Glide distance at various weights

[jrh]

I feel kind of stupid for not understanding this, but I guess a good pilot is always learning...I've had this question in the back of my mind for a long time and never bother to ask anyone what the answer is.

I've read in several textbooks that best glide speed varies with weight. A heavily-loaded aircraft will have a faster best glide speed than a lightly-loaded aircraft.

This is because best glide speed is based off of flying at a specific angle of attack that produces the most lift for the least amount of induced drag. That AoA will occur at different airspeeds depending on weight.

I believe I understand all of these facts so far.

What I don't get is this concept--textbooks say an aircraft will travel the same horizontal distance across the ground while gliding, regardless of weight. It will get there faster at a higher weight, but the total distance a gliding aircraft can cover remains the same.

I don't agree with this concept in theory, or in practice. I don't think a heavy aircraft is able to cover as much ground as a lighter aircraft.

The way I've reasoned it out in my head is that if the AoA remains constant, the induced drag remains constant. However, if an aircraft is heavier, it must go faster, and if it's going faster, it must have more form/parasite drag. If it has more drag, by definition, energy is getting wasted. Therefore, a heavier aircraft's total energy (Kinetic + Potential) will reach zero sooner than a lighter aircraft's.

Another, less scientific way I think about this is that if a plane needs to go faster to maintain a certain AoA, the nose must be pitched down. If the nose is pitched down, the "aiming point" will fall closer to the plane than if the pitch were higher.


But all of this contradicts the textbooks saying, "A plane will glide the same distance at best glide AoA regardless of weight." So I know I'm probably wrong. But what piece am I missing? How can this scenario be proved mathematically?

 

[MidlifeFlyer]

I think this is one for a tgrayson analysis, but I don't have a problem with buying the concept that an individual airplane will cover 1.5 miles per thouusand feet of altitude will do so at 105 kts at a 3300 lbs and at 99 kts at 2940 lbs.

 

[tgrayson]

The way I've reasoned it out in my head is that if the AoA remains constant, the induced drag remains constant.

Not exactly. The induced drag coefficient remains constant. You have to combine that with airspeed to get the actual induced drag. If you have a light airplane and a heavy airplane at the same AoA, the heavier one will be going faster and incurring more induced drag.

However, if an aircraft is heavier, it must go faster, and if it's going faster, it must have more form/parasite drag.

Yes, but the coefficient of parasite drag remains the same. You're mixing your metaphors by talking about induced drag coefficients above and total drag now.

If it has more drag, by definition, energy is getting wasted.

Not necessarily. The total energy "wasted" will be the original kinetic energy of the aircraft and its altitude, which go from the same quantity to zero during the course of the flight, assuming the same original airspeed.

Therefore, a heavier aircraft's total energy (Kinetic + Potential) will reach zero sooner than a lighter aircraft's.

That much is true, but you aren't talking about energy wasted, you're talking about energy wasted per unit "time". Glides are all about energy wasted per unit "distance". The heavier airplane will surely be on the ground sooner, but at the same distance.

Another, less scientific way I think about this is that if a plane needs to go faster to maintain a certain AoA, the nose must be pitched down. If the nose is pitched down, the "aiming point" will fall closer to the plane than if the pitch were higher.

A conflation of AoA, pitch, and flight path. The faster airplane will have a lower AoA, lower pitch, and identical flight path.

How can this scenario be proved mathematically?

The dependency of glide angle of L/D ratio is a pretty simply proof. You might be able to pick it out from this diagram:

 

[jrh]

Yeah, I was hoping tgrayson would show me where I'm going wrong.

To me, it seems like the heavier aircraft is "getting something for free" so to speak. If this concept is accurate, we could stick an elephant in a 172 and glide the same distance as normal, as long as we pitched for 300+ knots. There'd be no advantage to keeping an aircraft lightweight.

A thought I just had--maybe there is increased drag from the higher airspeed, but it's offset by the higher potential energy of an aircraft with greater mass? I don't know.

 

[tgrayson]

If this concept is accurate, we could stick an elephant in a 172 and glide the same distance as normal, as long as we pitched for 300+ knots.

Gliders do that when they race.

Note that there IS a limit to how far you can take this; one thing the analyses don't take into account is Reynolds number. If you had a really, really large difference between the light and heavy airplanes, then the heavy airplane would be flying so much faster that some of the assumptions used when calculating drag lift drag curves would change.

