My initial thought would be that without the column of air pushing on the surface, it would decrease until it reached some new equilibrium with the terminal velocity of the water bucket. And that feels to be like it might be true for water just poured out, but in a constrained bucket, the whole question just confuses me. Plus, maybe even my original thinking is wrong.
Thanks!
Having trouble parsing this. Can you repeat with different words, or maybe being more specific about what you’re asking?
I have a bucket of water at sea level, and the water is at 1 atm at its surface. I then drop the bucket from some height. Does the air pressure on the water’s surface change?
Yes.
However, I don’t know what happens once that bottle/bucket reach terminal velocity - I’d imagine the pressure returns.
The column of air model is useful for figuring out what the air pressure would be at a given altitude, but it’s not a good model for how air pressure actually works. The pressure is pretty much uniform on all surfaces at a given location, because all of the air is pressurized, not just the “column” over your object.
Ignoring small local changes attributable to aerodynamic effects of the falling bucket, the air pressure on the water would remain pretty constant. You could test this by dropping a bucket of boiling water. If there was a significant decrease in the effective air pressure on the water, this would decrease the boiling point of the water, and at least some of it would turn into steam at a greater than expected rate while falling.
That’s gravity, not air pressure.
It changes from the air pressure at the height you drop the bucket until it reaches the ground where air pressure is higher. Depending on the bucket shape there might be a slight pressure drop over the water but I think more likely the air over the bucket becomes turbulent.
When an object moves through a fluid - whether the fluid is a gas (like air), or a liquid (like water) - the pressure on the front of the object increases, and the pressure on the back of the object decreases.
So assuming your bucket is upright, you’d see a decrease in pressure on the free surface of the water.
It takes quite a bit of speed to develop a significant increase in pressure on the front of the object:
Somewhere out there, there’s a similar plot showing what pressure drop you achieve as air accelerates to try to fill the void behind the advancing bucket. It takes a lot of velocity to get a noteworthy drop: you’d have to get the bucket up to several hundred MPH in order to experience a ~50% reduction in pressure.
And the air resistance of the bucket - assuming it stays upright - means the water itself is not in free fall. Indeed, at terminal velocity it will be experiencing 1G. So the only effects would be the turbulence of the air in the open (top) of the bucket. Note that liquids are essentially uncompressible (unlike gases) which means the water will not expand appreciably just because the pressure is lower on the top of the water.
Your original thinking is wrong.
If you slowly hoist the bucket to 1 or 2 or 5 miles above sea level, the air pressure on the water surface will slowly decline because the air column above that altitude is less as you go up.
Likewise the air pressure pushing up on the bottom of the bucket and in on the sides; all of those will slowly decline as you hoist the water up. As @Horatius said.
Now drop the bucket from some altitude. The pressure due to the ambient atmosphere (IOW the static pressure) will increase with decreasing altitude on the way down exactly as it had decreased on the way up. Same exact movie, just played in reverse and more quickly.
The water will experience the static pressure changes described above. Plus the pressure contribution from aerodynamics as the bucket moves quickly through the atmosphere powered by gravity.
As @Machine_Elf said, there is a small amount of ram air pressure building up on the bottom of the quickly falling bucket and a corresponding reduction in air pressure in the turbulent wake above the bucket where the water surface sits. But by and large, those are rounding errors on the static pressure from the prior paragraph.
Okay everyone, I think I got all that, thanks for the clear explanations.
You won’t get out of it that easy, this is The 'Dope, my friend.
My I fear uninformed thought is that it matters what the value of “some height” is?
Pressure doesn’t really change much over several hundreds of meters.
In that height of a drop the process above, which is utilized to draft when cycling or racing cars, seems likely more significant?
Which would essentially “suck” water out of the somehow consistently vertical bucket, yes?
From the responses above I’m thinking, no. But not understanding yet.
That’s all been pretty much covered above. To re-iterate:
If the bucket of water was dropped from a relatively short height where the difference in atmospheric air pressure between the top and bottom of the drop wasn’t significant – and also the bucket wouldn’t be traveling fast enough for aerodynamic pressure to matter – the static air pressure on all sides of the bucket including the top surface of the water would remain exactly the same whether it was stationary or falling. That’s pretty much the end of the story.
No. But since the falling bucket would be weightless some of the water might tend to leave the bucket due to turbulent air currents, but that’s rather beside the point. Excluding dynamic forces, the bucket of water in free-fall would behave exactly as it would in a space station with zero gravity. Falling doesn’t “suck” anything out of it. The water is maintained in the bucket by gravity. Static atmospheric pressure has absolutely nothing to do with it and it remains constant regardless of how the bucket is moving.
You need to separate 3 things:
Once you do you can predict most behaviors.
For most stuff (except near terminal velocity) you can mostly ignore the forces of the air since it comes from all sides and cancels out.
Well.
I took one for science one time. I can tell you what happened.
I was standing by my kitchen island eating my raisin bran flakes with exactly 1/2 cup of skim milk.
In some weird coincidence a moth flitted around the hanging pendant light. Two Siamese cats decided it would be fun to jump up on the island and up to the light. There-by causing me to jump.
In a physics related confabulation, my bowl released from my grip. Went up about 2 feet and crashed down to the granite countertop. Smashing into a million shards.
I promptly noticed milk/flakes in my face, hair and shirt. No shards of glass.
The pressure was 100% released.
Except raisins were on the floor with the shards of glass. I presume their weight or maybe stickiness kept them with the bowl.
At about 3 minutes later the pressure was released on my mouth. And curse words emitted. Again 100%.
My conclusion: the liquid will fly out into your face. Wear eye protection.
And don’t eat cereal with cats stalking the room for flying insects.
That’s the part that I question?
I mean if I can cycle to a velocity that can cause a measurable, even noticeable impact of that aerodynamic drag and benefit of drafting and of being in a peloton, then it seems likely a bucket can accelerate over a few hundred meters to such a velocity. Well faster.
Drafting behind a moving object prevents the ram air effect. It doesn’t easily result in lower than ambient pressure behind a moving object. Turbulent air on average will maintain the same pressure as ambient air. The reduction in altitude is the only thing guaranteed to result in changing air pressure as the bucket falls, changing to higher pressure. I don’t think an ordinary bucket will be stable while falling so it may just hit the ground before most of it’s contents which are no longer constrained by the bucket.
I’m not really sure what you’re saying here. The OP asked how “the column of air pushing on the surface” would affect a bucket of water that was in free fall. The answer is clear: the question is about static air pressure, and the answer is that it wouldn’t affect it in any way, and furthermore, atmospheric pressure isn’t just “pushing on the surface”, it’s pushing in all directions, and it has nothing to do with how the object is moving (that last bit is “dynamic” pressure).
Of course other factors are involved, too, and the most significant one is that gravity ceases to be a factor while the bucket is in free fall, so you’ll likely get spillage out of the bucket due to random motions caused by air currents, but that has nothing to do either with static air pressure nor with dynamic air pressure due to velocity through the airstream.
I thank @Beckdawrek for contributing the results of her scientific experiment involving cats, moths, and a bowl of cereal! 
What it mainly illustrates is that rapid changes in the direction and magnitude of the velocity vector of a bowl of milk-bearing cereal is likely to lead to a big mess!
I think you meant “higher air pressure”. Other than that quibble IMO you’ve nailed the whole scenario