A regular function has the ability to graph the height of an object over time. Parametric equations allow you to actually graph the complete position of an object over time. For example, parametric equations allow you to make a graph that represents the position of a point on a Ferris wheel. All the details like height off the ground, direction, and speed of spin can be modeled using the parametric equations.
What is the position equation and graph of a point on a Ferris wheel that starts at a low point of 6 feet off the ground, spins counterclockwise to a height of 46 feet off the ground, then goes back down to 6 feet in 60 seconds?
Applying Parametric Equations
There are two types of parametric equations that are typical in real life situations. The first is circular motion as was described in the concept problem. The second is projectile motion.
Thus the vertical parameterization is:
Thus parametric equations for the point on the wheel are:
Projectile motion has a vertical component that is quadratic and a horizontal component that is linear. This is because there are 3 parameters that influence the position of an object in flight: starting height, initial velocity, and the force of gravity. The horizontal component is independent of the vertical component. This means that the starting horizontal velocity will remain the horizontal velocity for the entire flight of the object.
The horizontal component is basically finished. The only adjustments that would have to be made are if the starting location is not at the origin, wind is added or if the projectile travels to the left instead of the right. See Example A.
The vertical component also needs to include gravity and the starting height. The general equation for the vertical component is:
When does the ball from Example 1 reach its maximum and when does the ball hit the ground? How far did the person throw the ball?
After about 4.2588 seconds the ball hits the ground at (-20.29, 0). This means the person threw the ball from (30, 5) to (-20.29, 0), a horizontal distance of just over 50 feet.
Kieran is on a Ferris wheel and his position is modeled by the parametric equations:
Jason throws the ball modeled by the equation in Example 1 towards Kieran who can catch the ball if it gets within three feet. Does Kieran catch the ball?
This question is designed to demonstrate the power of your calculator. If you simply model the two equations simultaneously and ignore time you will see several points of intersection. This graph is shown below on the left. These intersection points are not interesting because they represent where Kieran and the ball are at the same place but at different moments in time.
You can solve this system many different ways.
Nikki got on a Ferris wheel ten seconds ago. She started 2 feet off the ground at the lowest point of the wheel and will make a complete cycle in four minutes. The ride reaches a maximum height of 98 feet and spins clockwise. Write parametric equations that model Nikki’s position over time. Where will Nikki be three minutes from now?
Candice gets on a Ferris wheel at its lowest point, 3 feet off the ground. The Ferris wheel spins clockwise to a maximum height of 103 feet, making a complete cycle in 5 minutes.
1. Write a set of parametric equations to model Candice’s position.
2. Where will Candice be in two minutes?
3. Where will Candice be in four minutes?
One minute ago Guillermo got on a Ferris wheel at its lowest point, 3 feet off the ground. The Ferris wheel spins clockwise to a maximum height of 83 feet, making a complete cycle in 6 minutes.
4. Write a set of parametric equations to model Guillermo’s position.
5. Where will Guillermo be in two minutes?
6. Where will Guillermo be in four minutes?
7. Write a set of parametric equations to model the position of the ball.
8. Where will the ball be in 2 seconds?
9. How far does the ball get before it lands?
10. Write a set of parametric equations to model the position of the ball.
11. Where will the ball be in 2 seconds?
12. How far does the ball get before it lands?
13. Write a set of parametric equations to model the position of Riley’s ball.
14. Write a set of parametric equations to model the position of Kristy’s ball.
15. Graph both functions and explain how you know that the footballs don’t collide even though the two graphs intersect.
To see the Review answers, open this PDF file and look for section 10.5.