Modeling of Water Flow

Procedure Continued

8. Now that we have chosen a parameter, if our model is correct it should match ALL of our experiments! Just like filling the tube with more water for the second test, we will need to "fill the model with more water". Adjust the Water Depth knob as closely as possible to the height of mark #3. Click Run to regenerate the graph.
9. Now check the graph. This time you want to see how long it takes from the start at mark #3 until the finish at mark #2, so slide the line until '1:Water depth' reads very close to the mark #2 height. According to the model, what time did this occur? Does this match the experiment?
   
   Something appears to be wrong with the model! Luckily it is not a very complicated model, so there are only a few things to adjust to try and make it work correctly.
   
   Two of the model inputs are the Water Depth and the Flow Parameter, and both have to match the experiment. Water Depth comes directly from a measurement, and has to be the same for both the experiment and the model. We are allowed to adjust the Flow Parameter but only once. If we change it to fit the second test, it will no longer work for the first test!
   
   The only thing left to change is the algorithm i.e. the way the model does the calculations. In this case, the problem lies in the formula we used to calculate the exit flow.
   
   When water pours from the tube, it takes the shape of a parabola (although one that changes as the water pours out). Scientists and mathematicians know that parabolas come from a relationship where one value is squared, (like A = πr²). In this case the height of the water [from potential energy, m g h] is proportional to the square of how fast it comes out [from kinetic energy, .5 m v²]. Or, if inverting the equation, the speed of the flow ('how fast it comes out') is proportional to the square root of the height.

10. Adjust the Exponent slider to a value greater than 1. Click the Run button to regenerate the graph. How does the exponent affect the shape of the graph?
11. The exponent that corresponds to a square root is 1/2 (or 0.5). Adjust the Exponent slider to 0.5.
12. Set the Water Depth input back to the height of mark #2 and click Run to regenerate the graph.
13. Click and drag on the graph to find the time value that matches the height at mark #1. Is this time value still correct?
    
    Because we changed our algorithm, our input parameter is no longer correct, so we must again make it fit with experiment. This process is called parameter tuning