We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water. We offer several sources of simulations on YouTube and at this site from which we collect data and ask students to verify their model through parameter estimation.
SPANISH LANGUAGE VERSION We have placed in Supporting Docs both Student and Teacher Version (LaTeX and PDF Versions) with a Spanish LaTeX Class file, SIMIODE-SPANISH.cls. Names will be x-y-S-Title-StudentVersion-Spanish and x-y--T-Title-TeacherVersion-Spanish.
Joint Mathematics Meeting 2020 AMS SPecial Session talk PowerPoint talk by John McClain, Lecturer, University of New Hampshire, on 18 January 2020 in Denver CO USA is included in Supplemental Docs as well.
We model the height of a falling column of water in a right circular cylinder (radius = 4.17 cm) emptying through a small hole on the side of the cylinder. Indeed, in all the videos below where are referred to in this Modeling Scenario the cylinder is a 2 liter soda pop container with a metric ruler affixed to it for reading height in cm. The hole through which the water exits varies a diameter = 13/64 " = 0.257969 cm) in the bottom of the column.
We list videos which can be used for collecting data with appropriate information on the configuration. The YouTube version streams faster than the SIMIODE download version. Best viewed in 720p or 1080p HD version on YouTube.
(1) This is an mp4 video in which the exit hole for the water is 5/64 in.
(2) This is an mp4 video in which the exit hole for the water is 7/64 in.
(3) This is an mp4 video in which the exit hole for the water is 9/64 in.
(4) This is an mp4 video in which the exit hole for the water is 11/64 in.
(5) This is an mp4 video in which the exit hole for the water is 13/64 in.
Addenda: On 27 February 2019 we added the file 1-015-T-Mma-Torricelli-TeacherVersion-Addenda.nb and its pdf version in which we offer yet another approach to estimating the parameter in the model. This new approach uses symmetric differences to approximate the derivative so you do not have to actually solve the differential equation and then estimates the parameter. We compare the two approaches.
Addenda: John McClain, University of New Hampshire, Durham NH USA gave a fine talk at the Joint Mathematics Meetings 2020, Denver CO USA, in January 2020, in an AMS Special Session, Wall to Wall Modeling Activities in Differential Equations Courses. The title of the talk was "Modeling the Draining of a Bottle" and the talks' focus was using this Modeling Scenario. There are issues raised here that users might wish to incorporate in their course use of this Modeling Scenario. We include the PowerPoint and PDF version of his talk in the Supporting Docs under the name 0930-McClain-BottleDrain.
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