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MATH 570

Mathematical Modeling

  • Assignment 1. (Due Tuesday, September 16)
    Problem Alpha. Propose a method by which Galileo might have been able to get the value 32 ft/sec^2 for the gravitational acceleration constant g.
    Problem Beta. Model the situation of a ball being released from rest and allowed to roll down an inclined plane that makes an angle theta with the ground.

  • Assignment 2. (Due Tuesday, September 30)
    Problem Gamma. Solve the differential equation for a falling raindrop subject to air resistance as in the Stokes model. Show that as t->Infinity, the velocity converges to what we called the terminal velocity.
    Problem Delta. Solve the velocity-squared model for a falling raindrop (by whatever method you prefer -- partial fractions, a sine substitution or a hyperbolic tangent substitution). Compute the time that the model predicts for a raindrop of diameter .02 feet to fall 1024 feet, and comment on your answer.
    Problem Epsilon. Test the predicted terminal velocities from both the Stokes model and the velocity squared model against enough of the experimental data provided in class to be able to make a reasonable statement about the applicability of each model.

  • Assignment 3. (Due Tuesday, October 6)
    Problem Zeta. Use the Maple code on page 19 of the course text to produce plots of the type given in Figure 2.2. What choices of the original problem parameters have been made in the code found in the text?
    Problem Eta. All parts of this problem refer to the boundary value problem (BVP) for the amplitude function, chi(xi), found in (2.16) and (2.17) in the course text.
    a. Complete the exact solution for chi(xi) begun in class.
    b. Use the Maple code found on page 23 of the course text to solve the BVP. Compare your answers to parts a and b.
    c. Use Maple to implement a large-J approximation scheme for the BVP. In particular, compute the first four approximations, chi_0, chi_1, chi_2 and chi_3. Plot these solutions together with the exact solution for J=100, J=10, J=1, and J=.1 (one plot, or set of plots, for each value of J), and comment on what you observe.
    Problem Theta. Interpret in words what is happening to the three flagploes whose amplitudes are graphed in Figure 2.3b in the course text. Use Maple to produce your own plot(s) along the lines of Figure 2.3b.
    Exercise 2.2 on pages 29, 30 of the course text.

  • Assignment 4. (Due Thursday, October 30)
    Problem Iota. a. Repeat the dimensional analysis of the terminal velocity of a raindrop neglecting the viscosity of the air.
    b. Repeat the dimensional analysis of the terminal velocity of a raindrop neglecting the viscosity of the air, but taking into account the density of the drop.
    c. Repeat the dimensional analysis of the terminal velocity of a raindrop including both the viscosity of the air and the density of the drop.
    d. Discuss your answers to parts a, b and c.
    Exercises 2.3-2.5 on pages 30-32 of the course text.

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    This page last modified 10/29/97