Look Out! Space Junk!
Team Name: _________________________
If you saw the movie Gravity, then you know that space junk can be a real problem for astronauts. In
the movie, testing a space-weapon creates a cloud of fast-moving pieces of scrap metal that zoom
around outer space, and end up destroying the space shuttle that George Clooney and Sandra
Bullock were riding in.
Space junk isn’t just a problem for astronauts, either. If you have Dish or DirecTV, use GPS on your
phone, or listen to Sirius radio, then you are relying on satellites. If those satellites have to maneuver
to avoid space junk, or get smashed into smithereens by space junk, that means our costs for TV,
smart phones, or satellite radio will go up.
And! Not all of that debris stays in space. On average, at least one piece of large (>10cm) space
debris enters the Earth’s atmosphere every day. Some of it burns up in the atmosphere, but some of
it lands here on Earth. So far, no one on Earth has been hurt by space debris. But as we keep
putting more and more junk into space, will it become more likely that we have to dodge flaming
hunks of space junk hurtling from the sky?
Just how much junk is in outer space? Is it getting worse? How much worse? Space scientists work
hard to both count the number of observable pieces of space junk, and to keep track of the mass of
the debris that’s spread out in Low Earth Orbit. That way, even as bigger pieces of junk smash into
each other and break up into smaller pieces, we still have a sense of just how much junk is out there
causing havoc.
[Note: This packet draws heavily from NCTM’s “Modeling Orbital Debris” lesson plan:
http://illuminations.nctm.org/Lesson.aspx?id=1386]
1) In 1992, scientists estimated that there were 4,000,000 pounds of space junk. The space junk is
all different sizes and made of different stuff, but it’s mostly little pieces of metal. About how much
space would 4,000,000 pounds of pennies take up?
Info:
•
•
•
•
One penny weighs about 2.500 grams = 0.0882 ounces.
The height of one penny is 1.52 mm
The diameter of one penny is 19.05 mm
There are 2.54 centimeters in an inch.
2) Relate your answer to something we might know the size of:
• 4,000,000 pounds of pennies would fill an Olympic Swimming Pool (25 meters by 50 meters)
to a depth of: ________________
3) Actual space junk is spread out in space. The Earth’s radius is about 6,371 km and space
stations orbit about 350 km above the surface of the Earth. Would 4,000,000 pounds of pennies,
arranged end-to-end, 350 km above the surface of the Earth, completely circle the Earth? How
many times?
The 4,000,000 pounds of space junk we’ve been exploring was from 1992. You might be
wondering, “How much space junk is there now? Is the amount of space junk growing? How is it
growing?” Well, in 1990, when scientists estimated that a total of 4 million pounds of debris was in
Earth orbit, they also estimated that at that time we were adding 1.8 million pounds per year to the
already serious problem, which in a few years would result in 9.5 million pounds of orbital debris.
The 1990 prediction also stated that the amount of debris being added was anticipated to increase
to a rate of 2.7 million pounds per year by the year 2000.
4) When creating models, mathematicians favor the simplest model that will account for the
phenomena in question. Generally, a linear model gives the simplest case. So, using the reported
1990 rate of increase of 1.8 million pounds per year and assuming 4 million pounds of existing
debris at the beginning of 1990 (end of 1989), write a linear model to predict the number of pounds
of orbital debris at the end of any given year, t. Assume that t = 1 represents the end of the year
1990.
5) Use your linear model to predict when the weight of space junk will be 9.5 million pounds.
6) Write a second linear model using the predicted 2.7 million pounds per year rate of increase and
the initial 4 million pounds for 1990.
7) Use your second linear model to predict when the weight of space junk will be 9.5 million pounds.
8) Does either model likely match what’s actually happening to the rate of increase of the weight of
space junk? Why or why not?
9) On the next page, graph a model that has a rate of increase of 1.8 million pounds per year from
1990 to 2000, and a 2.7 million pound per year rate of increase after 2000.
YA
>X
Math-Aids.Com
10) Use your graph (or equations) to predict in what year the total amount of space junk will be 25
million pounds.
11) In your graph above, does the slope (or rate of change) change gradually between 1990 and
2000, or abruptly in the year 2000?
12) Do you think the actual rate of change changed gradually between 1990 and 2000, or abruptly in
the year 2000?
Again, let's make the simplest assumption: the rate at which we are adding debris increases at a
constant rate (goes up by the same amount each year) from 1.8 million pounds per year in 1990 to
2.7 million pounds per year in 2000.
13) Complete the following table to show the amount of debris added each year and the total
amount in orbit at the end of the year.
