Altitude prediction VS Airspeed

[simurq]

Well, the more I delve into the topic, the more are chances to ask noob questions, as you can see if you follow up my posting statistics lately. :) So, please bear with me!

First things first... I'm writing a program which needs to calculate altitude of an aircraft at each checkpoint until it reaches the cruising altitude. Let's assume that a pilot chooses to climb at best rate-of-climb speed (Vy). And here is a sample scenario:

Airfield's pressure altitude (QNH) -> 1,500 ft
Distance to the 1st checkpoint (S) -> 10 nm
Airspeed taken from POH (Vy) -> 78 KIAS
Climb rate -> 1010 fpmNow, to find the altitude at next checkpoint I would simply do the following and wouldn't bother posting here:

Time to reach the checkpoint -> 10 nm / 78 KIAS = ~7.69 min
Altitude -> 1,500 ft + (1010 fpm x 7.69 min) = ~9,270 ftBut... for accuracy sake (http://williams.best.vwh.net/avform.htm#Mach), I need to know the ground speed (GS) of an aircraft which, in turn, depends on TAS derived from IAS using... altitude (!) as one of the variables. :confused: Looks like a chicken-and-egg problem to me, doesn't it!

Again, I could keep the calculations simple using the famous "2% increase per 1K of altitude" rule-of-thumb for the speed. But I need your help to explore all available methods to get rid of altitude-speed dependency before I move on.

So, I will highly appreciate any help!!! Thanks!

[Capn Bloggs]

There's probably a flight computer app that will work out the TAS as you climb.

Don't forget to allow for the decreasing rate of climb as the engine runs out of puff whilst climbing...

I have the link to Ed's great circle nav web form calculator on all my devices. Very handy! :D

[ATCast]

For an simple approach to this problem one could cut up the 10 NM into a number of segments.
Suppose you would use 20 segments of 0.5 NM to represent the 10 NM

Now calculate the ground speed at the start of the segment and assume that it is constant over the whole segment. Then calculate the time it takes to reach the end of the segment and calculate the new altitude. With this new altitude calculate the speed for the next segment. And so on until the end of the 20th segment is reached.

This is called the Euler method (http://en.wikipedia.org/wiki/Euler_method)
To improve accuracy it's easiest to increase the number of segments. However that become computationally very costly soon.

For more accuracy, you could improve by using the backward Euler method or go for higher order models like RK4 (http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_method).
These methods are more work to implement than just adding more segments but they give the required accuracy more efficiently.

[simurq]

After some research I am seriously thinking about introducing climb gradient in calculations. For example:

Climb gradient (θ) = 1,010 fpm / [(78 knots x 6,076 ft)/60 min] = ~12.78%
Climb angle = ASIN(climb gradient) = ~7° (0.122173 rad)
Height to climb = TAN(climb angle) x distance = 7,834 feet
Altitude = 1,500 + 7,834 = 9,334 feet But still... in both cases I have to use KIAS, not the ground speed! It seems I'm stuck again... :ugh: :ugh: :ugh: