Artifact
Scene from Mel Brooks’ classic, Spaceballs. Start off by simply showing it (or any combination of the split scenes) to your students (apologies if there are advertisements, you can skip them in like 5 seconds):
Now, I’m not sure if you’d want to show the entire scene in a classroom setting, both because it’s rather long and because there are some, *ahem* cruder moments (“sir, she’s gone from suck to blow“). Personally, I think starting off the first five minutes of class with the entire scene might engage the kids, make them laugh, wake up, etc. But then, I don’t have any administrators or parents to answer to at the moment. Regardless, I’ve broken up the scene into five pieces, which I share below.
Guiding Questions (GQs)
Personally, this scene brings up a ton of questions for myself. Hopefully after watching the scene there will be several GQs from your students. Here are the two primary GQs (a.k.a. “Need to Knows” for you PBL types) that will lead to the mathematics behind this scene that I have.
- How much air is in Planet Druidia?
- How big is Planet Druidia?
- How far above the surface is the “air shield”?
- How quickly can Mega Maid suck the air out of a planet?
- Did Mega Maid blow out the air faster than she sucked it in?
Solutions to GQs
Here is Planet Druidia again.
The two questions we really can’t answer for certain are “how big is Druidia?” and “how far above the surface is the ‘air shield’?”. But, you know what? Planet Druidia looks a lot like Earth to me, so let’s run with that.
What we need to find for the volume of air is the volume of a spherical shell. Or, the volume of two concentric spheres of differing radii.
So for this particular problem, we have
or,
where R=radius of Druidia/Earth+altitude of air shield and r=radius of Druidia/Earth.
The radius of the earth is about 6300 km (or 4000 miles). But how far up is the shell ceiling? Or, where does space begin?
According to famed astro-physicist and Nova Science Now host Neil de Grasse Tyson,
So, that’s 100,000 additional meters, or 100 km. So now the volume equation is
Which comes out to a volume of 50671795107 cubic km of “air”, which is how big Mega Maid’s vacuum bag would have to be.
However, I did leave out one potentially crucial fact that would add a whole extra level which we’ll revisit in the future: air is less dense as you ascend in the Earth’s (and presumably, Druidia’s) atmosphere. So I gave the volume of “air” in quotation marks, but as any good Coloradoan knows, there’s a lot less air at higher altitudes. At this point it becomes a density problem involving integration of the shell. As I said, we’ll revisit this problem in the future, but for now, since we’re just starting out, let’s stick with the volume problem. We’d hate to make a critical math mistake already.
Also unanswered is the rate of sucking/blowing portion of the problem. A stopwatch and some division should do the trick. Although, once we tackle the calculus portion of the problem we could get a nice interesting plot of Air in Mega-Maid’s Vacuum Bag vs. Time.
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Here is the same scene, split up into 5 pieces. They were split into pieces that would allow for a stop-and-start method of showing the scene. Also, some pieces are crucial to the problem, others just for humor.
emergent math 