Analysis of depth vs y for coke bottle (quiz form with solution)

Give the following results from your analysis of the experiment done with the coke bottle:  

• Copy your table of depth vs. clock time, and make a quick sketch of your graph of depth y relative to hole vs. clock time t, labeling the greatest depth on the y axis and the greatest clock time observed on the x axis. 
• Give your corresponding estimate of the average velocity of the water surface over each measured time interval, and quickly sketch your graph of average velocity of surface vs. midpoint t.
• Give a table of the velocity of the exiting stream vs. t.
• Give your estimate of the ratio of container diameter to hole diameter.

Your analysis should result in a depth vs. clock time graph that decreases at a decreasing rate (i.e., decreasing and concave upward).

The speed v1 of the water surface is related to the speed of the exiting water by the ratio of the cross-sectional areas of the container and the tube through which water exits.

Our data will give us direct information about v1 and v2:

• A graph of exit velocity vs. clock time should consist of v vs. t points that fit close to a decreasing straight line, as incidated below.
• A graph of the magnitude of the velocity v1 of the water surface vs. t should also give us results that fit close to a decreasing straight line, as indicated.

Since v2 = v1 * ( A1 / A2) we expect that the slope of the v2 vs. t graph will be A1 / A2 times that of the | v1 | vs. t graph.

For the given container the cross-sectional diameters of container and tube are 6.8 cm and .32 cm, so that the ratio A1 / A2 of areas will be (6.8 / .32)^2 = 450, approx..

v2 vs. t and v1 vs. t results from one group indicate a v2 / v1 ratio around 500, with a margin of error of +-150.  This is consistent with the measured area ratio of 450.