Chapter 14.4, Problem 50E

Variable speed on a circle Consider a particle that moves in a plane according to the equations x = sin t² and y = cos t² with a starting position (0, 1) at t = 0

1. a. Describe the path of the particle, including the time required to return to the starting position.
2. b. What is the length of the path in part (a)?
3. c. Describe how the motion of this particle differs from the motion described by the equations x = sin t and y = cos t.
4. d. Consider the motion described by x = sin tⁿ and y = cos tⁿ, where n is a positive integer. Describe the path of the particle, including the time required to return to the starting position.
5. e. What is the length of the path in part (d) for any positive integer n?
6. f. If you were watching a race on a circular path between two runners, one moving according to x = sin t and y = cos t and one according to x = sin t² and y = cos t², who would win and when would one runner pass the other?