Examples of Parametric Equations

Examples of parametric equations

Tanya, who is a long distance runner, runs at the average velocity of 8 miles per hour. Two hours after Tanya leaves her house, you leave in your car and follow the same path. If your average velocity is 40 miles per hour, how long will it be before you catch up with Tanya? Use a simulation of the two motions to verify your answer. Set your calculator to simultaneous graphing.

This problem lends itself very nicely to parametric equations: one to describe Tanya’s motion, and one to describe the motion of you in your car. Let’s choose t=0 to be the

time Tanya left the house. Let’s also choose y1  2 as the path she will run. (This value as we will see is not that important.) (There is a constant vertical distance here)

Let us call the path of the car as y2  4 —a parallel path to Tanya.

Let the other equation represent the distance covered:

Tanya: x1  8t

Car: x2  40(t  2) (There is a t-2 since time started when Tanya left—so the car traveled for 2 fewer hours.)

The car will catch up when x1  x2 , so 8t  40(t  2) .Therefore, t=2.5 hrs after Tanya leaves the car will catch up.

Graph the parametric equations. Set T min to 0 and T max to 3 and T step to .1

1. Explain how this demonstrates the situation:

Example 2

It can be shown with calculus that the parametric equations of a projectile fired at an inclination of  to the horizontal, with an initial speed of v0 , from a height h above the 1 horizontal are x  v cos t y   gt 2  v sin t  h 0 2 0

(g is 9.8 m/s2 or 32ft/s2) Suppose Jordan hits a golf ball with an initial velocity of 150 feet per second at an angle of 30o to the horizontal. Work with a partner to

A. Find the parametric equations to describe the position of the ball as a function of time.

B. How long is the ball in the air?

C. When will the ball is at its maximum height? (notice the y equation is a quadratic of t)

D. Determine the distance the ball traveled

E. Simulate the motion on your calculator. Choose an appropriate T max and min (see question B)

Explore the motion of a ball thrown directly up (theta= ?) at a speed of 100 ft/sec from a height of 5 ft above the ground. Set Tmin =0 ; Tmax =6.5; T step =.1; Xmin=0 ;Xmax=5 ;Ymin=0; Ymax =180

A. What happens to the speed at which the graph is drawn as the ball goes up and back down?

B. How do you interpret this physically?

C. Repeat the experiment using different values of T step. How does this affect the experiment?