How Does Water Velocity Change When Filling a Necked Cylindrical Container?

[BugMeNot_dude]

Homework Statement

Conceptual question. I have a cylindrical container with a hole at the bottom and a water level filled to just before a sloping neck towards the top (area at the top of the neck is smaller than before it slopes). How would the velocity of the water leaking through the bottom hole change if water is added so that it fills up to the neck?

Homework Equations

(P1)+1/2ρ(v1)^2+ρg(h1) = constant (P2)+1/2ρ(v2)^2+ρg(h2)
with the # in (variable#) indicating subscript.
1 indicates the top of the container, 2 indicates the hole near the bottom of the container.

The Attempt at a Solution

Besides velocity increasing due to v=sqrt(2gh) (which is not what I'm looking for), I don't really know what other effects there would be.
The only idea I have is that the velocity of the water level decreasing (v1) would no longer be negligible due to the decreased volume.
Pressure remains constant, so I don't see how changing area would affect anything, especially since force is directly related to it.

Any ideas? Thank you.EDIT: I've figured it out. Please close this thread.

 

[anathema]

I would like to offer some insights and suggestions to help answer this question. First, let's consider the Bernoulli's equation that you have mentioned:

(P1)+1/2ρ(v1)^2+ρg(h1) = constant (P2)+1/2ρ(v2)^2+ρg(h2)

This equation describes the relationship between pressure, velocity, and height of a fluid in a container. In this case, P1 and P2 represent the pressures at the top and bottom of the container, v1 and v2 represent the velocities at the top and bottom, and h1 and h2 represent the heights at the top and bottom.

Now, let's apply this equation to the situation described in the forum post. Initially, the container is filled with water up to just before the sloping neck. This means that the height at the top (h1) is larger than the height at the bottom (h2). As a result, the velocity at the top (v1) will be lower than the velocity at the bottom (v2). This is because the water at the top has a larger potential energy due to its higher position, and therefore has a lower kinetic energy (velocity). This is in line with what you have mentioned in your attempt at a solution.

Now, when water is added to fill up to the neck, the height at the top (h1) and the bottom (h2) will be the same. This means that the velocity at the top (v1) will be equal to the velocity at the bottom (v2). This is because the potential energy of the water at the top is now equal to the potential energy of the water at the bottom, resulting in equal kinetic energies (velocities).

In summary, as more water is added to the container, the velocity of the water leaking through the bottom hole will increase because the height at the top and bottom are now equal, resulting in equal velocities at the top and bottom according to Bernoulli's equation.

I hope this helps to answer the conceptual question. Keep up the good work in exploring and understanding scientific concepts!