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Water comes out of a horizontally stationed hose and creates an arc as it heads towards the ground. Can I determine the speed the water was traveling in when it exited the hose by the measuring the arc which is created? For example, let's say that I measure that the water has dropped 2 inches vertically when measuring 1 foot horizontally away from the nozzle - at what speed did it exit the nozzle?

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To a reasonable approximation, sure. At least if you wait until a steady state condition is attained. How close to you need to get? Is the flow uniform to that level? – dmckee♦ Dec 5 '12 at 19:56
This is something I was wondering about while taking a shower, not my thesis. So assume any simplifying assumptions... – ytoledano Dec 5 '12 at 20:00

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To first order you can ignore air resistance1 and treat it as a perfect ballistics problem. So the vertical deviation of the stream from it's initial straight line, $\Delta y$ tells you the time elapsed sine leaving the nozzle by $$t = \sqrt{\frac{2 \Delta y}{g}} .$$

The the distance along the initial stream direction to point from which the $y$ measurement was made is $\Delta x$ ad we have $$ v_i = \frac{\Delta x}{t} = \Delta x\sqrt{\frac{g}{2 \Delta y}} .$$

For problems like this the largest errors are likely to be the mechanics of the measurement and the non-uniform initial velocity of the stream rather than air resistance.

A common place to see this in action is at any "jumping jets" fountain. If you watch closely you will see that the initial part of any particular jet has a lower trajectory than the rest, and that the main body generally has a beautiful parabolic trajectory.

1 Because once the stream is established no air is being displaced and at "hose" or "shower" velocities there is little viscus friction in the boundary layers.

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Nice thinking in the footnote! Do you have a qualitative idea of how will surface tension affect this? I'm thinking it will tend to straighten the jet, and make it go slightly further. – Jaime Dec 5 '12 at 20:38
Not really sure--this isn't my field of expertise--but I think that surface tensions biggest contribution will be in holding the stream together. Now, if it did come apart the claim about no air being displaced would probably fail and we'd lose the near-zero friction resulting in the streaming slowing and falling below the ideal trajectory. – dmckee♦ Dec 5 '12 at 21:34
Actually, surface tension is the cause of it braking up in droplets. en.wikipedia.org/wiki/Plateau–Rayleigh_instability Also, air will continuously be accelerated by the jet. However, I think the ballistics approach is good enough here. – Bernhard Dec 6 '12 at 10:21
@Bernhard Thanks. Something to add to my reading. – dmckee♦ Dec 6 '12 at 15:55

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