Math in the Movies: Up
Let me say right off the bat that I am in no way insulting or belittling the new Pixar film Up, which was released in theatres this weekend; it was one of Pixar’s better outings, and that’s saying something. However, after reading a ridiculous article on PopularMechanics.com that actually defends the physical plausibility of Up, I couldn’t help but write a thing or two about the (im)plausibility of the movie myself.
The absurdness of the article appears as early as its fourth sentence:
Given the fact that one cubic foot of Helium can lift 60 pounds, with even small, 1 x 1-foot balloons, Carl’s rig has the capacity to lift about 618 tons—enough to lift about 150 Hummers.
Let’s think for a second about the numbers in that sentence, shall we? One cubic foot of helium can lift 60 pounds. Really, PopularMechanics? When was the last time that you were at a fair and saw a full-grown man get carried away by four helium balloons? How about instead of putting window washers and construction workers on scaffolding, we just tie a handful of balloons to their waist? Seems like a more economical solution to me.
Indeed, since PopularMechanics.com can’t seem to be bothered, let’s do some actual math to figure out how plausible the situation in Up really is. Helium’s density is about 5.06 grams per cubic foot and the density of air is roughly 36.11 grams per cubic foot. Thus, one cubic foot of helium can lift roughly 31.05 grams (this number will vary slightly depending on what figure you use for air density and whether or not you include the weight of the balloon itself, but I think that we can all agree that it is slightly less than 60 pounds).
Assuming that the average balloon indeed holds a cubic foot of helium (this seems like a reasonable assumption to me), we then find that it would take some 2630 balloons just to lift a 180-pound old man. To pick up his house (assuming a weight of 50 tons, which seems reasonable to me) as well would then require some 1.46 million balloons—a far cry from the reported 20622 balloons that were actually used. But hey, 20622 balloons would be enough to lift about 1400 pounds, or just under 3/4 of a ton. That’s pretty close to what you said, right PopularMechanics?
Update [June 1, 2009]: PopularMechanics has now corrected their article by replacing the offending text with “Given the fact that Helium can lift over six times its weight, Carl’s idea isn’t entirely fiction.”