Mathematically Speaking...

[cyhu]

I looked for a while but there are too many variables that I could not find...

What would be the difference (other than negligible...) of the distance travelled for an aircraft at 10, 000 feet ASL and 30, 000 feet ASL, assuming no climb or decent.

To make it somewhat easier, Point "A" (lat, long) to Point "B" (lat, long).

Had this question from a pax recently, figured someone on here might have an idea how to calculate it...

Thanks

[Hedley]

The earth's radius at the equator is 3,963.189 sm.

At 10,000 feet, the radius will be 3,963.189 + 1.894 sm and therefore the circumference (2 x PI x R) will be 24,913.33 sm.

At 30,000 feet the radius will be 3,963.189 + 5.682 sm and therefore the circumference will be 24,937.13 sm

The ratio between 30k vs 10k is 24,937.13 / 24,913.33 = 1.00095
or 0.095 percent - almost one tenth of one percent farther.

[Lost in Saigon]

Sure, but how much less gravity would there be way up there?

[bandit1]

holy crap.

[cyhu]

I figured this would take a little while to answer but like he said above...

HOLY CRAP !!

Thanks Hed

[Crazymax]

Well, as you know, the air gets thinner with the altitude. We need less power to make the plane go forward at a certain speed as we get higher.

Here is how we calculate the range difference for different altitude:

SR = V * Ex / (SFC*W)

Where SR is the Specific Range (ft/lbf)
V is the Velocity (ft/s)
Ex is th aerodynamic efficiency coefficient at x altitude (unitless)
SFC is the Specific Fuel Consumption (lbf/(hr*lbf))
W is the Weight of the aircraft (lbf)

Ex = CL/CD

CL = Lift Coefficient
CD = Drag Coefficient

CL = 2*W/(ρsl * σx * S *V^2)

Where ρsl is the air density at sea level (lbf*s^2/ft^4)
σ is the air density coefficient at x altitude (unitless)
S is the surface of wing (or lifting body) (ft^2)

CD = CDo + K*CL^2

Where CDo form drag coefficient (unitless)
K is the induced drag coefficient (unitless)

If you have CDo, K, S and SFC and for a given W and V, you can easily find SR for 2 different altitudes and find out the difference. For example, for a BAe 146, CDo = 0.019, K = 0.04275 and S = 832 ft^2 and SFC = 0.324 lbf / (hr*lbf). So, for a weight of 85 000 lbf and a velocity of 700 ft/s :

At 10 000 ft (σ10K = 0.7382) : CL = 0.2377
CD = 0.02141
E10K = 11.10
SR10K = 1016 ft/lbf

So, for every pound of fuel you burn, you will travel 1016 ft

At 30 000 ft (σ10K = 0.3736) : CL = 0.4696
CD = 0.02843
E30K = 16.52
SR10K = 1512 ft/lbf

So, for every pound of fuel you burn, you will travel 1512 ft (50% increase from 10K).

The SR is a good approximation for range calculations. Note that there is no consideration for weight reduction due to fuel burn. This means that those values will increase when we take into consideration the fuel burn (which is less at higher altitude than at lower altitude, so the gap between 30K and 10K will be even bigger than it is now).

To sum up, it depends on the aircraft but it generally, going up will increase you SR significantly.

Max

EDIT : OOPS I just re-read the question, I'm totaly out in the field lol. I'll leave it anyways...

[Hedley]

Holy crap to CrazyMax

how much less gravity would there be way up there?

If you're serious, I'll calculate it.

[Pilot_adam]

Where do you guys learn all this ????
At the local flight school ??!!! How come no one tough me that ??

[Crazymax]

Where do you guys learn all this ????
At the local flight school ??!!! How come no one tough me that ??

Nah.. I have a Mechanical Engineering Degree. I cheated

Max

[husky]

Since gravity is proportional to 1/r^2 and the earths circumference is proportinal to r, gravity would be less by a greater percentage than the greater distance travelled.

[Crazymax]

Since gravity is proportional to 1/r^2 and the earths circumference is proportinal to r, gravity would be less by a greater percentage than the greater distance travelled.

Gravity can be determined by the following formula :

g = 9.78(1 + 0.005302*sin^2(L) - 0.0000058*sin^2(2*L)) - H*3.086 × 10^-6

Where g is the Gravitational Acceleration
L is the Lattitude (deg or radians)
H is the altitude (meters)

Lets say we are at 45 degrees latitude :

At Sea Level, g = 9.8321 m/s^2
At 10K, g = 9.8227 m/s^2
At 30K, g = 9.8039 m/s^2

So there isn't much of a difference (in fact, engineers neglect them when they design airplanes, they usually use 9.81 m/s^2 or 32.2 ft/s^2)

Max

[niss]

How fast does superman have to fly to counter the earths rotation and turn back time?

[rejd]

How fast does superman have to fly to counter the earths rotation and turn back time?

Here is my educated guess on this one

"Really fast"

Thanks

[trey kule]

Let me see if I got this all straight

Cyhu has way way to much time on their hands,

Hedley, actually knew how to calculate the answer....

Crazy Max jumps in with a calculation on range which begs the original question but demonstrates his genius.

I never cease to be amazed at some of the thinking that you guys do...and the things you know about aviation.

Keep it up.

[sprucegoose]

does this mean were all getting hooped for our milleage !

[bandit1]

like I said, HOLY CRAP. End of story. Stop wasting your time figuring this stuff out. There is alot of great porn to be found on the net. NOW GET TO IT!

[Pilot_adam]

like I said, HOLY CRAP. End of story. Stop wasting your time figuring this stuff out. There is alot of great porn to be found on the net. NOW GET TO IT!

I finally found someone one the same page as I !!!!!

[Strega]

Where do you guys learn all this ????
At the local flight school ??!!! How come no one tough me that ??

Nah.. I have a Mechanical Engineering Degree. I cheated

Max

Phew,, at least there are some other educated people here!

its always fun to explain to pilots that range has something to do with "ln" and "e"

Cheers

[MUSICMAAN]

Okay... so there's a Bee flying along, minding his own business along the highway... along comes a car and SPLAT goes the Bee on the windshield.

Does the Bee come to a complete stop before traveling in the opposite direction on the windshield, or does the Bee instantly start traveling in the oposite direction????

[Strega]

neither,, the bee ceases to be the moment it start to go splat!

[airway]

Here's another angle:

TAS increases with altitude for the same IAS. Assuming no wind your ground speed would also be higher at at a higher altitude.

BTW, What was the last thing to go through the bee's mind as he hit the windshield?

[bandit1]

Easy. The last thing to go through the bee's mind is his A S S

[mellow_pilot]

I looked for a while but there are too many variables that I could not find...

What would be the difference (other than negligible...) of the distance travelled for an aircraft at 10, 000 feet ASL and 30, 000 feet ASL, assuming no climb or decent.

To make it somewhat easier, Point "A" (lat, long) to Point "B" (lat, long).

Had this question from a pax recently, figured someone on here might have an idea how to calculate it...

Thanks

Just to throw a logical wrench into your mathematical works...

Both planes travel the same distance over the ground. The linear distance would obviously be different, as demonstrated by Hedley, but the actual ground distance covered is the same...

So the question must say, to get the answer you want, whether you're looking at linear distance (air miles) or the distance travelled, A to B.

HA HA!! Now I feel smart like the other people!!!

[ocnek]

The answer is quite clear: "An African Swallow"

[cyhu]

Man, you guys really poured your hearts into this one... Thanks again!

I love the additional workouts, it's rare we consider what aeronautical engineers have to consider in the design of any said aircraft...

Awesome guys !