Saturday, October 17, 2009

physics, helium balloons, 6-year-olds, and Lawn-Chair Larry

PHYSICS ALERT! PHYSICS ALERT!! (I post about actual physics so rarely that I thought a disclaimer warranted. But don't worry - this is a high-school-level problem.)

Unless you've been hiding under a rock for the past few days, you heard about the reality-TV-"star" family drama this week in which 6-year-old Falcon Heene was alleged by his 9-year-old brother to have flown away in a homemade helium balloon. (Yes, his name was Falcon; you can't make this stuff up. Well, I couldn't, anyway - I won't speak for you.) The story exploded when the balloon finally fell to earth and rescuers found no one inside. After a long search, the boy was found hiding in the attic over the family garage; he had never been in the balloon.

Reporters were admittedly unsure whether the boy was actually in the balloon as it rose to 7000 feet and drifted some 50 miles. It seems, however, that no one in most newsrooms or in the police department tried to actually figure out, scientifically, whether the boy could have been in the balloon at all and have it fly. Surely no blame should be placed on the police; they are obligated to act to prevent the worst-case scenario. That said, the calculation isn't terribly difficult, and is worth noting for future reference. In this case, the calculation doesn't give a definitive answer, at least not with the information some physicist on the other side of the country reading a sketchy AP article has access to. But we can draw a few conclusions.

Here's how to determine whether the balloon could lift the boy. There are two forces at work in the problem: 1) gravity pulling down, and 2) the buoyant force of the atmosphere pushing up the less dense helium-filled balloon. It's really this simple; if the gravitational weight force of the boy + balloon is greater than the buoyant force, he won't get off the ground. If the buoyant force is bigger, he flies. Here's what you need to know for the calculation:
  • The weight of the boy: A typical 6-year-old probably weighs 50 pounds. For calculations, it's more convenient to use metric (SI) units, which in this case means that the boy's mass is 22.5 kg (kilograms) and the gravitational force on the boy is 222 N (Newtons).
  • The weight of the balloon: This is something we don't know, but can estimate. The balloon's weight comes from the Mylar and the carriage. We'll come to this again in a few minutes.
  • The volume of helium in the balloon: We know this because we know that the balloon was "saucer-shaped", about 20 feet wide and 5 feet tall. A saucer shape is mathematically approximated by an "oblate ellipsoid", which is just a squashed sphere. The formula for the volume of an oblate ellipsoid is V = 4/3 * pi * a^2 *b, where a and b are the half-width along the long and short dimensions respectively. Here a = 10 ft and b = 2.5 ft. Plugging in these numbers, we compute a volume of 880 cubic feet of helium. In metric units, that's 24.9 cubic meters.
  • The densities of helium and air: We know this from the Internet, or from a chemistry textbook. The density of helium is about 0.18 kg/m^3, and the density of air is about 1.25 kg/m^3. Helium is much less dense than air; that's why it floats - or really, why the air sinks around it. (Interestingly, the buoyant force is just gravity in disguise.)
To calculate the buoyant force, we use the famous relation from Isaac Newton: F = ma (that's force = mass x acceleration, where in this case the acceleration is that due to gravity, 9.8 m/s^2.) We calculate the mass by multiplying the difference between the densities of helium and air (1.07 kg/m^3) by the volume of helium in the balloon (24.9 m^3), and we find the force by multiplying by 9.8 m/s^2. Multiplying these three numbers together, we get 261 kg*m/s^2, which is the same as 261 N (Newtons, the SI unit of force, was named for Isaac for obvious reasons).

We already know that the weight-force of the boy is about 222 N. With just the weight of the boy accounted for, there would be about 40 N of lift force left over in this equation, because 261 N - 222 N = 39 N. To translate back into English units, that's just about 9 pounds.

That means that if the Mylar balloon and whatever carriage or box the boy would have been sitting in weighed only ten pounds, the boy would not leave the ground.

