# Bernoulli Equation and Velocities

1. Jan 25, 2017

### XanMan

1. The problem statement, all variables and given/known data

A cubic wine box of dimensional length $h$ has a small tap at an angle at the bottom. When the box is full and is lying on a horizontal plane with the tap open, the wine comes out with a speed $v_0$.

i) What is the speed of the wine if the box is half empty? (Neglect the speed of the liquid at the top of the box.)
ii) What is the speed of the wine if the box is tilted by 45 degrees? (See attached figure)

[Assume the pressure at the top and bottom is equal]

2. Relevant equations

Bernoulli Equation: $\cfrac{p}{\rho} + \cfrac{v^2}{2} + \phi =$ const.

where $\phi$ is the potential energy for a unit mass as a function of the height $z$.

It follows from the Bernoulli Equation that is $p_0$ is constant at the top and bottom, and the initial height is $h$, we get: $$v_0 = \sqrt{2gh}$$

3. The attempt at a solution

i) By Bernoulli Equation, we get:

$$\cfrac{p_0}{\rho} + \cfrac{0}{2} + \cfrac{gh}{2} = \cfrac{p_0}{\rho} + \cfrac{v_1^2}{2} + 0$$
(since at the bottom $z = 0$, and if I understand right, the velocity of the liquid at $h/2$ is 0 (?)).

Simplifying, we get:

$$\cfrac{gh}{2} = \cfrac{v_1^2}{2}$$

Answer: $$v_1 = \sqrt{gh}$$

ii) I am not quite sure about this part of the question. I tried using Pythagoras' Theorem to find the height of the liquid in terms of $h$, and got $\cfrac{\sqrt{2}h}{2}$.

Following a similar procedure as in i), I got the following result, which I think is incorrect, and would like your help on it (thanks!):

Answer: $$v_2 = \sqrt{\sqrt{2}gh}$$

Note: I am unsure where to use $v_0$ in the problem, or the fact that the tap is "at an angle"!

2. Jan 25, 2017

### Staff: Mentor

Looks good to me! But I assume they want the answers in terms of $v_0$, not $h$. (You can convert between the two.)

The tap being at an angle should not matter.

3. Jan 25, 2017

### XanMan

Ah, how silly of me...I neglected to use $v_0 = \sqrt{2gh}$! Thanks DocAl : - )

4. Jan 26, 2017

### akshay86

sir I have a doubt why component of gravitational force doesn't matter here when tap is at an angle
considering a fluid part it has 2 forces on it 1 by the pressure or weight created by fluid part above it and gravitational force on it.
please coeerct me if I am wrong

5. Jan 26, 2017

### XanMan

Where do forces come in however?

6. Jan 26, 2017

### XanMan

Also, why does the angle of the tap not matter?

7. Jan 26, 2017

### Staff: Mentor

The effect of gravity is included in the potential energy term.

All that matters is the height difference (review the derivation). Of course, the subsequent motion of the fluid does depend on whether it is sent out at an angle (like any other projectile).

8. Jan 26, 2017

### XanMan

Yes, that's what I was thinking as well - it's the velocity *just* as it is leaving the pipe. Sorry for all the questions lately, but this was in my first university physics exam and I'm a bit scared about it to be honest! Never actually properly had physics before - quite a challenge but I'm enjoying it! :-)

9. Jan 26, 2017

### Staff: Mentor

Exactly.

Never apologize for asking questions! I'm glad you're enjoying your physics journey and I'm sure you are up to the challenge.

10. Jan 26, 2017

### XanMan

Doc Al, did anyone tell you how awesome you are? Thanks for the support! :D