# Flash Illustrations

Each of the following was a modeling assignment for a second semester calculus course. The assigment is designed to require students to recognize common shapes and match them with known functions. The assignment is given after the topics using parametric form have been studied.

The assignment has two section. First the students must find a parametric model for the action described. This is often broken into multiple parts which they put together in a foreshadowing of vector techniques. The second is to use their model to find useful parameters and/or calculate derivatives. Note that the illustrations below are answer key versions: the yellow line tracing the curve is not provided for the students. They are required to use their own model. Note that the students have Mathematica available for graphing their model equations.

Traditionally textbooks have asked parametric model questions such as the cycloid and Witch of Agnesi by giving verbal descriptions. This does not match reality in which motion based problems are typically seen not read. Further my students seem almost incapable of transferring words into images. This skill is tackled through other assignments. This modeling project strips away the interpretation aspect allowing me to develop and assess solely their ability to match functions to motions.

Please also note, in an attempt to teach students that solutions are slowly and iteratively developed in life, projects involve revisions. This means that if a student's model has flaws, which would effect the second half of the project, they can submit, receive feedback, improve, and resubmit for substantial credit.

## Swinging Ferris Wheel

Guido is at the fair and decides to ride on a ferris wheel. Being an adventurous sort he begins swinging his gondola back and forth. Use the illustration to understand the description.

• Find a model for the position of the bottom, middle of Guido's gondola. The model should include as variables the rotation speed of the wheel (given as times per minute), the maximum angle Guido swings, and the speed of Guido's swinging (in terms of cycles per minute).
• If Guido swings quickly enough the resulting curve ``wiggle'' left and right as it moves around. Find a pair of wheel and swinging speeds that do not have any wiggling.
• If a bird sits on the wheel and slips, in which direction would it be thrown into flight?
• If a butterfly sits on the bottom (outside) of the gondola and slips, in which direction would it be thrown into flight?

## Helicopter Rescue

Guido did not read the Turnagain Arm tide tables and found himself stranded on a rock as the tide came back into the arm. A rescue helicopter was sent to pick him up. Unable to land, they let down a rope for Guido to grab. After rising the helicopter began moving forward. Guido, not this wisest person on earth, began swinging back and forth on the rope.

Find a model for the position of the end of the rope. The model should include as variables the forward speed of the helicopter, the length of the rope, and the speed of Guido's swinging (in terms of seconds per radian). Use the illustration to understand the description.

• Find a representation for the location of the point to which the rope is attached.
• Find a representation for Guido's position.
• Find conditions under which Guido's path does not cross itself.
• Find conditions under which Guido never hangs over the same spot twice.

## Helicopter Flight

A helicoptor is flying in a straight line at v miles per hour. The rotor, which is s feet long, is rotating at a rate of r rotations per minute. Use the illustration to understand the description.

• Find a representation for the location of the center of the rotors.
• Find a representation for the tip of the rotor.
• Find conditions under which the path of the rotor tip does not cross itself, i.e., your graph will not cross itself.