Polynomials and the Path of a Space Lab Crash

I frequently use current events in math class.  This is our latest endeavor!

China’s Tiangong-1 Space Lab is expected to crash sometime this month.  Typically, reentries of large satellites are controlled so that they crash into the ocean.  In this case, China’s space agency lost control of the satellite two years ago.

The internet is full of attention grabbing headlines about this including:

 

Chinese Space Station carrying TOXIC chemicals to CRASH into Europe or New York NEXT WEEK

 

Chinese Space Station: Tiangong-1 could crash into Earth and hit Europe next week

 

And this one that sounds like a movie ad:

 

China’s Out of Control Space Station Coming to a City Near You!

 

Don’t panic!  Despite the dire warnings, you are not at risk of being obliterated by a space lab crashing through your roof. Most of the lab will burn up as it re-enters the Earth’s atmosphere with remaining debris scattering over a large area.

 

But scientists around the world are tracking Tiangong-1 and making models to predict when and where it will land.  Here’s one article that explains how they are doing it:

 

https://www.livescience.com/62077-tiangong-re-entry-date.html

 

While the models being developed to pinpoint the re-entry and landing sites must consider an incredible number of variables and depend on sophisticated modeling tools, the basics of understanding the path of this or any other satellite or projectile only require an understanding of the polynomial functions that we’ve been studying in Algebra this month.

Path of a Space Lab in Orbit

While in orbit the Tiangong-1 Space Lab, like all other satellites, travels in an ellipse. We won’t work with ellipses until Pre-Calculus, but the equation is just a two-variable version of the equations we’ve been working with this week:

where

​Although we haven’t studied this in class yet, you can try to graph an ellipse using the same technique of picking values for one variable and solving for the other then plotting the points.  Try finding points. Then, graph these ellipses:

Look at your graphs and use the analysis tools we learned in class to think about what happens when you change the value of any of the constants in the equation.
Path of a Space Lab when it Slows Down

Many things will make a satellite, such as the Tiangong-1 Space Lab, change course or slow down.  Normally, ground control corrects the path by activating one or more engines on the satellite.  In 2016, China’s space agency lost its connection to the Tiangong-1 Space Lab, so it hasn’t been able to correct its path for two years.
 
As a satellite slows down, its path becomes less elliptical and more circular.  Mathematically, the values of and in the equation for the ellipse become closer to each other.  Try graphing these to see what happens:

Notice that the first equation gave you a long, stretched out ellipse.  Each subsequent equation gives you a rounder ellipse until you end up with a circle in the last equation. (Mathematicians call this degree of stretchiness “eccentricity” and is usually calculated using the variable e.)
 
Notice that if A = B, the equation for the ellipse become the same as the equation for a circle with : C = r^2

If a satellite traveling in a circle slows down further, its path become a spiral and eventually follows the path of a projectile.
Path of Space Lab Debris
This week we worked on solving equations in the form Ax^2 + Bx + C = 0 where A, B and C were meaningless coefficients and x was a meaningless variable.  We can apply our understanding of this basic quadratic equation by assigning values to each of these as follows:

​Substituting these variables gives us:

​Since the altitude of any object on the ground is zero, we can find the time it takes for a projectile to land by solving the following equation:

​This is the standard equation used to figure out how long a baseball or rocket will be airborne before it lands on the ground.  Then the time can be used to find the distance traveled by putting it into a simple equation that normally looks like this:

Where:

​We can use this equation for debris from Tiangong-1 Space Lab after it re-enters the Earth’s atmosphere.
 
Last year, in Pre-Algebra we played with the equation for the Force Due to Gravity:

We used this to confirm the acceleration due to gravity close to the surface of the Earth and to understand why there is very little gravitational force in the space station.  When we get to Calculus we’ll be able to work with cases where g is a variable this is constantly changing over time.

​Solving Mathematical Problems of Space Travel

Using the quadratic equations that we learned in class, it seems like the path of the Tiangong-1 Space Lab has three distinct parts.  But we can see that the circle is just a special type of ellipse.
 
To see how projectile motion and orbiting are related think about this:

  • Imagine that you throw a baseball.  It will travel some distance following the path of a parabola and land somewhere on the ground.
  • Imagine that you threw the same baseball with more force.  It will travel further before landing on the ground.
  • Now imagine that you are superman.  You throw the ball so hard that the distance it travels before landing on the ground is greater than the radius of the Earth.  There is no ground for it to land on.  It will keep falling toward the Earth but never land.  It goes into orbit.


In “Hidden Figures” Katherine Goble wrestled with the problem of how the path of a rocket would change from an elliptical orbit to a parabolic fall to Earth.  Rewatch the movie and to find her “aha” moment when she realized that these two paths could be described with one equation.  That equation involves trigonometry and polar coordinates.  You can see Ms. Goble work out the mathematics of this in the movie.  The unifying equation is:

Further Study

I know what you’re going to ask me and the answers are:
 
“No, we can’t spend our entire next class meeting playing with these equations.  You don’t yet have that math background to derive all of them. BUT we can spend part of the class period playing with projectile motion since it ties into the quadratic functions that we worked on.”
 
“Yes, some of these equations are the ones that Dylan derived and wrote on the window last year.  As sad as it was to erase his work, I’ve very proud that you were able to add the derivation of the quadratic equation to the window last week.”
 
“Yes,I recommend that you spend some of your time off of school doing your own research on this.”  I suggest the following:
 
http://www.physicsclassroom.com/class/circles/Lesson-4/Circular-Motion-Principles-for-Satellites
 
Rewatch the movie Hidden Figures.

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