Height of Milk in a Carton with Two Pierced Holes

[TeTeC]

Hello everyone.

'A carton of milk is placed on a table. It has 2 holes pierced through a vertical side. At one point in time, the 2 jets of milk flowing out are reaching the table a the same point. Determine the height of milk in the carton as a function of the respective heights $$y_{{s}}$$ and $$y_{{i}}$$ of the superior and inferior holes.'

Here is my work:

The equations of motion first, used in this situation:
$$x=v_{{x_{{s}}}}t_{{s}}$$ (1)
$$x=v_{{x_{{i}}}}t_{{i}}$$ (2)
$$0=1/2\,g{t_{{s}}}^{2}+y_{{s}}$$ (3)
$$0=1/2\,g{t_{{i}}}^{2}+y_{{i}}$$ (4)

Where $$h$$ is the height of milk in the carton, assuming that Torricelli's Theorem can be used here, I have in equations (1) and (2):

$$x=\sqrt {2g \left( h-y_{{s}} \right) }t_{{s}}$$ (5)
$$x=\sqrt {2g \left( h-y_{{i}} \right) }t_{{i}}$$ (6)

Then, when replacing (5) and (6) in equations (3) and (4):

$$0=1/4\,{\frac {{x}^{2}}{h-y_{{s}}}}+y_{{s}}$$ (7)
$$0=1/4\,{\frac {{x}^{2}}{h-y_{{i}}}}+y_{{i}}$$ (8)

Using equation (8), I find:

$${x}^{2}=-4\,y_{{i}} \left( h-y_{{i}} \right)$$

Then I replace in equation (7) and follows a few lines of development:

$$-{\frac {y_{{i}} \left( h-y_{{i}} \right) }{h-y_{{s}}}}+y_{{s}}$$
$$-y_{{s}}h+{y_{{s}}}^{2}=-y_{{i}}h+{y_{{i}}}^{2}$$
$$-y_{{s}}h+y_{{i}}h={y_{{i}}}^{2}-{y_{{s}}}^{2}$$
$$h={\frac {{y_{{i}}}^{2}-{y_{{s}}}^{2}}{y_{{i}}-y_{{s}}}}$$
$$h=y_{{i}}+y_{{s}}$$

Actually, I don't have obvious problems with this exercice, I just find the result quite amazing, for this is really a simple answer... Maybe I'm just unable to deal with such easy answers.

The question is : Is this seems to be right?

Thanks a lot!

P.S: the wording is translated from French... Excuse me for any misunderstanding or English mistake.

 

[OlderDan]

The question is : Is this seems to be right?

Assuming you are satisfied that the algebra is correct, think about what must happen to h relative to the height of the upper (superior) hole as the height of the lower (inferior) hole goes to zero. What about other changes in the heights of the two holes, such as when they get very close together? Do the corresponding changes in h seem reasonable based on how you expect x to vary for each hole as h varies? Does your result suggest that h ever changes in a way that seems unrealistic? If the algebra looks correct, and the result does not imply anything unreasonable, believe it.

 

[geosonel]

$$h=y_{{i}}+y_{{s}}$$

i agree! this result is very interesting. (i checked your work, and it seems correct.)

 

[TeTeC]

Thanks for the answers.

OlderDan, I've been trying to answer your questions to myself, and I've found nothing paradoxal... So I believe it. :shy:

As the algebra seems correct, the best thing I could do is trying to experimentally check the result... A bottle of water should give me something similar.