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Could airplanes, in theory, be designed to takeoff by first tying the tail to a post, throttling the motors to full power, and then letting loose?
Could it even take off while tied, if the cable is attached to a point that's higher than the tail of the aircraft?
As already mentioned, aircraft cannot take off without moving through the air, because it is the airflow around the wing that generates the lift required for take-off, and the only practical way of generating sufficient airflow around the wing is to move the wing through the air. This is typically done by moving the wing, and thus also the fuselage it is attached to, forward.
There is, of course, another way to look at that: aircraft can take off without moving relative to the ground, if exposed to a strong enough wind from the front of the aircraft.
That said, keeping the aircraft stationary, "throttling the motors full power, and then letting lose" is a very common short-field take-off technique at least for propeller aircraft; there's nothing theoretical about that. The aircraft is, however, typically kept stationary by applying full brakes while applying full engine power, rather than by using an external tie-down.
The tail likely isn't designed to take the force of holding the aircraft stationary while the engine is running at full whallop, although there's no reason why such an anchor point couldn't in theory be installed (see motor aircraft towing gliders to altitude, scale up somewhat, and you're close).
Since the brakes are already there and need to be powerful enough to at the very least hold the aircraft stationary during the engine run-up prior to take-off, which happens at a significant fraction of full power and can quite well happen at a hold short point just before the runway, there's no reason to spend weight on reinforcing the tail section to that point. Instead, just use the brakes for the few seconds needed.
No, an airplane cannot takeoff while tied to a post.
What creates lift is the speed of the wings through the air.
(or the alternate frame of reference view: The speed of air over the wings.)
If the plane is tied to a fixed point, like a post, it is not moving in the air, and the wings will not have any lift.
Well that’s part of a catapult launch from a carrier.
The aircraft is held motionless on the cat by a holdback bar attached to the nosewheel or a holdback bridle which is also attached to an anchor point on the flight deck. The link between the bar or bridle and the aircraft is a frangible metal link, sometimes called the ‘dog bone’ because it is shaped something like a dog biscuit. It is strong enough to hold the aircraft still, even under full power but fractures when the force of the catapult or EMALS is applied to the airframe during the launch stroke.
For terrestrial takeoffs, I suppose one could do something like that but it is much simpler to just hold the brakes, run up to full power, then release the brakes and perform the takeoff roll. This is the preferred method of takeoff for light twins and turboprops, as it minimizes the length of the takeoff roll.
