I understand that tailwind is typically a good thing during your flight as it increases your ground speed and gets you to your destination faster. Despite that I get a feeling that pilots typically get very uncomfortable when there is a significant tailwind present during landing. Sadly, recent proof of this danger was in Aspen, Colorado accident
Why are tailwinds bad during landing?
Given "extra long" runway for extended rollout, are there any dangers with landing at a higher ground speed?
How are tailwinds mitigated? Would ATC just switch the landing direction?
Just an additional human factors flavored aspect:
Your speed relative to the runway is greater than usual. If you start adjusting your relative speed by your usual (headwind) visual reference, this might well lead to a stall situation - never a good thing close to the ground.
It's not so much a matter of "pilots typically get very uncomfortable" as it is "pilots recognize that it is an inherently less safe situation", and pilots (at least the ones you want to fly with) tend to be somewhat safety obsessed.
So Why are tailwinds during landing "bad"?
The same reason tailwinds in cruise flight are good: you're moving over the ground at a higher speed.
The amount of energy the aircraft has (and thus its landing distance) is roughly proportional to the square of the groundspeed.
Let's consider the plane I fly as an example (a Piper Cherokee) because I know the numbers:
if I'm flying by the book then when the wheels of my plane hit the ground my indicated airspeed should be around 45 knots.
If there's no wind my ground speed and my indicated airspeed will be about the same - my brakes will have to slow me from 45 knots to a stop before I run out of runway.
The brakes on the Cherokee aren't Porsche brakes (they're tiny little things), but they're up to the task and a book-perfect landing would have me stopped in 600 feet.
A 15 knot headwind reduces my groundspeed to 30 knots - at 2/3 the speed I can stop much sooner with the same braking force, which means less wear on the brakes and a greater safety margin of runway remaining.
A book-perfect landing in these conditions would have me stopped in about 270 feet.
A 15 knot tailwind increases my groundspeed to 60 knots. The tiny little brakes on the Cherokee CAN stop me, but they're going to take much longer to do so: A book-perfect landing would take over 1000 feet to stop (nearly twice the "normal" no-wind landing roll).
The big consideration above is landing distance -- as you noted in your question you can solve that with a longer runway (more pavement means more room to stop), but there are other dangers with tailwind landings, mainly in the fact that Landings don't always go according to plan.
The slower your groundspeed at landing the less energy you have to get rid of if something goes wrong
Consider the three landings I outlined above, but let's add a problem to the mix: One of my tires blows right at touchdown, and the plane veers off the runway into a ditch.
There is much less energy to dissipate with a 15-knot headwind (30kt groundspeed) than there is with a 15-knot tailwind (60kt groundspeed) - it's roughly the difference between crashing your car driving through a school zone versus speeding on a highway.
Fortunately it's really easy to avoid landing with a tailwind -- most runways are bidirectional (Aspen, which you mentioned in your question, is a notable exception due to geography).
ATC (or at uncontrolled fields, the pilots themselves) will "turn the airport around" once a certain tailwind limit is reached. Precisely when that happens depends on the airport and the traffic mix (at my home field, with mostly small piston aircraft, ATC will typically switch the runway when the tailwind component exceeds 5 knots - at JFK they might land a jet with a 10-knot tailwind rather than messing up the arrivals).
Of course there's nothing that says a pilot has to accept a landing with a tailwind: If the pilot is not comfortable with the conditions they can request a runway aligned so the winds are more favorable. (There is a somewhat famous case of an airline pilot at JFK who ultimately decided to declare an emergency to ensure a landing on a favorable runway).
The primary concern is the increased landing distance due to the increased ground speed. Landing & stopping distances increase more-than-linerally with each knot increase in ground speed. Given other dynamics of a landing aircraft this can be the straw that breaks the camel's back.
The perfect storm
True story. C-141 on approach with a tail wind - not so bad by itself. Add in a 7,000-ish foot runway and indicated airspeed a little high.
And the worst part of all of this?
To answer in part the question of how tailwinds are sometimes mitigated, an air carrier's op specs will typically specify how much of a tailwind is allowed for their operations. The two 747 carriers I flew for allowed a max tailwind of 10 knots.
Whether a captain should choose to land given the max tailwind can be a complicated call. For example, runway 02 at Nadi, Fiji is about 9,000 feet, which makes it a bit of a short field operation for a 747 landing at or near max landing weight, a common occurrence with freighters. In the instance I'm thinking of, we were well inside the outer marker when the tower told us the wind had switched and that there was now a 10 knot tailwind.
I chose to land and here's why:
747 tires have 225 psi tires filled with nitrogen, and each tire has a thermal plug that will blow before a tire does. The QRH for handling hot brakes included instructions to warn the ground crew not to stand to the side of the tires, but the Nadi people were well aware of that.
All brake temperature gauges went into the red. I got out of the airplane as quickly as possible to take a look. They already had a huffer on each side of the airplane blowing on the brakes.
As others have highlighted, the question is with what ground speed you touch down. You might think that plus/minus a few knots doesn't really matter, but:
So, to adapt voretaq7's example, if your approach air speed is 45 knots, with a 15 knots headwind you land with 30 knots, while with a tailwind you land with 60 knots - now you touch down with four times the energy you need to dissipate, and need four times as much runway to stop (which, happily, lines up nicely with his POH numbers).
Now, finally, one more example:
Suppose the runway is such that with no wind, you can land with approach speed v and just stop prior to the end.
With a (mild) headwind of 10% of your airspeed, you touch down with 0.9v, thus you now use only 81% of the available runway, and have gained quite a buffer.
Whereas, if you have a 10% tailwind, you touch down with 1.1v. Thus, your required stopping distance now is 1.21 times the available runway. When you hit the end now, you'd need 21% of the runway length extra to stop. Now, this gets a bit tricky, but feel free to do the maths, this means that you hit the wall at the end with nearly half of your approach speed, still having about 21% of your (1v) landing energy left to dissipate.
So - what would you rather have, a 20% buffer of runway remaining (with head wind), or hitting the end of the runway with over 40% of your landing speed (with tail wind)? :-)
With a really good headwind and a little bitty plane (Cessna 150) you can get your groundspeed down to a trot and and just set it on the ground. Have done it at PDK in Atlanta, in March.
Do not do this with passengers. Safety is one reason. Not terrifying them is another.
I understand that tailwind is typically a good thing during your flight as it increases your ground speed and gets you to your destination faster. Despite that I get a feeling that pilots typically get very uncomfortable when there is a significant tailwind present during landing. Sadly, recent proof of this danger was in Aspen, Colorado accident Questions: Why are tailwinds bad during landing? Given "extra long" runway for extended rollout, are there any dangers with landing at a higher ground speed? How are tailwinds mitigated? Would ATC just switch the landing direction?
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