Keep in mind a couple of things: First, if you saw the balloon on the TV, it looked like the balloon wasn't quite fully inflated. This was probably because it had a small leak that allowed helium to escape so the balloon could descend. However, if the balloon had been filled only to 80% of capacity, it definitely could not have taken off with the boy on board. (Tip o' the cap to a chemistry professor friend of mine who commented on Facebook!)

Second, let's assume for the sake of argument that the balloon was full and the box and balloon weighed only five pounds, leaving 4 pounds of lift force (or in SI units, 18 Newtons). We can calculate how fast the boy would have left the ground, again using the equation F = ma. In this case, we know the force (18 N) and the mass (24.9 kg), and we want to know the upward acceleration of the balloon. Rearranging the equation to read a = F/m, we compute a = 0.72 m/s^2.

That means that the boy's speed would increase by 0.72 meters per second every second. So after one second, his upward speed would be 0.72 m/s, which is 1.6 miles per hour - a pretty gentle start. After two seconds, he would be a few feet off the ground and his speed would be 3.2 miles per hour. But after ten seconds, assuming he didn't jump off after two seconds when he realized what was happening, he would be a couple of stories up and going at the healthy rate of 16 miles per hour. (Now, admittedly, we've assumed that the balloon box was overly light in this analysis - but the point is, it doesn't take much of a difference between the boy+balloon weight and the buoyant force to send someone traveling up pretty quickly.)

So to sum up, it's not terribly likely that the boy would be able to float away with that balloon - but it's just close enough to give a police chief heartburn, even if he had done the calculation, in this case.

One of the reasons why people are inclined to believe that anyone could float away in a balloon is the famous case of Lawn Chair Larry, winner of an Honorable Mention in the Darwin Awards. In 1982, Larry Walters strapped a bunch of weather balloons to his lawn chair and, miscalculating, ended up floating to 16,000 feet and trespassing on LAX Airport airspace! You can read all about it at the links. What I want to point out is that the Darwin Awards write-up is particularly interesting in light of the above calculation - because if you believe what they have to say about the balloons, Larry never would have gotten off the ground.

The Darwin Awards story says that the 45 weather balloons Larry strapped to his chair were 4-foot-diameter balloons. They even calculated for you how many cubic feet of helium one of the balloons contains (33). Here's the problem: If you go through the math, 45 x 33 is 1485 cubic feet, or 42 cubic meters. The buoyant force from 42 cubic meters of helium is 440 N, which in English is 99 pounds. Judging from the picture, Larry was not anorexic. Between him and his chair and his cooler, there had to be at least 200 pounds. There's no way he ever would have left the ground if he had been using 4 foot balloons, much less "taken off like a shot", as the writeup claims.

Here's the mistake: Look at the picture of the chair and the balloons: the balloons were considerably larger than Larry in his chair. This Wired article corrects the record - there were 42 balloons and they were eight feet in diameter. Okay, wrong by a factor of two, no big deal, you say? Wrong! A factor of two in diameter means a factor of eight in volume, and therefore the buoyant force (subtracting 3 balloons) was 740 pounds! Let's calculate the upward acceleration like we did for the boy, assuming Larry + chair weighed 200 pounds: a = F/m = 2402 N / 90 kg = 26.7 m/s^2.

By comparison, the acceleration due to gravity is 9.8 m/s^2, sometimes referred to as 1 "G". So when Larry's friends cut the ropes, he was pulling almost 3 Gs. The Darwin Awards article says "he streaked into the LA sky as if shot from a cannon"; indeed!

HowStuffWorks has a nice synopsis of helium balloons, too.

UPDATE 10/18: The balloon story is now believed to have been an intentional hoax. One of the reasons that police arrive at this conclusion (aside from the boy's inadvertent confession on Larry King Live) is that, while Falcon weighed only 37 pounds (!), the balloon weighed 18 pounds more than what the publicity-hound Heene dad originally told police. After more precisely measuring how big the balloon is, investigators have concluded that Falcon never could have gotten off the ground after all. Physics 1, Reality TV 0!

1 comment:

Tidbits of Torah said...

I'm glad that I am not a Physics major! all that time could be spent learning a new language - like loshon kodesh


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