Basic buoyancy question If I have a cup of water filled with air at the bottom of a pool, then when the cup is "upside down" the air doesn't leave because the water pressure is pushing it up against the top of the container.
But then when I flip the cup, such that it is no longer upside down, the air rushes upwards. But why on earth does this happen? I mean the water's pressure is pushing this air downwards against the bottom of the container. If I were to draw an FBD which force is causing this air bubble to rush upwards?
Thanks
 A: I think I see your question, if the surface of the air bubble were perfectly flat, and the air + cup didn't float, then the surface would have an equal pressure across it and it would not move. The system, however, is in an unstable equilibrium, the slightest perturbation will cause the bubble to rise out of the cup.
Consider this, since the pressure of the water is proportional to the height of the water above it, a bump up in the surface of the bubble would have less water pressure above it. The air pressure is essentially constant pressure throughout the bubble, so it would push into the bump and expand it. The bigger it gets the more the air can expand it. The water in the areas around the bump move down, increasing the pressure and so increasing the rate that they move down. 
In addition, the water adheres to the cup walls, creating a meniscus along the edge of the bubble, drawing the water down and starting the rest of the air bubble to rise.
A: When the cup is tilted up, the water wants to flow into the cup. That is what water does - it attempts to flow downstream. In doing so it displaces the air. Now the air experiences the force of the water (pressure below bubble > pressure above)
A: Like you said, the water is pushing UP. It is always pushing the air up (it is the definition of buoyancy). Therefore, when you flip the cup, the buoyancy force continues to push the air upwards. Before you flipped the cup, the cup is exerting the downward force that counteracts the buoyancy force. 
Therefore in the FBD, the buoyancy force (pointed upwards) is what causes the air bubble to rush upwards. There is also a downward force of gravity, but it is negligibly small compared to the buoyancy force.
A: It is due to Archimedes' principle. The total force on the 'air bubble' under the water is the force of gravity on it (downwards) plus the force of gravity on the water that the 'air bubble' displaces. Since the same volume of water that replaces the air is more massive (because of higher density) there is a net upward force on the bubble and causes it to move up. The actual equations of motion of the bubbles in fluids are quite complicated to calculate however.  
Lastly when the cup is upside down as you wrote, the air bubble stays inside because the normal force provided by the bottom of the cup.
