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 y 2 = 4
a parallel path to Tanya.
Let the other equation represent the distance covered:
Tanya: x1 = 8t
Car: x 2 = 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 = x 2 , 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 θ to the horizontal, with an initial speed of v 0 , from a height h above the
1
horizontal are x = (v0 cosθ )t
y = − gt 2 + (v0 sin θ )t + h
2
(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 be 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 =0.1;
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?