It really does have to do with air pressure. Bernoulli’s principal states that as the velocity of a fluid (air) increases, its internal pressure decreases. So, when you try to inhale air that is moving at a high velocity across your face, you are not dealing with the usual14.7 psi atmospheric pressure, it is much lower. Side note- that’s also how airplanes fly, wings are shaped so the air on top is moving faster than air on bottom, making the pressure below the wing higher than that above; aka “lift”.
Your diaphragm contracts and relaxes to let air in and out of your lungs. Ambient air pressure is what allows air to be driven into your lungs. According to the Bernoulli principle, air moving with speed has a decreased static pressure, so the faster the air is moving, the more your diaohragm has to work to inhale.
Your body’s respiratory system does not just suck air in and push air out, it also regulates the amount of flow on those two activities because your air filtration system is only efficient at certain flow rate, so if you inhale while air is being blown at you your body will try to limit the speed of air flow proportional to the speed of the air being blown at you by involuntarily resisting how fast you suck air and it will make you feel as if inhaling is hard.
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