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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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.
Even a bit of tailwind can be a hazard. Tailwind conditions can have adverse effects on aircraft performance and handling qualities in the critical flight phases of takeoff, approach and landing. Tailwind, for instance, increases the required runway length to land on or takeoff from.
If you are already in the air, going to another airport to land may be the better option. There are many reasons to avoid a takeoff or landing with a tailwind. These range from the fact that takeoff and landing distance requirements will be longer to reduced climb gradient due to higher ground speed over distance.
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.
As the aircraft approaches the runway, the pilot uses the rudder to point the aircraft's nose into the wind, counteracting the crosswind and maintaining the desired flight path. This means the aircraft will be angled slightly sideways, or 'crabbing' as it comes in for landing.
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.
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.
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.