Monday, October 8, 2012

Launching Dragon: Not Everything is Linear

We've been doing a ton of work lately around slope and linear equations. As I've mentioned in class, slope is probably the most important concept we'll talk about all year, and you'll see it throughout your high school mathematics and science courses. But I wanted to take a moment and remind you that not everything is linear (we'll talk about non-linear equations second semester).

A great example of this is last night's launch of a Falcon 9 rocket with the Dragon capsule. If you're interested, watch and listen to the first 7 or 8 minutes or so of the flight (note that I've linked to the 9:47 mark in the video - liftoff is just after that, then watch and listen for 7 or 8 minutes after that). As you listen to mission control you'll occasionally hear them talk about the velocity. For example, at the +1:05 they say its velocity (another word for speed, but it includes direction) is 230 meters per second.

We've done enough work with slope and dimensional analysis that you should be able to picture what that velocity means: 230 meters per second means that for every second that passes, the rocket travels another 230 meters. So picture the slope triangle for that. So if you were graphing this, distance would be on the y-axis and time would be on the x-axis. You would plot a point at 65 seconds (1:05 equals 65 seconds, right?)  representing the distance the rocket had traveled, and then you would draw a slope triangle that went up 230 meters on your y-axis, and over 1 second on your x-axis.

This is where it's going to be different from what we've done in class. In class we've worked with linear equations, which means the slope is constant - stays the same throughout the situation. So that means that at any point in the rocket's flight you could draw a slope triangle of up 230 meters and over 1 second to find the next point on your graph. But if you continue to watch and listen to the launch, you'll note that at about the 4:26 mark (266 seconds) the mission controller says that the rocket is traveling 3.1 km/sec (which is equivalent to 3100 m/sec). So if you were to draw a slope triangle now, at 266 seconds, it would have to be up 3100 meters and over 1 second.
So if you picture those two slope triangles at those two places on the graph, you can hopefully see that this can't possibly be a straight line. If you continue to watch and listen, you'll hear that at 5:20 the velocity is 3.5 km/sec, at 6:23 it's 4 km/sec and at 7:38 it's 4.9 km/sec. At each of those points if you drew your slope triangle you would see that the graph has a steeper slope in each instance. In other words, the rocket is accelerating. When an object is accelerating, then its distance vs. time graph will not be linear, but instead will be curved.

Later, when you get to calculus, you'll learn about finding the "second slope." This is an equation where instead of plotting distance vs. time, you actually plot velocity (which is your slope on a distance vs. time graph) vs. time. This "second slope" is called the second derivative and represents acceleration (regular slope is called the first derivative and represents velocity). This rocket launch is complicated and involves lots of different variables, but if we look at it in a somewhat simplified form, and plot velocity vs. time, that graph is somewhat linear. Which means that while the velocity (slope) is constantly changing, the acceleration (the slope of the slope) was (more or less) constant (linear).

You may not completely understand this post, but my hope is that you're getting more and more comfortable with the idea of slope, and that through seeing examples such as this one, you can see the real-world applications of the mathematics we're learning.

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