Calculating Bird Airspeed from Ground and Wind Speed

[KLB]

Homework Statement

Please bear with me. I'm a PhD student biologist struggling to remember some maths and physics, so I fear that this question may seem very simple to people using this forum.

The problem I have is with some data I have on bird flight, measured using a radar. I have bird ground speed (in metres per second) and heading (in degrees from north) and wind speed (in metres per second) and heading (also in degrees from north). I want to calculate airspeed. This would be simple if I had the angles between the two forces, but using bearings from north doesn't give me any of the internal angles of the triangle that I need to construct.

I can obviously use charts and marine navigational aides to work this out in practise, but I have 500,000 bird tracks, so need to be able to program the calculations.

One example track has bird ground speed of 15 m/s, with a heading of 136 degrees from north. Wind speed is 5 m/s with a heading of 295 degrees from north.

Homework Equations

Sine Law
sin a/A = sin b/B = sin c/C

Cosine Law
c^2=a^2+b^2+(2abcosc)

The Attempt at a Solution

I'm afraid that this is as far as I've got. My problem is really finding one of the internal angles of the triangle, in which case I can use the cosine law to calculate the length of the third side.

I've considered treating the two vectors seperately and looking at the displacement in x and y for each, but this doesn't deal with tracks and wind speeds that are in the opposite 180 degrees (you get problems with whether a speed should be added or subtracted).

Many thanks for any guidance you can give on how to treat this problem or on methods for finding one of the internal angles.

 

[tiny-tim]

My problem is really finding one of the internal angles of the triangle, in which case I can use the cosine law to calculate the length of the third side.

Hi KLB! Welcome to PF!

You already have one of the internal angles - it's the ground speed heading minus the wind speed heading!

And remember cos(90º + x) = -sinx.

[size=-2](if you're happy, don't forget to mark thread "solved"!)[/size]​

 

[Moetasim]

Hello there, it's great to see a biologist delving into the world of physics and math! Don't worry, your question is not simple at all and it's perfectly understandable that you may be struggling with the calculations. Let's break it down step by step.

First, let's define some variables for easier understanding:
- Bird ground speed = Vb
- Bird heading = θb
- Wind speed = Vw
- Wind heading = θw
- Airspeed = Va

To calculate airspeed, we need to use the law of cosines to find the length of the third side of the triangle formed by the bird ground speed, wind speed, and airspeed. However, as you mentioned, we need to find one of the internal angles of the triangle first.

To do this, we can use the law of sines. Since we know the bird ground speed and wind speed, we can use the following equation:

sin θb/Vb = sin θw/Vw

Rearranging this equation, we can solve for the angle θb:

θb = sin^-1(Vb/Vw * sin θw)

Now that we have the angle θb, we can use the law of cosines to find the airspeed:

Va = √(Vb^2 + Vw^2 - 2VbVwcosθb)

Plugging in the values from your example track, we get an airspeed of approximately 19.5 m/s.

As for dealing with tracks and wind speeds that are in opposite 180 degrees, we can simply take the absolute value of the difference between the two headings (i.e. |θb - θw|) and use that value in our calculations. This will ensure that we are always using the correct angle.

I hope this helps you with your calculations and programming. Best of luck with your research!