An experiment was conducted to investigate the time it takes water to pour out of a can through a little hole and the amount of water in the can. To find the dependence on the size of the hole, five large cylindrical containers of water of the same size were emptied through relatively small circular openings of different diameter. To find the dependence on the amount of water the same containers were filled to different heights.
Each measurement was repeated several times using a handheld stopwatch to measure the times to the nearest 1/10th of a second. The averages of the times are recorded in the following table.
Height cmDiameter (cm)
30 15 8 4 11.5
97.4 sec 68.9 sec 50.3 sec 35.6 sec 17.8 sec2.0
54.8 sec 38.7 sec 28.3 sec 20.0 sec 10.0 sec3.0
24.3 sec 17.22 sec 12.6 sec 8.9 sec 4.4 sec4.0
13.7 sec 9.7 sec 7.1 sec 5.0 sec 2.5 sec5.0
8.8 sec 6.2 sec 4.5 sec 3.2 sec 1.6 sec
Find the relationship between the time and:
a) the height of the water in the can
b) the diameter of the hole
c) using the results of a) and b), formulate a general expression for the time y as a function of both the diameter and the height
d) Use the expression from c) to predict the time required to empty a can with a 3.5cm diameter to a height of 20cm
Knowledge of log-log and semi-log graphs..
The Attempt at a Solution
a) We plotted a graph using a constant diameter (8cm) of time vs. height.
Time on y axis and height on x axis. The graph appeared to be logarithmic, so we linearized the graph by log"ing each side. Our equation: y=.5logx+b
b) We basically did the same thing as question a, using a constant height (8cm) and plotting time vs. diameter. Our equation: y=-2logx+b
c) Our answer is y=kh^1/2d^-2 We are not exactly sure what our reasoning is
d) We aren't sure about c) so we haven't ventured here yet