Can you measure the speed of water coming out of a hose by its arc? 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?
 A: 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.
