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. This is shown in Figure 1 – The Triangle of Velocities.
Figure 1 - Triangle of Velocities
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?
Calculating the Required Heading
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 – Figure 2.
Figure 2 - The Sine Rule
We simply substitute in the parameters from Figure 1, and rearrange to solve for our heading (HDG). Thus,
Calculating the 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 – Figure 3.
Figure 3 - Ground Speed
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.
[...] and subsequent ground speed to my spreadsheet I first had to derive the solution to the Triangle of Velocities. I thought a page showing the solution here might be useful to the next person trying to solve the [...]
Thank you for this. I tried to work it out from first principles and being an engineer, started with cartesian co-ordinate geometry which leads to some tricky simultaneous equations. Then I realised that Euclidean geometry was the answer but got in a muddle again by trying to use the cosine formula. I hadn’t used the sine formula since school (> 40 years) and had forgotten it ! A happy rediscovery.
One thing I would add to your excellent monograph and that is to remind your readers that they must convert wind angles from “from” to “to” for consistency with the Heading and Track vectors – then everything falls into place. Since you mention spreadhseets, it might also be worth warning that they work in radians (or at least the ubiquitous Excel does). A further picky point, you don’t define Wd and Ws although it’s pretty obvious to anyone who has thought about the problem.
Best wishes,
John Medhurst
Thanks for your comments John. Good to know it helped someone!
[...] solution uses the law of sines, which states that the ratio of the length of a side to the sine of the [...]
this does not help me at aallla soo take it down and yall know that stuff don not help haha so stupid ughh
I’m sorry this post did not help you Lisa. But tell me, why should I take it down just because it didn’t contain the information you needed? Other people have found it useful so I think I will leave it here. Thanks for your comments, and good luck!
Finding this post sovles a problem for me. Thanks!
All very interesting! You take me back to my school days. But are all these calculations necessary? If your looking to look clever in the clubhouse, I would suggest that your looking to have your ego stroked!
Let me ask the question… How long will it take you to calculate 5 legs that your trip may require?
We conducted an experiment with the above scenario. Using my CRP-1 to calculate my drift and not forgetting variation and lastly my GS, all 5 legs took 6 minutes! I ask you, all who read this to try doing it the ego method. And we used a maths teacher, he laughed and told us “don’t be so f****** stupid.
THIS METHOD IS OF NO VALUE!
Thanks for your comments Gill.
Method is of no value? How do you think people worked out how to make a CRP-1? And do you never want to know something, just for the sake of knowing and understanding?
Working this out certainly had value when I wanted to put all the calculations onto a spreadsheet.
But thanks anyway. Happy flying
Your missing the point Steve. We all learnt this at school. My dear Steve, it seems your arrogance is indeed only preceded by your encyclopaedic opinion of yourself. Yeh,, keep it simple… stupid!! But no doubt you’ll try to justify your superior intellect.
Must go… no flying today. xx
Nope, not justifying anything. Just happy to help. If someone finds it useful, that’s great.
Thanks for your comments.
Steve;
It is useful, as you said, to those who prefer to understand. I am familiar with the calculations but was looking for a shortcut – someone who put it together into a flight planning spreadsheet, or at least a cross check that I’ve not made a careless error in entering my own excel formulas.
I, like most, are totally comfortable using an E6-B but prefer a less tedious means to getting a good knee board flight plan together. Your explanations are excellent and clearly stated, and with diagrams to boot.
I don’t quite get the comments from those who attacked you, but applaud the restrained politeness in your responses. You’ve gone to the trouble to share information that is of interest to you, and I, unlike some others, really do appreciate it.
Thank-you.