### 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:

## 2 Comments:

NICK! YOU ARE AMAZING! how do you do it all?

Hey Nick..helpful blog as usual. =) That animation thing is really good. The only thing is that I think the first link is broken..unless it's just me.

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