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90 Exploration 5.3
3. How Jong is the boxcar?
Use computer simulations of Indiana Newton jumping onto the boxcar and estimate the length of the boxcar.
4. Indiana Newton floats down.
First, explore Indiana’s fall-times in the viscous- and Newtonian-drag cases where the coefficient k/m has magnitudes ranging from 0 to 0.5. The units for the coefficient are sec-1 (viscous damping) and ft-1 (Newtonian damping). Then find a formula in terms of h, k/m, and t for Indiana’s position after he jumps: first in the viscous-drag case, then in the Newtonian-drag case. Suggestion: In the viscous case, first solve V = —32 — kv/m, v(0) = 0, and then integrate and use y(0) = h to get y(t). In the Newtonian-drag case proceed similarly but with V = —32 — kv2. (This one is hard!) Choose values of the parameters in each case, and compare the graphs of the height function y(t) from your formula with the graphs obtained by the ODE Architect. Any differences?
Answer questions in the space provided, or on
attached sheets with carefully labeled graphs. A
notepad report using the Architect is OK, too.
Exploration 5.4. Ski Jumping
1. Tilt the edge of the ski jump upward.
Use the first Things-to-Think-About on Screen 3.7 of Module 5 to see what happens to the ski jumper’s path if the edge of the ski jump structure is tilted upward at 0.524 radians (about 30°). Set a = 0.01 sec-1, and vo = 85 ft/sec, and sweep on the lift coefficient b from 0 to 1.0 in 20 steps. Compare your graphs of the jumper’s path with the chapter cover figure. Then animate your graphs. Now fix b at the value 0.02 sec-1 and sweep on 6 (in radians) to see the effect of the tilt angle on the jumper’s path. Explain your results.
The second Things-to-Think-About on Screen 3.7 of Module 5 asks you to use the ODE Architect to estimate the smallest value of the viscous damping coefficient b that will allow the ski jumper to loop-the-loop. If a = 0.01 sec-1 and V0 = 85 ft/sec, estimate that value. Then increase b by increments from that value upward all the way to the unrealistic value of 5.0 sec-1 and describe what you see.
You need to know about matrices and eigenvalues to complete this part. See also Chapter 6.
Explain why the eigenvalues of this matrix are complex conjugates with negative real parts if a and b are any positive real numbers. Explain why you are more likely to see loop-the-loops if a is small and b is large. Do some simulations with the ODE Architect for various values of a and b that support your explanation. If you plot a loop-the-loop path over a long enough time interval, you will see no loops at all near the end of the interval. Any explanation?
Complex eigenvalues and loop-the-loops.
The system matrix of the viscous drag/lift model for ski jumping is
—a —b ba
4. Newton on skis.
The fourth Things-to-Think-About on Screen 3.7 of Module 5 puts Indiana Newton on skis with Newtonian drag (of course!). This situation takes you to the expert solver in the ODE Architect, where you are asked to explore every scenario you can think of and to explain what you see in the graphs.
First-Order Linear Systems
Oscillating displacements x\ (t) and x2(t) of two coupled springs play off against each other.
This chapter outlines some of the main facts concerning systems of first-order linear ODEs, especially those with constant coefficients. You'll have the opportunity to work with physical problems that have two or more dependent variables. Such problems can be modeled using systems of differential equations, which can always be written as systems of first-order equations, as can higher-order differential equations. The eigenvalues and eigenvectors of a matrix of coefficients help us understand the behavior of solutions of these systems.
Linear systems; pizza and video; coupled springs; connected tanks; linearized double pendulum; matrix; system matrix; Jacobian matrix; component; component plot; phase space; phase plane; phase portrait; equilibrium point; eigenvalue; eigenvector; saddle point; node; spiral; center; source; sink
Chapter 5 for definitions of vector mathematics.
Many applications involve a single independent variable (usually time) and two or more dependent variables. Some examples of dependent variables are:
the concentrations of a chemical in organs of the body the voltage drops across the elements of an electrical network the populations of several interacting species the profits of businesses in a mall
Applications with more than one dependent variable lead naturally to systems of ordinary differential equations. Such systems, as well as higher-order ODEs, can be rewritten as systems of first-order ODEs.
How to convert a Here’s how to reduce a second-order ODE to a system of first-order ODEs
second-order ODE t° a system of (see also Chapter 4). Let’s look at the the second-order ODE