Year
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Amount Added During Year
(Millions of Pounds)
1.8
Total in Orbit at End of Year
(Millions of Pounds)
5.8
2.7
14) Since we assumed that the increase in the velocity of littering was achieved in equal annual
increments, you can write a linear equation that describes the increase in the amount of debris
being added each year (i.e, the increase in the annual velocity of littering as a function of the number
of years since 1990. In this case, we let a = 0 in 1990 because we are assuming that the 1990 rate
of 1.8 million pounds per year is our baseline rate). Then we can write an equation in terms of a that
represents the rate of littering a years after 1990.
(In other words, write an equation for Column 2 of your table in terms of years after 1990).
15) The situation described in your equation, where the rate of increase of litter is itself increasing at
a constant rate, is analogous to a vehicle that accelerates at a constant rate from an initial speed to
a final speed. Use the data generated in the table you made to create a graph of the total number of
pounds of orbital debris that have accumulated relative to the year. You graph should cover the
period from 1990 through 2003.
YA
>X
Math-Aids.Com
15) Does this data appear linear to you? Why or why not?
16) Follow the instruction below to use your calculator to generate a quadratic model for the table:
(Note: if you need help on the calculator part, raise your hand and a volunteer will come to your
table)
(Note 2: you don’t need to use the calculator for this part, if you know other ways to generate a
quadratic equation to match data points)
Data for this example:
To enter the data:
• Press [STAT] [1] to access the STAT list editor.
• Input the data in the L1 and L2 lists, pressing [ENTER] after each number.
• Press [2nd] [MODE] to QUIT and return to the home screen.
To calculate the quadratic regression (ax2+bx+c):
• Press [STAT] to access the STAT menu.
• Scroll right to highlight the CALC menu.
• Press [5] to select QuadReg(ax+b).
• Press [2nd] [1] [,] [2nd] [2] to input L1,L2.
• Press [,] [VARS], scroll right to highlight Y-VARS, then press [1] [1] to input ,Y1.
• Press [ENTER] to calculate the quadratic regression. This will also copy the quadratic regression
equation to the Y= Editor.
To graph the data and the quadratic regression equation:
• Press [2nd] [Y=] [1] to access the STAT PLOTS menu and edit Plot1.
• Press [ENTER] to turn On Plot1.
• Scroll down to Type: and press [ENTER] to select the scatter plot option.
• Scroll down to Xlist and press [2nd] [1] to input L1. Scroll down to Ylist and press [2nd] [2] to input
L2.
• Press [GRAPH] to graph the data and the quadratic regression equation.
• If the graphs are not displayed, press [ZOOM] [9] to perform a ZoomStat.
[Calculator tips from: https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=76122]
Quadratic Model:
17) How well do you think the quadratic model fits the assumption that the rate of change increases
by the same amount each year? Why?
18 - 21) Use your two linear models, your model that changes slope in 2000, and your quadratic
model to predict the accumulation of debris after twenty years, thirty years, and fifty years.
After 1 year (end
of 1990)
After 5 years (end
of 1994)
After 10 years
(end of 1999)
After 20 years
(end of 2009)
Linear Model 1
Linear Model 2
Model that
Changes Slope
Quadratic Model
22) For the period from 1990 to 2000, the graph of the quadratic model lies between the graphs of
the two linear models.
a) Explain why this result is reasonable.
b) Will the quadratic graph always lie between the two linear graphs? Explain.
23) Explain why the quadratic model for the debris problem can be described as a "uniform
acceleration" model.
So far, you have looked at two models, a "constant velocity" linear model and a "uniformly
accelerated" quadratic model. Let's look at one more model.
24) Suppose that the amount of litter added each year grew not by a fixed number of pounds but by
a fixed percent of the amount already in space - a situation analogous to an investment of money
with interest compounded annually. For example, what would happen to the original 4 million
pounds if the litter added each year was 20 percent of the amount already in orbit?
Complete the following table to determine the amount of debris that would accumulate over the
period from 1990 to 2000.
Year
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Amount Added Each Year
(20% of Previous Amount)
(in Millions of Pounds)
-0.8
0.96
Total in Orbit at End of Year (in
Millions of Pounds)
4
4.8
25) Does the exponential regression model Total in Orbit at End of Year = 4(1.2)a where a = number
of years since the end of 1989 match the values in your table? How do you know?
If not, adjust the model or the table so your table and rule match.
26) The scientists believed that 1.8 million pounds of space junk was added in 1990, but the
exponential model above only has 0.8 million pounds being added. Adjust the model so that it adds
1.8 million pounds of space junk in 1990.
27) In your adjusted model, how many pounds of space junk are added during the year 2000?
28) Adjust your model again so that 2.7 million pounds of space junk are added in the year 2000.
29) Which of the three exponential models that you made do you think is most reasonable? Use
whichever one you think is more reasonable to finish the table:
After 1 year (end
of 1990)
After 5 years (end
of 1994)
After 10 years
(end of 1999)
After 20 years
(end of 2009)
Exponential Model
30) Of the Linear, Quadratic, and Exponential models, which one do you think best fits the
scientists’ predictions? Why?