Algebra Applications: Ballistic Missiles

Narrator: During World War II the Germans rained down V2

rockets on many civilian targets especially in England.

Thousands were killed through this devastating attack.

The V2 was one of the first ballistic missiles.

Its rocket fuel propelled it to a height of fifty miles,

where the rocket would descend to Earth hitting the ground at

incredible speed.

Little could be done to defend against the V2 rocket except

to hide in underground bunkers.

Thus began the age of the ballistic missile.

During the Cold War the US and the former Soviet Union had

thousands of intercontinental ballistic missiles, ICBM's,

aimed at each other.

These missiles could travel long distances, reach

altitudes of over a hundred miles, and had the potential

for far more devastating damage because these missiles

were nuclear.

It was only the prospect of what was termed mutually

assured destruction, known by the acronym MAD, that prevented

a nuclear war.

But during the 1980's President Reagan proposed a

ballistic missile defense system that would stop these

missiles in mid-air.

Since that time the technology for a ballistic missile

defense has steadily improved.

While the technological challenges were significant,

the mathematics behind this technology relies on a

quadratic system of equations.

Basically, the path of a missile is a parabola, and the

path of the intercepting missile is also a parabola.

Where the missiles intersect is the solution to the

quadratic system.

Let's explore a quadratic system graphically using the

TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a graph window.

Let's graph two parabolas.

One parabola will represent the ballistic missile, and the

other parabola will represent the anti-ballistic missile.

At the F1 entry line, input this function.

Press the DOWN ARROW, and at the F2 entry line input

this function.

Press ENTER.

As you can see the graphs intersect.

To find the intersection point press MENU and under POINTS

AND LINES select INTERSECTION POINT.

Use the nav pad to move the pointer above one of the

graphs and press ENTER.

Then use the nav pad again to hover over the second graph.

Press ENTER again.

The coordinates of the intersection point

will appear.

The intersection point is a solution to the quadratic

system of equations.

This intersection point is where the anti-ballistic

missile will impact the ballistic missile.

In reality the path of the ballistic missile varies from

the parabolic path.

There are a number of reasons for this.

Such factors as weather, the missile guidance system, and

other factors will change the path of the missile from one

parabola to another parabola.

What the anti-ballistic missile does is read the

coordinates of the path of the missile and projects where the

missile will be.

This involves instantaneous reading of data in order to

make adjustments.

But the underlying principle still holds.

When the anti-ballistic missile hits the ballistic

missile a solution to our quadratic system has

been determined.

[Music]