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'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 [tex]y_{{s}}[/tex] and [tex]y_{{i}}[/tex] of the superior and inferior holes.'

Here is my work:

The equations of motion first, used in this situation:

[tex]x=v_{{x_{{s}}}}t_{{s}}[/tex] (1)

[tex]x=v_{{x_{{i}}}}t_{{i}}[/tex] (2)

[tex]0=1/2\,g{t_{{s}}}^{2}+y_{{s}}[/tex] (3)

[tex]0=1/2\,g{t_{{i}}}^{2}+y_{{i}}[/tex] (4)

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

[tex]x=\sqrt {2g \left( h-y_{{s}} \right) }t_{{s}}[/tex] (5)

[tex]x=\sqrt {2g \left( h-y_{{i}} \right) }t_{{i}}[/tex] (6)

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

[tex]0=1/4\,{\frac {{x}^{2}}{h-y_{{s}}}}+y_{{s}}[/tex] (7)

[tex]0=1/4\,{\frac {{x}^{2}}{h-y_{{i}}}}+y_{{i}}[/tex] (8)

Using equation (8), I find:

[tex]{x}^{2}=-4\,y_{{i}} \left( h-y_{{i}} \right)[/tex]

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

[tex]-{\frac {y_{{i}} \left( h-y_{{i}} \right) }{h-y_{{s}}}}+y_{{s}}[/tex]

[tex]-y_{{s}}h+{y_{{s}}}^{2}=-y_{{i}}h+{y_{{i}}}^{2}[/tex]

[tex]-y_{{s}}h+y_{{i}}h={y_{{i}}}^{2}-{y_{{s}}}^{2}[/tex]

[tex]h={\frac {{y_{{i}}}^{2}-{y_{{s}}}^{2}}{y_{{i}}-y_{{s}}}}[/tex]

[tex]h=y_{{i}}+y_{{s}}[/tex]

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. :uhh:

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.

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# Homework Help: Milk and fluid dynamics

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