Algebra Applications: Parabolic Paths

A rocket is launched straight up, as you can see from the

launch pad.

SO why does it end up going in a curved path?

Why doesn't it go straight up?

And of all paths, why a parabolic path?

It depends on your frame of reference.

From the point of view of the spaceship it is a straight

path up.

But from the point of view of Earth it is a

curved path.

The Earth is rotating on its axis and so is the rocket.

But once the rocket lifts off it is no longer rotating with

the Earth.

This diagram shows that the rocket is moving in two

directions that are perpendicular to each other.

It is moving sideways at the constant speed of the Earth's

rotation, which is about four hundred and fifty meters

per second.

It is accelerating upward at an initial speed of about

twenty seven hundred meters per second.

These two independent movements define the motion of

the rocket.

Let's use the TI-Nspire to investigate this motion.

We will be using the parametric function

capabilities of the calculator.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a new graph window.

To activate the parametric equation capability press

MENU, and under graph types select PARAMETRIC.

Notice that unlike the function graph capability two

equations are shown, one for X and one for y.

With a parametric equation both X and Y are dependent on

another variable, in this case t.

In other words, the distance traveled along the x-axis

depends on how much time, t, has elapsed.

The same goes for the distance traveled along the Y axis.

The rocket is moving horizontally at a speed of

four hundred and fifty meters per second.

It's equation of motion is X of T equals 450t .

Vertically the rocket starts with an initial speed of

twenty seven hundred meters per second, but it slows down

with a constant acceleration.

The equation of motion in the vertical direction is Y of T

equals twenty seven hundred T minus 4.9 T squared.

Input these equations into the appropriate fields.

Input the equation for X of T and press the DOWN ARROW

to move to the Y of T field.

Then input the equation for Y of T and press the DOWN ARROW

to move to the field for setting the range for t.

We want the graph for the first ten minutes of the

flight, or six hundred seconds.

Change the value to the right of T to six hundred.

Press the CLEAR key to remove the previous value and replace

it with six hundred.

Then press ENTER.

To see all of the graph press MENU, and under WINDOW ZOOM

select the ZOOM FIT option.

Now you can see the whole graph.

And you can now see that this parametric graph is a parabola.

Notice that this graph passes the vertical line test so it

is a function.

But not all parabolas are functions.

This parametric graph, while still a parabola, does not

pass the vertical line test.

And this is why conic sections are known as

quadratic relations.

Returning to the path of the rocket, place a point on the

graph and follow its trajectory.

Press MENU and under POINTS AND LINES select POINT.

Use the nav pad to move the pointer so that it is on

the parabola.

press ENTER, then press ESCAPE.

Notice that a point is on the parabola with the

coordinates displayed.

Now make sure the pointer is above the point.

The pointer should change to an open hand.

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

closed hand.

Use the RIGHT ARROW to move the point along the graph.

Notice how the coordinates change.

When the point reaches a Y coordinate of around three

hundred and seventy thousand meters, or three hundred and

seventy kilometers, it reaches the altitude of the

International Space Station, which is the first stop on the

trip to Mars.

Since this is the only portion of the parabola that we need

you can shorten its length.

Press tab to bring back the equation entry area and press

the UP and DOWN ARROWS as needed to get to the range of

t values.

Recall that you had set this to six hundred.

Press the CLEAR button to replace it with a value such

that the parabola ends at the point corresponding to the

point you added to the parabola.

Try to get your screen to look like this.

So, the next stop is the International Space Station.