In order to have motion, you need to have a force applied to the object.
F = ma
No, in order to change motion, you need to have a force applied.
It is F=ma, not F=mv.
All that I am asking is for you to place the force on the FBD that supposedly keeps the airplane is sync with the earth rotation, when it is flying in the sky.
You have already been told what that force is, why do you need a diagram?
It is simply air resistance.
Write down the equation of the conservation of momentum with all components, x, y and z to prove me wrong
No. You write it down to prove your baseless crap.
If order for the airplane to get synchronized with the rotation of the Earth, there has to be a force that is applied to it. Remember the airplane takes off from a runway that faces East to West, with the supposed spin of the earth which by the way at that location was 94 m/s and then it land on a runway the is facing North to south with a supposed spin of the earth of 393.33 m/s
And guess what? The same is basically true if Earth was stationary.
The plane takes off in one direction and has to land in another.
Do you know the force responsible for it? AIR RESISTANCE, or a combination of air resistance, thrust, weight and lift.
I even asked you to draw a free body diagram, but you didn't. I guess that means you accept Earth is round?
Somewhere in between the flight, there was a force acting on the airplane that caused the airplane to get synchronized with the new earth velocity which my the way, overcame the airplanes inertia.
Do you mean during?
If so, yes, there was a force acting for the entire duration of the flight. It is called AIR RESISTANCE (or more technically a combination of air resistance, thrust, lift and weight). What magic extra force do you think you need?
For the most part, the atmosphere moves with Earth. That means when the plane takes off with Earth moving at (just using your numbers) 94 m/s, the air is moving at that speed.
As the plane travels around Earth to a spot where Earth is moving at 393.33 m/s, the atmosphere is also changing speed and as the plane is in the air, it will experience air resistance to make it travel at that speed.
All that I am asking is for you to place this force on a Free Body Diagram.
And I am saying that is entirely pointless. Are you going to reject air resistance? If not are you going to explain why it can't synchronise the plane?
You even have that in your diagram to some extent labelled "drag".
Guess what? That drag force is dependent upon the plane's velocity through the air. It can point in any direction (however if it is pointing forwards you will have a problem flying).
If I would have said that there is no gravity and a block slides on a plain is do to God. If I would also challenge to draw me a Free Body Diagrom of the block and to place the force of gravity on it, you would have no problem in doing it.
The issue isn't having a problem, the issue is going to the effort of making one for you to just ignore it.
Even then I likely wouldn't draw you one.
But in the airplane example, how can you place a nonexistent force on the FBD!!!
The force exists. It is already in yours. Why do we need to make another?
As for your video, there was a side wind (pressure) that pushed the airplane to the side. This force can be calculate by multiplying the airplanes Surface Area that was perpendicular to the wind with the Pressure, but you must be consistent in your units.
And guess what? The atmosphere moving with Earth would cause the same effect.
Are you saying that air resistance causes the airplane to get synced with the earth supposed rotation?
Are you also saying that
The air is moving with Earth.
Yes I am.
The air moves with Earth for the most part. There is some minor wind as the friction is not infinite.
The plane is moving through this air, which results in friction AKA air resistance AKA drag causing the plane to match the speed of the air and thus the speed of Earth.
In the case of winds, the thrust component as well as the lift component can be manipulated to fine tune the matching as occurs in the crosswind landing I provided.
Again, WHAT DO YOU THINK IS MISSING?
WHY DO YOU THINK THIS CAN'T SYNC THE PLANE?
Now how about you draw me the Free Body Diagram of the Forces that keep the airplane in sync with the supposed earth spinning.
Here one is:

Tell me what's missing?
Remember that this is just a generic one not showing any specific direction.
The drag component will be in the direction of the air's velocity relative to the plane.
Assuming it is a simple plane without thrust vectoring thrust will be forwards.
Weight will be down.
Lift will be complex based upon the specific orientation of the flight surfaces and the speed (and direction) of the airflow over it.
The force you are looking for is predominately that drag force.
So happy, you have your diagram. Now going to admit there is no issue for a spinning RE?
OK, with your logic, the Wind velocity on the earth will be a gradient that will have a constant increase from the one airport to the second airport do to that at one location the earth velocity is 94 m/s and in the second location the earth velocity is 393 m/s, but this is not true.
No, that is true. If it wasn't there would be massive friction between Earth and the air causing it to happen anyway.
In convention heat transfer we can calculate the thermal layers and motion layers of air do to a moving plate. These boundary layers have a finite height of impact and over height the effect gets lost. Let’s say that I was in the metro dome and I placed a walker in the center of the field. I then jacked up the speed of the walker to 10 m/s, would the air on the top of the dome which was located right over the walker go the way the walker was working. NO
Is this anything like the situation you are attempting to discuss? No.
How about this:
Just have your walker and a small amount of air around them. Don't have anything else like a stationary dome or the like to trap the air.
Then the air at the top will eventually start moving with the walker.
Convection heat transfer disproves this notion that the atmosphere is rotating with the earth.
No it doesn't.