Homework #4
Instructions. This is due on February 28
at the beginning of class. It is worth 20 points. Late papers lose 3 points per
day late. You must work by yourself, but feel free to consult with me. Explain
what you did and why you did it: good explanations are essential! Your paper
should have an introduction, a main body and a conclusion. Start early, ask
questions and do well.
Introduction. In this assignment, you will use the ODE Architect
software that came with the book to explore a Winter Olympics event, ski
jumping.
Problems. Start by running
“Multimedia ODE Architect” and choosing “Models of Motion” and then “Ski
Jumping.” The opening screen has a demonstration with narration. Listen to it
and use the information in it and other sources to briefly explain how a jumper
changes the lift coefficient. In particular, describe body and ski position for
a good jump. Then go to the next page.
On the new screen, click on each component of the
model. In the first two assignments, you explored the difference between linear
and quadratic drag. Which model is being used here? Also, in which direction is
the drag force? In which direction is the lift force? Hit the Solve button. How
far does this jump go? Go to the next page.
Start on this new page by clicking Solve to see how
far the jump goes. Then increase the value of the initial velocity by 5 ft/s
and see how far the jump goes. Then reduce it 10 ft/s (down to 80) and see how
far the jump goes. In general, approximately how many feet does an increase of
1 ft/s add to the jump? Use this estimate to predict the initial velocity to
reach 300 feet. Check whether your answer is correct, and adjust the initial
velocity if necessary. Go to the next page.
Start on this new page by clicking Solve to see how
far the jump goes. Then increase the value of a to first 0.2, then 0.3 and record the distances. How small does
the drag coefficient have to be to reach the flag? Go to the next page.
Start on this new page by clicking Solve to see how
far the jump goes. Then increase the value of b to first 0.02 then 0.03 and record the distances. What happens if
b = 0.1? Is it true that larger b-values always produce larger
distances? What happens if b = 0.8?
Go to the next page.
On this last
page, explain the difference between “Newtonian drag and lift” and the model
used previously. Click the Solve button to see how far the jump goes. Briefly
describe the shape of the skier’s trajectory and how it differs from the shape
produced by the previous model. Find the initial velocity needed to reach the
flag. If the skier can increase the lift coefficient to 0.025, how much can the
velocity be reduced and still reach the flag? Does this seem very realistic to
you? Go to the next page.
On the page of Things to Think About, notice that
there is an option (lower right) to explore Newtonian drag and linear lift. In
many situations, this is the most accurate model. Exit the program.