Relation of Weight for Best Gliding Speed

[Pugachev Cobra]

Hi there, my first post! Little background, I'm a pilot studying some flight theory, and came across a question which I honestly cannot answer. The problem goes like this: The pilot wants to increase the range of an aircraft in a gliding flight. If the "optimum" flight speed is 100kt for a weight of 500kgf, and the aircraft is weighing 845kgf, what's the speed it should descend? The answer given to me was 130kt, but I cannot find the relation in these numbers. Any help is kindly appreciated! Thanks in advance.

[Jimmy Do Little]

The relationship between the numbers can be found here...

Lift-to-drag ratio - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Lift-to-drag_ratio)

Cheers

[Pugachev Cobra]

Actually I've already read that Wikipedia article, but it doesn't say anything about weight, except the common knowledge that weight does in fact increase speed. But, I want to know the calculations, if anyone here can help. I've googled and read some articles and some books trying to find a relationship, and I know it's not complex, but I just cannot see how the answer of 130kts came to be.

[Coldbear]

The answer is simple math. It's a relative....

(I don't know the mathematical background)

sqrt(new weight/old weight) <=> sqrt(845/500) is 1.3 which is then multiplied by 100kts(old speed) to get 130kts(new speed)

hope this helps

Martin

[Port Strobe]

The angle of attack for best lift to drag ratio is fixed, so the mathematical background is to resolve a free body diagram

CL(L/D max) = LcosA/½p(V^2)S for glide angle A at best glide speed.

As I'm sure you'll know LcosA approximates L which is approximately W, and then you end up with the expression above.

As a rule of thumb if you know the percentage increase in weight you can work out the percentage increase in speed as half of that, the accuracy however starts to fall away not too far beyond the ratios involved in your examples but give it a shot, it's a useful check for bull$hit in = bull$hit out.

[bfisk]

The key to solving this problem is the fact that best glide speed is, in fact, best gliding AoA. Since the aerodynamic shape of the aircraft remains unchanged, so does the AoA.

Because L=0,5*rho*CL*IAS^2*
and rho, CL(AOA(bestglide)) and S being constant,
Lift is proportional to IAS squared, thus,
IAS is proportional to the square root of lift.

Given that the amout of lift needed is proportionate to the weight of the aircraft we can state that

(NewIAS/OldIAS)=(root(NewWeight)/root(OldWeight)), you get
NewIAS=OldIAS*root(NewWeight/OldWeight)

and with numbers that turns out to be
NewIAS=100*root(845/500)
NewIAS=130

as stated by the previous poster :ok:

[Pugachev Cobra]

Sorry for the late reply, just had the time to read it now. Many thanks to everyone that helped answering. Really nice to be able to exchange and increase my knowledge with all of you. It helped a lot, thanks again! The only reason I didn't tried to work with the Lift formula, is because I thought rho wouldn't be a constant. As you all stated before, S, CL and Rho are constants, but I thought since it was for a gliding flight, the air density wouldn't be constant. Perhaps someone could tell me why it applies? Thanks folks!

[Port Strobe]

The scenario you described in the 1st post was for a heavier aircraft in the same conditions so the atmospheric properties would not change. If you're talking about extended glides from altitude then the EAS will be the same all the way down, and if it's slow speed compressibilty won't matter so I guess that makes the CAS and most likely IAS constant. TAS of course will vary. Unless you're in a glider it's rather academic though I'd have thought since in a (previously) powered aircraft you'll have bigger things to worry about than correcting for small ASI errors etc. You'll probably gain a better performance increment by having a ball park figure in mind and correcting it for head or tail wind components (add or subtract half the wind respectively) to gain best glide angle over the ground.

[Tinstaafl]

Rho is fixed in this scenario because you're comparing like-for-like. That is, at any instant, what speed corresponds to the AoA for best L/D ratio. Imagine it this way: You're in a glide at a certain weight and the correct speed for the AoA to achieve max L/d. A few skyivers land on you. At the very instant the weight increased when they er...boarded the aircraft, what speed is needed to maintain the AoA.

Conversely, while in the same glide, all the occupants exited or you dumped stuff overboard instantly (dropping bombs would work). What speed is now needed for the best AoA?

In both these scenarios, the weight is considered to have changed with negligible change to the other factors. Another way to get to grips with the problem is in this thought experiment: Imagine an aircraft able to have multiple glides (eg with a climb to altitude between each one) at different weights. The aircraft would then experience the same rho at some point in each descent.