Linear Versus Non-Linear
Counting arithmetically is defined as counting in constant intervals. For example: 1, 2, 3,
4…, or 5,10,15,20… A graph of an arithmetic function makes a straight line. A graph of
y = x +2 is a straight line. Counting exponentially is defined as counting in inconstant
(not constant) intervals. For example: doubling or tripling, 2, 4, 8, 16…, or 3,9,27, 81…
A graph of an exponential or logarithmic function makes a curved line on a graph of
arithmetic number lines. A graph of y = x2 curves upward as you discovered with your
double trouble graph. Nature has countless examples of non-linear functions from the
doubling of populations to the exponential increase in energy released from a star when it
collapses. Try graphing one such real world example below.
When we launch a rocket from Earth typically some material falls off and ends up in
Earth orbit as space junk. For example pieces of foam tile routinely fell off the Space
Shuttle (now retired). For many years space junk increased arithmetically each time a
new rocket was launched, but today the amount of space junk has reached a critical
threshold where the amount of space junk is increasing non-linearly. How can this be
unless we are increasing the frequency (number) of space flights? Assuming that space
flights remain constant through time, the amount of space junk has increased to a point
where pieces in orbit now hit each other and break into 2 or more pieces. So if a rocket
flight loses one piece during flight but collides later with an older orbiting piece and they
each break into 2 pieces we now have an increase of 4 pieces after that flight. Why do
we care about space junk and how much is in Earth orbit? Consider that the Space
Shuttle Columbia disaster of 2003 was caused when a piece of lightweight foam tile fell
off the craft during launch and punched a hole in the side of the vehicle that ultimately
destroyed it and the crew during reentry. NASA must track each and every piece of
space junk to make sure that future space craft do not hit any debris due to the
tremendous speeds that such craft go and the destruction that might otherwise occur.
On the back of this page use the graph paper to make a graph of space junk pieces
growing over time. Follow the instructions for labeling your x and y axes and then graph
the data below.
Label your x axis: number of space flights and starting at the origin as zero, label every
five lines as 1,2,3 etc. (starting from 0, count fives lines 1, then count fives lines 2, etc.).
Label your y axis: pieces of space junk and starting at the origin as zero, label every line
by ones (0, 1, 2, 3 etc. don’t skip any lines – your top line should be 39).
Graph the following data: space flight 1 = 1 piece of junk, space flight 2 = 2 pieces of
junk, space flight 3 = 3 pieces of junk, space flight 4 = 8 pieces of junk, space flight 5 =
18 pieces of junk, space flight 6 = 38 pieces of junk. Connect your data points to make a
line.
At which flight does junk increase non-linearly? Can you tell how many pieces of junk
are being created due to collisions after space flight 3?