Algebra Applications: Circular Paths

As the Ares rocket leaves the Earth's atmosphere it has

discarded its rocket boosters, and what is revealed is the

Orion capsule, which is what will travel to Mars.

But first the Orion needs to dock at the International

Space Station.

In docking with the International Space Station

the Orion needs to change its motion to a circular orbit

around the Earth.

A circle is another conic section, and we can overlay a

circular orbit around a point on the parabola that

corresponds to docking with the International

Space Station.

We want to create a circle whose center is at the origin

and that intersects the point on the parabola that

represents where the Orion docks with the International

Space Station.

Before constructing the circle.

Change the window setting of the graph window.

Use a nav pad to move the pointer near the origin.

Press and hold the CLICK key until the pointer changes to a

closed hand.

Then use the nav pad to move the origin to the middle of

the screen.

Press ESCAPE when you are done.

Now you want to rescale the graph window so that the

circle that you are about to draw will fit.

Use the nav pad to move to the horizontal axis.

Move the pointer so that is above one of the tick marks.

You will see an open hand.

As before, press and hold the CLICK key until it changes to

a closed hand.

Press the LEFT ARROW key and watch how the size of the

parabola shrinks as the graph window rescales.

Try to get your screen to look like this.

You are now ready to construct the circle.

Press MENU and under shapes select CIRCLE.

Use the nav pad to move the pointer to the origin,

press ENTER.

This defines the center of the circle.

Now use a nav pad so that the pointer is on the point on

the parabola.

Press ENTER.

You should now see the circle.

Because of the way the graph was scaled your circular orbit

probably looks like an ellipse.

To make it look like a circle press MENU, and under WINDOW

ZOOM select ZOOM SQUARE.

The orbit should now look like a circle.

A circle is a conic section, and since all circles do not

pass the vertical line test the equation of a circle is

not a function.

The equation of this circle was found by pressing MENU,

and under ACTIONS select COORDINATES AND EQUATIONS.

Use the nav pad so that the pointer hovers over

the circle.

Press ENTER once, then use the DOWN ARROW to position the

label for the equation.

Press ENTER again.

The equation of this circle is X squared plus Y squared

equals three hundred and eighty seven thousand.

A circle can also be written as a pair of

parametric equations.

Press TAB to bring back the equation entry area.

Make sure the X two and Y two equations are shown.

Input the equations.

Press ENTER.

The equations for X and Y are trigonometric functions and

notice that the parametric circle overlaps the

first circle.

At any point in its orbit the coordinates of the spacecraft

are shown.

Because the parametric version uses sine and cosine, this

captures the idea that the orbit around Earth

is periodic.

In fact, the Orion and the International Space Station

complete one orbit around the Earth, or one period, every

ninety minutes.

Motion that causes you to retrace your steps creates a

relation, not a function.

As the Orion capsule orbits the Earth the crew is

preparing for the long trip to Mars.