Pressure in Bernoulli's equation

Question

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I've been reading an introductory physics textbook. It has a chapter on fluids which I'm finding quite confusing. Specifically, I don't understand the meaning of the pressure terms $P_0$ and $P_1$ in the the equation:

$P_0 + \frac{1}{2} \rho v_0^2 + \rho g h_0 = P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1$

I have been thinking about a particular problem which I have thought up (see image). The problem is as follows: Imaging a big lake 100m deep. At the bottom, there exists a small pipe which allows water to flow from the lake into a vacuum. I wish to find the velocity at which the water leaves through this small pipe. My idea to do this is to consider a fluid particle at the surface of the lake. This particle will experience atmospheric pressure $P_0 = P_{atm}$. Since the lake is large, $v_0 \approx 0$ and since the lake is 100m deep, $h_0 = 100$. Now consider a fluid particle in the pipe at the bottom of the lake. It would experience a pressure of $P_1 = 0$ since the pipe empties into a vacuum. We can set $h_1 = 0$ since I am free to choose the point from which I measure height. Now I can simply solve for $v_1$. Is this correct?

The reason I am unsure is that I don't understand how to choose the values of $P_0$ and $P_1$. What I would like to understand the nature of the pressure terms in Bernoulli's equation shown above. For example, why is/isn't $P_1$ equal to $100 \rho g $, which would be the pressure exerted on the liquid in the tap since it is at depth 100m. Or is this equation for calculating pressure at a particular depth only valid in fluid statics?

Finally, what is pressure (maybe this is too broad a question)? I believe it is a scalar since we can simply talk about the "Pressure" of a fluid at a particular depth, yet I use the equation $P = \frac{\mathbf{F}}{A}$ where $\mathbf{F}$ is a vector. Wouldn't this suggest that P is a vector?

Any answers to any of my questions would be much appreciated.

Answer 1

Your way is correct. About your doubt it's important to see the two points on the streamline where you want to apply the Bernoulli Equation. Taking point just inside and just outside the the pipe will matter a lot.

$P_1=0 $ when the second point you choose the point outside the pipe ,which is open to vacuum so pressure is zero. Whereas if you choose a point just inside the pipe at the bottom of the tank the pressure is $P_1=P_0+\rho g h$ .Also this case will give us the velocity of water at the bottom of tank and not the velocity of water leaving the pipe .

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Also pressure is a scalar and can be seen as $$P=\dfrac{\vec F.\vec A}{|\vec A|^2}$$ Where area vector is a vector pointing perpendicular to the arwa of the surface and $|A|$=area of surface