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?