There's no doubt that while you can land with a tailwind, it increases your risk of things going wrong. But at the same time, most GA airplanes have performance charts that let you calculate takeoffs and landings with up to 10 knots of tailwind.
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In the landing case, tailwinds may also result in higher number of go arounds due to the increased probability for an unstable approach. This increases both pilots' and controllers' workload. Additionally, landing aircraft would be likely to remain longer on the runway and vacate via later-than-normal taxiways.
The airplane will still fly; it just takes considerably more runway to depart. Most Pilot's Operating Handbooks (POHs) will use a phraseology in their takeoff roll performance charts such as increase ground roll 10% for every 2 kts of tailwind or similar.
Because headwind increases the lift, pilots prefer to land and take off in headwind. Tailwind is wind blowing from behind the aircraft. It reduces the lift and aircraft generally avoid taking off or landing in tailwind.
Because headwind increases the lift, pilots prefer to land and take off in headwind. Tailwind is wind blowing from behind the aircraft. It reduces the lift and aircraft generally avoid taking off or landing in tailwind.
Other special cases include a tailwind, where the plane and wind are acting in the same direction. As you may suspect, the speed of the aircraft increases when there is a tailwind and decreases when there is a headwind.
On take-off and landing, headwinds are good because they allow you to use a shorter runway. In flight, headwinds are bad because they slow you down and require that you use more fuel to get to your destination. Conversely, tailwinds are bad on take-off and landing, but are good in flight.
Operate in accordance with Manufacturer's Limitations: Aircraft manufacturers publish a tailwind component limit for both takeoff and landing in the AFM. In most cases, it is in the order of 10 knots but may be as high as 15 knots.
That flight set a record for the fastest subsonic transatlantic commercial airline flight — 5 hours and 13 minutes, thanks to a 202 mph tailwind. For reference, the speed of sound is 760 mph; but the plane did not break the sound barrier.
With this in mind, horizontal winds (also known as “crosswinds”) in excess of 30-35 kts (about 34-40 mph) are generally prohibitive of take-off and landing. As far as how this happens, it depends on where you are in flight.
As the plane descends into ground effect, it may actually accelerate if the engines are producing enough thrust, since in ground effect the plane requires much less power to keep flying. Power from the engines will translate into speed, if not height.
The Boeing 747 Jumbo Jet reached a top speed of 825 miles per hour which was boosted by a strong jet-stream and tailwind of over 200 miles per hour. Aircraft that cross the Atlantic between North America and Europe flying eastbound are usually boosted by a tailwind that can shave several hours off their flight time.
In this case of takeoff, the fast air bearing down on the plane generates an upward force on the wings (analogous to a gun's recoil), which helps lift the aircraft. In short, pilots like to take off into a headwind because it helps them achieve wheels up faster.