Pre-AP PreCalculus Unit 11 Day10 Simulating Motion and Parametric Application

Example 1: Application---Ferris Wheel Problem

Eric is standing on the ground 75 feet from the base of the Ferris wheel that has a radius of 20 feet and the lowest point is 3 feet above the ground. Janice is on the Ferris wheel, which makes one revolution counter-clockwise in 12 second. At the instant she is at point A, Eric throws the ball to her. The release height is at the bottom of the Ferris Wheel and he throws the ball at an angle of 58 degrees with a velocity of 60 feet per second.

Assume that g= 32 ft/sec and neglect air resistance.

A is directly right of center Picture not to scale

75 feet

a. Write the parametric equations for the path of the ball.

b. Write the parametric equations for the path of the Ferris wheel.

c. Graph the parametric equations simultaneously on your calculator to simulate the motion.

d. Will the ball come close enough for Janice to catch?

Example 2. Astronomers detect a meteor approaching Earth. They determine the meteor’s path is represented by and

where x and y are in thousand miles. The Earth’s center is at the origin.

a. Find the rectangular equation and sketch without a calculator. You do not need to note direction at this time.

b. Using the calculator graph the parametric equations using and

What do you notice?

Change to

However, the ___________ ____________does not apply to the situation because the

meteor is __________________Earth with center at (0, 0).

c. Using the calculator, find the x and y coordinates

d. How far is the meteor from the center of the Earth at ?

e. At what time is the meteor closest to the Earth?

This occurs when the meteor arrives at the _____________ of the graph.

f. At , how far is the meteor from the surface of the Earth if the earth’s diameter is 7920 miles?

g. Before Earth’s gravity deflected the meteor into its curved path, the meteor was traveling straight along one asymptote of the hyperbola. Write the equation of the straight path.

h. What do you suppose is the physical significance of the “other” asymptote?