Inertia on Planes: Pressure Differences Inside?

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Blacklukes
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Hello, I have a question regarding inertia. When we are on a plane, when it increases its speed we feel a force that goes on the opposite direction pushing us against the seat right? What about the air inside the plane, are there any difference in pressure due to inertia?
 
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Absolutely! The pressure decreases towards the front of the cabin & decreases towards the rear. You can observe the effect if you have a helium balloon on a string: When the aircraft accelerates, the balloon will drift towards the front of the cabin. You can see the same effect in an automobile. (Given that, which way will the balloon drift when an automobile makes a right turn?)
 
When you talk about you being "pushed against the seat", the plane is really pushing the seat into you. The moment the engines produce thrust, the plane (and by extension, the seat) is being accelerated forward and it's really pushing on you because you aren't moving due to your inertia. The sensation in the plane you are describing would a type of fictitious force, akin to feeling a "centrifugal force" inside of a car that's turn sharply.

I think the air would be compressed slightly when the plane is accelerating, but the compression would be negligible and only last for a very short amount of time.
 
At the moment, which you experience force which means acceleration, air would have stayed at the rear. However, when plane reaches constant speed air isgoing to experience pressure difference and net force, which means diffusion would happen. Thus pressure is almost constant when it is not accelerating so fast.