[Agent_4573]

ok I was wrong with my first equation but let me correct myself. Let me start with this

Obviously the air does not become denser as it compresses

I'm not understanding this because if you compress air you make it more dense. Also if you compress a gas it heats up, it doesn't cool down. Think about a nitrous bottle. When you shoot the juice you decompress the bottle by letting liquid nitrogen escape. This is what makes the bottle so cold.

Now as for the pressure being dictated by velocity, I was originally wrong with my equation. If you look at the equation of continuity and bernoulli's equation, you get two things, a1v1=a1v1 and that work is equal to the difference of two pressures time the volume of liquid(air) to pass over a section. putting these two equations together and substituting back in, ,you get pressure change*volume=.5*mass*velocity at point 2 squared + .5*mass*velocity at point 1 squared. This means that velocity and pressure are related.

Think of it this way. You have a garden hose. You put your thumb over the end of the hose and the water squirts out faster. You decreased the area of the end of the hose and that made the water move faster. But in order for the water to move faster, you must increase the pressure. Thats why the more of the hose end you cover, the harder it is to keep your thumb over the hose. if you think im wrong doublecheck me in "Serway and Beichner's Physics for scientist and engineers" volume 1 fifth edition pages 470-472