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Use ODE Architect to find the position of the ball at several different times t for several different initial velocities. Assume no air resistance and that the ball moves in a vertical line. What is the name for the shapes of the solution curves in the ty-plane? Does it take longer for the ball to rise or to fall? Show and explain the difference (if there is one!).
2. Hand-to-hand motion of the ball.
For a given initial speed v0, find the range of values of the angle 90 so that the ball goes from one hand to the other. Now increase the initial speed. What happens to the range of successful values of 90? Explain. [Suggestion: First take a look at Screen 3.5 (Experiment 2 in Module 2); then enter the equations in ODE Architect and vary d0 with fixed v0 to find the ranges. You may also want to take a look at Screens 1.2 and 1.3 in Module 5.]
40 Exploration 2.3
3. Raise your hand!
Suppose the juggler raises his catching hand one foot higher. Repeat Problem 2 in this setting.
4. Juggling two balls.
Construct model ODEs for tossing two balls, one after the other, from one hand to the other. Use ODE Architect to find the positions of both balls at time t.
Answer questions in the space provided, or on
attached sheets with carefully labeled graphs. A
notepad report using the Architect is OK, too.
Exploration 2.4. The Sky Diver
Second-order ODEs of the form y" = f(t, y, y) are to be solved in Explorations 2.3 and 2.4. Since ODE Architect only accepts first-order ODEs, we will replace /' = f by an equivalent pair of first-order ODEs. We do this by introducing v = y as another dependent variable:
y = v
v' = f(t, y,v)
1. Terminal speed of a falling body.
Use ODE Architect and determine the sky diver’s terminal speeds for several different values of the viscous damping coefficient (use m = 5 slugs and g = 32 ft/sec2). Is there any difference if the sky diver jumps at 25,000 feet instead of 13,500 feet? [Suggestion: After entering the ODE and solving, click on a Data tab in either of the two graphics windows and use approximate data you find there.]
2. Slow down!
If a sky diver can survive a free-fall jump only if she hits the ground at no more than 30 ft/sec, what values of the viscous drag coefficient k make this possible? Are these k-values realistic? (Use m = 5 slugs and g = 32 ft/sec2.)
A step function is one of the engineering functions. You can find them by going to ODE Architect and clicking on Help, Topic Search, and Engineering Functions.
A Modeling Challenge!
Let’s construct a model for a parachute that opens over a 3 second time span. The ODEs for this model are given on Screen 4.5 (Experiment 2), but we have to define k( t). Assume that the sky diver has a mass of 5 slugs and that she jumps from 13,500 ft. The parachute starts to open after 65 seconds of free fall and the damping coefficient changes linearly from kff = 0.86 slugs/ft to kp = 6.71 slugs/ft as the chute opens. In other words,
kff, t < 65
êff + bL±lL(t-65), 65 < f < 68
kp, t > 68
(a) Write an expression for k( t) using the properties of step functions. Hint:
1, 65 t 68
StepU 65) - Step(t, 68) = . otherwise
(b) Use ODE Architect to plot the sky diver's acceleration, velocity, and height vs. time, using your expression for k( t).
3 Some Cool ODEs
Key words See also
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A room heats up in the morning, and the air conditioner in the room starts its on-off cycles.
In this chapter, we'll use Newton's law of cooling to build mathematical models of a number of situations that involve the variation of temperature in a body with time. Some of our models involve ODEs that can be solved analytically; others will be solved numerically by ODE Architect. We'll compare the analytical solutions and the numerical results and see how both can be used to verify predictions made by the models.
Modeling; Newton's law of cooling (and warming); initial conditions; general solution; separation of variables; integrating factor; heat energy; air conditioning; melting