13.5 - Parametric Equations for Moving Objects

The main idea of this section is to separate what's going on with the x-axis and what is going on with the y-axis

The best way of describing this is using an example. Let's pretend that a firetruck is able to navigate around at 40 mph eastward and 60 mph northward to reach the next city. The firetruck is currently at the point (50, 20)-- 50 miles east and 20 miles north of the origin (we'll say the town center).

First, we'll find a parametric equation for the path of the firetruck, using t for the hours.

To do this, we simply take our coordinates and our velocities to get:

x = 50 + 40t
y = 60 + 20t

When we graph this, we get a line that can help us answer the following:

Predict the time that the firetruck will be 80 miles north of its current location.

To do this, let y = 80:
80 = 60 + 20t

It will take it about an hour to get 80 miles north of its current location.

Here's an image to help visualize this:




The next idea that we learned is: Parametric Equations of a Cycloid

The reason that we learn this is to calculate where a point is on a rolling wheel along the x-axis at any point in time.

In class, we learned how to derive all of this, but here are the equations:

x = a(t - sin t )
y = a(1 - cos t )
where 't' is the number of radians the wheel has rolled since the point was at the origin.

Here's a really really good explanation on how to derive this, if you weren't in class today:

Another Example

Let's take a look at another example of parametrics in action:

Rocky the flying squirrel:

is flying at 1,000 m/hr west and 800 m/hr vertically. At t=0, he is at point (100, 50).

a) Write parametric equations for his flight, using t hours as the parameter.
Solution:
x = 100 - 1000t
y = 50 + 800t

b) When will Rocky reach the top of the eiffel tower, which is 270 m higher than his current location?
270 = 50 + 800t
220 = 800t
t = .275 hours -- which is 16.5 minutes.

Here's two helpful websites:

http://www.libraryofmath.com/parametric-equations.html
is a summary which has this entire explanation reworded on a single page.

and

http://archives.math.utk.edu/visual.calculus/0/parametric.6/

which has the flash animation I posted above as well as a couple others. And something called LiveMath which I don't have, but you may.

In celebration of the recent talent show that we had, here is a clip from it:

http://www.youtube.com/watch?v=4G5XvqmRfSA