 

[jrh]

....

Ok, thanks for all that. I made my last post before I saw your first response.

So would it be accurate to say an airplane with greater mass has more "thrust" from it's weight during a glide? What I mean is, the forward component of weight would be greater in a heavier aircraft, which is what allows it to glide faster while maintaining the same angle through the air?

I'm still trying to wrap my head around this.

 

[tgrayson]

So would it be accurate to say an airplane with greater mass has more "thrust" from it's weight during a glide?

Absolutely. If you do the vector math, you will see that the component of gravity along the flight path equals the drag.

BTW, I can't help readdressing something I said above, because it's so important:

A conflation of AoA, pitch, and flight path.

This is the whack-a-mole of flight mechanics. A huge number of conceptual problems start here and cannot be resolved until people can be convinced to use the proper terminology. Wrong words produce wrong thoughts, so I'm not just being pedantic in making this correction. This is why I continually point out that it's incorrect to say that pitch controls airspeed, it's AoA, and saying one to mean the other will be an obstacle to growth in this body of knowledge.

 

[jrh]

Absolutely. If you do the vector math, you will see that the component of gravity along the flight path equals the drag.

Got it! That was the missing link for me. I couldn't figure out how a scenario with more drag could result in the same total glide distance, until I thought about this element.

BTW, I can't help readdressing something I said above, because it's so important:

...

This is the whack-a-mole of flight mechanics. A huge number of conceptual problems start here and cannot be resolved until people can be convinced to use the proper terminology. Wrong words produce wrong thoughts, so I'm not just being pedantic in making this correction. This is why I continually point out that it's incorrect to say that pitch controls airspeed, it's AoA, and saying one to mean the other will be an obstacle to growth in this body of knowledge.

Yeah, I totally agree and understand this part. I threw in my original statement about pitch without thinking it through. I'm usually careful to keep pitch attitude and AoA in seperate discussions, but tossed out an example off the top of my head too quickly in this case. I'm more careful when I'm actually teaching

 

[shdw]

This is because best glide speed is based off of flying at a specific angle of attack that produces the most lift for the least amount of induced drag.

I didn't see this specifically corrected, only indirectly, and think it is important. Not the least amount of induced drag, but the least amount of total drag results in best glide. I would reread what tgray said with that knowledge, it might shed some light.

Do you know the easy formula to determine the change in best glide?


Tgray: Would it be incorrect to say that AOA controls speed and pitch/trim acts to change AOA.

 

[tgrayson]

Not the least amount of induced drag, but the least amount of total drag results in best glide.

Correct, good catch.

Do you know the easy formula to determine the change in best glide?

sqrt(Weight_actual/Weight_maxGross) * BestGlide_maxGross

Would it be incorrect to say that AOA controls speed and pitch/trim acts to change AOA.

Correct, assuming by pitch you mean pitch control, rather than the angle the aircraft's longitudinal axis makes with the horizon. (And I'm sure you do, just wanted to keep it clear in print.)

 

[shdw]

sqrt(Weight_actual/Weight_maxGross) * BestGlide_maxGross

I said easy! As in something you could guestimate in your head.

Gross weight - actual weight = change in weight

[(Change in weight/gross weight) * 100] / 2 = % change in best glide


Looks more complicated, but it is all basic math.

Total weight = 2500
Actual weight = 2000
Best glide = 65

2500 - 2000 = 500

500/2000 = .25 * 100 = 25 / 2 = 12.5 percent change

So about 61? No calculator.

Edit: You will find this is off by maybe 5 percent, which with under 100 knots best glide is a matter of a couple 1/10ths of a knot. Nothing to be concerned with and something you can do in your head or teach to even the most math illiterate students.

 

[shdw]

Total weight = 2500
Actual weight = 2000
Best glide = 65

2500 - 2000 = 500

500/2000 = .25 * 100 = 25 / 2 = 12.5 percent change

So about 61? No calculator.

I screwed this up pretty bad, was in a rush and not thinking. Woops.

Anyways, here is it done right:

2500 - 2000 = 500

[(500 / 2500) * 100] / 2 = 10%

10% of 65 is 6.5, round to 7 | so 65 - 7 = 58 knots best glide


Done the way tgray said:

sqrt(2000/2500) * 65 = 58.13

The percentage this comes up with is 10.55%.