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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.
Best Answer
$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 .
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