How Do I Solve for Zero Velocity Using the Bernoulli Equation?

[CollectiveRocker]

A large tank of water has a hose connected to it. The tank is sealed at the top and has compressed air between the water surface and the top. When the water height h has the value 3.5 m , the absolute pressure p of the compressed air is 4.20 * 10^5 Pa. Assume that the air above the water expands at cosntant temperature, and take the atmospheric pressure to be 1.0 * 10^5 Pa. When the h=3.5 m, velocity = 26.2 m/s. When h=3.0, v=16.1m/s. When h=2.0 m, v = 5.44 m/s. How do I solve for when the velocity equals 0? I realize that I have to rearrange the Bernouilli Equation. Do [(1/2)pv1^2] and [(1/2)pv2^2] cancel when this happens?

 

[Nate810]

Yes, when the velocity is 0, the terms [(1/2)pv1^2] and [(1/2)pv2^2] cancel. You can rearrange the Bernoulli equation as follows:h2 - h1 = (1/2)*(p1/ρg - p2/ρg)*(v1^2 - v2^2)where h1 and h2 are the initial and final water heights, respectively, p1 and p2 are the pressures of the compressed air in the tank at the initial and final heights, respectively, ρ is the density of the water, g is the acceleration due to gravity, and v1 and v2 are the velocities at the initial and final heights, respectively. In your case, set v1 = 26.2 m/s and v2 = 0 m/s. Solve for h2. This will give you the water height when the velocity is 0.

 

[wais]

To solve for when the velocity equals 0, you can rearrange the Bernoulli equation to solve for v, which is the velocity. When the velocity equals 0, the equation will look like this:

v = √(2gh + (p1-p2)/ρ)

Since v equals 0, we can remove it from the equation, leaving us with:

0 = √(2gh + (p1-p2)/ρ)

To solve for h, we can square both sides of the equation:

0 = 2gh + (p1-p2)/ρ

Next, we can isolate h on one side of the equation by subtracting (p1-p2)/ρ from both sides:

- (p1-p2)/ρ = 2gh

And finally, we can solve for h by dividing both sides by 2g:

h = -(p1-p2)/(2ρg)

So when the velocity equals 0, the water height will be equal to -(p1-p2)/(2ρg).

To answer your question about the terms [(1/2)pv1^2] and [(1/2)pv2^2], they do not cancel out when the velocity equals 0. These terms represent the kinetic energy of the water at two different points in the tank, and they will be equal to each other when the velocity is 0, but they will not cancel out.

I hope this helps you solve for when the velocity equals 0 in this problem. Keep in mind that the Bernoulli equation is only applicable in certain situations, so make sure to check if it is appropriate to use in your specific problem.