# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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Lift force = — Xy/i + XXj

Newton’s second law in the i - and j-directions gives us

mX' = — SX — Xyf X'(0) =v0 , x(0) = 0

my" = —mg + XX — Sy y (0) =0, y(0) = 0

where m is the skier’s mass and S, X, and V0 are positive constants. Integration of each of these ODEs yields

mX — mv0 = —Sx — Xy

my = —mgt + Xx — S y

Divide by the mass to get the system IVP

X = —ax — by + V0 x(0) = 0

y = ~gt + bx — ay y(0) = 0

where a = S/m and b = X/m are the drag and lift coefficients, respectively.

When Newtonian drag and lift occur, S and X are not constants, so we can no longer integrate once to get X and /, and we must treat the original

second-order ODE differently:

X = v x(0) = 0

V = —Sv/m — Xw/m v(0) = vo

y = w y(0) = 0

w' = —g + Xv/m — Sw/m w(0) = 0

where v and w are the velocities in the i - and j-directions, respectively.

We have assumed that the bottom edge of the ski jump is horizontal, but everything can be modified to accommodate a tilt in the launch angle (see the chapter cover figure and Exploration 5.4, Problem 1).

Halliday, D., and Resnick, R., Physics, (1994: John Wiley & Sons, Inc.)

True, Ernest, “The flight of a ski jumper” in C-ODE-E, Spring 1993, pp. 5-8, http://www.math.hmc.edu/codee

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date______

Course/Section

Exploration 5.1. Dunk Tank

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r? J ? r* r-':r-Tr~~‘i, .r- — K

1. How big is the target?

Play the dunk tank game on Screen 1.6 of Module 5 and use various launch angles and speeds to help you determine the heights and diameters of the Einstein, Leibniz, and Newton targets, given that the ball has a 4-inch diameter.

2. One speed, two angle ranges for success.

Use the ODE Architect tool to find two quite different launch angles that will dunk Einstein if the launch speed is 40 ft/sec. Repeat for Leibniz and Newton.

86 Exploration 5.1

3. Launch angles and speeds that dunk Einstein.

Find the region in the 60 V0 plane for which the ball hits the target and dunks Einstein. Hint. Start with V0 = 40 ft/sec and determine the ranges for 60 using ODE Architect by playing the dunk tank game. Then repeat for other values of V0.

4. Solution formulas for the dunk tank model.

The position and velocity of the ball at time t is given by formula (2). Find a formula that relates the launch angle to the initial speed and the time T needed to hit the bull’s eye. If you had to choose between using your formula and using ODE Architect computer simulations to find winning combinations of speed and launch angle, which would you choose? Why?

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date______

Course/Section

Exploration 5.2. Longer to Rise or to Fall?

1. Throw a ball up in the air.

Do just that, and determine as best you can the time it takes to rise and to fall. You can use a whiffle ball for slower motion. Explain your results. (No computers here, and no math, either!)

2. Longer to rise or to fall in a vacuum?

What if there were no air to slow the ball down? Use ODE Architect to determine whether it takes the ball longer to rise or to fall. Try various initial speeds between 5 and 60 ft/sec. [Suggestion: Use the Sweep feature.]

88

Exploration 5.2

3. Longer to rise or to fall with viscous drag?

Suppose that air exerts a viscous drag force on a whiffle ball (a reasonable assumption). For various initial speeds, use the ODE Architect to determine whether it takes longer to rise or to fall. Does your answer depend on the initial speed? What physical explanation can you give for your results?

4. Longer to rise or to fall with /our own drag?

Repeat Problem 3, but make up several of your own formulas for the drag force. Include Newtonian drag as one case. This isn’t as outlandish an idea as it may seem, since the drag force depends very much on the nature of the moving body, e.g., rough or smooth surface, holes through the body, and so on. Discuss your results.

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date______

Course/Section

Exploration 5.3. Indiana Newton

1. Indiana Newton lands on the boxcar (no drag).

Indiana Newton jumps from a height h of 100 ft and intends to land on the boxcar of a train moving at a speed of 30 ft/sec. Assuming that there is no air resistance, use Screen 2.6 of Module 5 to find the time window of opportunity for jumping from the ledge.

2. Indiana Newton Jands on the boxcar (Newtonian drag).

Repeat Problem 1 but with Newtonian drag (coefficient k/m = 0.05 ft-1). Compare fall-times with the no-drag and also with the viscous-drag (k/m =

0.05 sec-1) cases. Find nonzero values of the coefficients so that Indiana Newton hits the train sooner with Newtonian drag than with viscous drag. How do the fall-times change as Indy’s jump height h varies?

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