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Air compression via acceleration?

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When we are sitting in our cars, and the car begins accelerating we are "pushed back" due to acceleration. Or if you have a ball in an empty vehicle, when it starts accelerating, the ball will roll to the back of the car.

I was wondering, is this also true in air? If there is an air-tight vehicle that undergoes massive acceleration will the air be compressed to the back of the vehicle? How much acceleration is needed to compress the air?

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