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I know that wind is taken into account when we compute the TAS(true airspeed) and GS(ground speed). However what is the relationship between the TAS and altitude?

For example: What is the true airspeed on the level flight if the headwind is 20 mph and ground speed is 100 mph at 30,000 feet altitude?

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    $\begingroup$ Learn to make the mental calculation of adding 2% to indicated airspeed for every 1000 ft above SL, and you can get a TAS value that is accurate enough for most situations. $\endgroup$ – John K Apr 18 at 14:13
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If you know GS and local wind speed, the TAS always the same, regardless of altitude. The true airspeed is called "true" because it is exactly how fast you are moving relative to the air. Picture it as a person in a weather balloon (which does not move relative to the surrounding air, i.e., TAS=0), pointing a radar gun at your aircraft to measure its speed. The results would be the same regardless of altitude, temperature, pressure, etc. This is what sets TAS apart from pretty much any other speed like IAS, CAS, EAS and Mach.

Only at truly ridiculous altitudes you would have some problems with calculating TAS, due to the increased orbital radius from the center of the Earth. However, at this point, you're well into the near vacuum of space which means that the whole concept of 'air' speed is quite silly.

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  • $\begingroup$ Thanks for valuable answer.. $\endgroup$ – user38468 Apr 18 at 12:01
  • $\begingroup$ Not to mention, in re your last paragraph, that you are dealing as much in plasma dynamics as fluid dynamics at those altitudes. $\endgroup$ – KorvinStarmast Apr 18 at 12:40
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The true airspeed (TAS) can be calculated from the indicated airspeed (IAS), which is derived from the pitot tubes and static ports, as follows:

$$ \mathrm{TAS} = \mathrm{IAS} \sqrt{\frac{\rho_0}{\rho(a)}} , $$

where $ \rho_0 $ is the air density at sea level and $ \rho(a) $ the air density at altitude $ a $, which depends on pressure $ P $ and temperature $ T $:

$$ \rho(a) = \frac{M \cdot P(a)}{R \cdot T(a)} , $$

where $ M $ is the molar mass of air and $ R $ is the universal gas constant.

Using the international standard atmosphere for $ P(a) $ and $ T(a) $, one can plot the TAS as a function of altitude: TAS as a function of Altitude for fixed IAS

The ground speed (GS) is then given by the vector addition of the TAS and the wind speed: $$ \mathrm{GS} = \mathrm{TAS} + v_\mathrm{wind} \cos(\alpha) , $$ where $ \alpha $ is the angle between the wind direction and the track of the aircraft ($ \alpha = 0^\circ $ for tailwind, $ \alpha = 180^\circ $ for headwind). This step is independent of pressure or temperature and as such independent of altitude. This means for a given TAS and headwind component, the ground speed is the same at all altitudes (for your example: 100 mph GS with 20 mph headwind implies 120 mph TAS at all altitudes).

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