FLOWLAW BRIEF ABSTRACT Dry lab data from an experiment in fluid flow using Torricelli's Law form the basis of this modeling activity. GENERAL INFORMATION FileName: FLOWLAW Full title: Torricelli's Fluid Law: A Dry Lab Approach Last Revision Date: 27 May 1996. Developer: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@Rose-Hulman.Edu. FAX: 812-877-3198. . Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. STATEMENT OF PROBLEM Sue and Bill have tried to bluff their physics lab on Torricelli's Law in their physics of fluids lab. In fact they are actually trying to dry lab it without performing any experiments. The experiment uses a cylinder of height 2 m filled with water with a sequence of plugged holes at a depth h where (h = .2, .4 , .6, .8, 1.0, 1.2) m from the top. For each depth h the cylinder is filled to the top (2 m), the plug at the depth h is pulled and the horizontal distance (maximum) x from the base at which the water hits is noted. Torricelli's Theorem says that the velocity of water emerging from the holes is proportional to the square root of the depth of water to the hole - say with constant of proportionality .2 for this configuration. The experiment asks them to see if there is a relationship between the depth h and the distance x for that height. Sue and Bill propose that the relationship between h and x is linear (increasing) for h going from 0 to 1 and decreasing linearly for h going from 1 to 2. They propose this without defense, hoping the professor will give partial credit. Write up a refutation of their conclusions which you post as a model solution. KEYWORDS Toricelli's Law, two dimensional projectile motion. TEACHER NOTES ISSUES RELATED TO THE PROBLEM Prerequisites Knowledge of two dimensional projectile motion. If the problem will invoke derivation of Toricelli's Law then more physics is neede. Time allotment - time management This problem could be a lab (really dry!) in class and otherwise could be a one night homework activity. Expectations Students will refute the postulated write-up using the velocity information offered and equations of projectile motion. Future payoffs Students will have seen Toricelli's Law for later work in fluids. Extensions Work on the derivation of Toricelli's Law from Conservation Principles. References and Sources Source idea: Modeling with Projectiles, Derek Hart and Tony Croft, John Wiley & Sons: New York. 1988. p. 28. Evangelestia Torricelli (1608-1647) was an Italian physicist and mathematician who also served Galileo as a secretary. He investigated the concepts of atmospheric pressure and vacuums and he invented the barometer. Torricelli was interested in fluids and their flow rates. He studied the following situation. Consider a cylindrical container (e.g., tin can open at top and closed at bottom) containing water. A small hole has been placed in the side of the can and the can is resting at the edge of the surface of a platform (on a table) with the hole in the can over the edge of the platform. Torricelli was interested in the rate at which the water would flow out of the hole both as a function of height and time. Equivalently he was interested in the change in volume of the water in the cylinder per unit time or the change in the height, h, (for a fixed radius, r, cylinder) per unit time, i.e. he was interested in dh/dt. Certainly one could reason that with more water in the cylinder, hence more weight on the water above the hole, there would be more force pressing down, thus causing more force on the water going out. This would mean there is more water per unit time exiting the hole when h is big than when h is small. Torricelli derived a differential equation for h(t) using Conservation of Energy arguments and found a relationship between depth of water and flow rate (and thus velocity of the escaping water). It was not linear!!! POSSIBLE SOLUTION We identify variables and functions of interest. h = depth of water from top of cylinder of water H = height of cylinder itself (here H = 2 m) a = constant of proportionality v is the velocity as a function of depth h. v[h_] = a Sqrt[h]; x and y are the respective horizontal and vertical position relative to the ground as a function of time, t, and depth of water from top of cylinder of water, h. x[t_,h_] = v[h] t; y[t_,h_] = - 1/2 9.8 t^2 + 0 t + (H-h); We determine when the water first hits the ground by setting the y coordinate equal to 0. sol = Solve[y[t,h]==0,t]; Next we determine the x coordinate of the water for the time it strikes the ground. xdist[h_] = x[t,h]/.sol[[2]] 0.142857 a Sqrt[h] Sqrt[-10. h + 10. H] Further we plot the horizontal distance x as a function of the depth h with two given parameters a = .2 and H = 2 m. xsol[h_] = xdist[h]/.{a->.2,H->2}; xplot = Plot[xsol[h],{h,0,2}, AxesLabel->{"Depth-h","MaxRange-x"}] -Graphics- And from this plot we can see that the maximum range x is NOT linear function of depth, but rather an arc shape, indeed it is from above a square root of a quadratic function and thus the students are in error. ISSUES IN SOLUTION Puts projectile motion in context and applies it to refute one theory based on Torricelli's Law. Students need to address the issue of when a projectile hits the ground.