Your challenge for this laboratory period will be to design and execute an experiment that will demonstrate that the relationship between pressure and depth in a static fluid is given by Bernoulli's equation. You will be expected to pay particular attention to the problems of accuracy and precision. The following information is presented to give you some guidance, on possible alternatives, but part of your grade will be based on your experimental design.

You might consider a container with three holes in the side. The pressure of the water at any point in the liquid is given by the equation

where P1 = atmospheric pressure, r = the density of the water, g = the acceleration due to gravity, & h = the depth of the water at the point being considered. According to this then, we should expect the pressure to be greatest at the lowest hole and least at the highest hole. If we then fill the container with water, we would expect to observe the water shooting furthest out of the lowest hole & going less far out of the upper holes.

According to Bernoulli's equation, for a steady, irrotational flow, the speed, pressure, and the elevation of a incompressible, nonviscous fluid are related by the equation:

P + rv² + rgy = constant

Or in other words

P1 + rv1² + rgy1 = P2 +rv2² + rgy2

Since our fluid will be at atmospheric pressure, the P's are the same & hence cancel leaving:

rv1² + rgy1 = rv2² + rgy2

The density of the water is the same throughout so r cancels out leaving

v1² + gy1 = v2² + gy2

If we now solve this equation for v22, we obtain

Now if we define y1 - y2 as h we get

At the surface v1 is very slow - to the point of being negligible, therefore

v2 = 

If we now consider the water streams using the kinemetric equations, we see in the x direction

vi = vf = = 

And in the y direction

v0 = 0 & a = g

so s = at²

t = 

Assume that the density of water is 1.000 X 10³& that atmospheric pressure is

1.01 X 10⁵ Pa. Set up the 3 holed container & fill with water. Keep adding water to the top and measure the horizontal (d) & vertical (s) distances the stream goes. Also measure the distance of the hole from the water level (h). Determine the v of each of the 3 streams both ways & compare using a % error calculation. Determine the pressure at each of the 3 levels & relate it to the velocity at that point. From which hole does the water shoot out the farthest? Do you see any correlation between the distances from the other two holes? Why? Be sure to address these points in the conclusion section.