these properties when given the other four. For example, most of the time we know what track and air speed we would like and we also have the forecast wind velocity. What heading do we need to steer to follow that track, and what ground speed will we achieve?

The usual method of solving this problem is with a “Flight Computer” such as the E-6B, also known as a “whiz wheel”. The wind side of the flight computer consists of a circular rotating compass rose which is marked with an index at the top, and has a transparent screen in the middle to allow viewing of the slide plate underneath. The slide plate is marked with concentric speed arcs and radial drift lines. The computer allows you to physically visualise the triangle of velocities, and read off the answer you require. But what calculations is the flight computer performing? How do you solve the problem mathematically?

In order to find the heading required, we need to make use of the sine rule. The sine rule states that for any triangle the ratio between a given side length and the sine of the corresponding angle is equal for each side of the triangle. We simply substitute in the parameters from the triangle of velocities, and rearrange to solve for our heading (HDG). Thus,

## Ground Speed

The ground speed is simply the magnitude of our track vector. The easiest way to determine this value is to divide the triangle of velocities up into two right-angled triangles. The length of the track vector is then just the sum of the x-component of our velocity through the air mass and the x-component of the wind velocity.

## Flight Planning

These equations are quite cumbersome, and if working out the solutions by hand then by far the quickest solution is to use the whiz wheel. However, now that we understand the mathematical solutions it is possible to enter them into a spreadsheet and speed up the flight planning process considerably.

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