A couple of weeks ago, I was at a birthday party for one of my son’s friends. It was a great day, at a little park a half hour drive north and east of where I live, situated on a tributary of the Ohio River. The kids all had squirt guns and the like, and got each other soaked down while the adults sat back and watched and took pictures and were grateful that they brought extra clothes and towels. There were balloons as well, because there were kids.
One child was super excited about the balloons. They were your ordinary latex kind, that you blow up with your own lungs, but he was bouncing them around and laughing. “Why don’t we all get a balloon?” he asked excitedly. “And then we can fly into the sky!”
So, yeah. It wasn’t my son that asked it. But it’s the kind of question he could have asked, so I’ll answer it.
How does a balloon float?
The same way a boat does.
Care to elaborate?
It’s tempting to say that things float because they’re light, but that’s not quite accurate. For example, an oil tanker floats but it is not light – they can carry anywhere from 1,500 to 550,000 deadweight tons, depending on size. No. Floating has everything to do with the mass of the object, and the fluid that surrounds it. See, all objects placed into a fluid displace some of the fluid (put a rock in a cup of water to see for yourself). If the mass of the fluid you displace is greater than your mass, you float. And air, for these purposes, can be considered a fluid.
But let’s look at some math, since the University of Chicago was kind enough to put together a document (Lighter Than Air: Why Do Balloons Float?) that explains all of this in some detail. There are two forces in play, the downward force (which is the pull of gravity) and the upward force (which is how much the fluid resists the downward force). The downward force (Fg) is the mass of the object (M) x gravitational strength (g), which is also how you calculate “weight” in physics. Weight, after all, is mass times gravity (which is why you weigh less on the moon, even though you retain the same mass). Upward force (Fb) is the mass of the fluid displaced (m) x gravitational strength (g).
Once you have Fg and Fb, you can calculate life=t. All that is is Fb – Fg. If the result is positive (meaning Fb is larger than Fg), you are sinking. If the result is negative (meaning Fb is larger than Fg), you are rising. And if the result is 0 (meaning the two forces are equal), you are floating immobile in midfluid.
Let’s do an actual example, shall we?
After consulting Google, I found an estimate that the average-sized party balloon masses 1.7 grams, and several notes that they can weigh more depending on the actual size, thickness, etc, etc. This will be important, momentarily.
Now, the density of air at sea level is about 0.0012 grams per cubic centimeter. So, if you inflate your hypothetical average-sized party balloon to a diameter of 1 foot (0.3048 meters, which means 30.48 centimeters), you get a sphere (for the sake of not making me crazy) containing 14,826.7 cubic centimeters of air. The inflated balloon weighs a total of (14,826.7 x 0.0012) + 1.7 = roughly 19.5 grams, and displaces 17.8 grams of air. So, it sinks. If you inflate the balloon to 2 feet in diameter (60.96 centimeters), you get a balloon containing 116,613 cubic centimeters of air. It weighs 141.6 grams, and displaces 139.9 grams of air.
Clearly, both balloons sink. And, in a vacuum, both would sink at the same rate because they have the same lift (-1.7).
But they don’t fall at the same speed. Not the ones I’ve played with, anyway.
Nope. Because we live in an atmosphere. And atmospheres create air resistance. I won’t go into the math there, because it made my head hurt a little, but it works like this: an object produces drag (a resistance to acceleration) based on the cross-section of the object perpendicular to the direction of movement. As the cross-section gets larger, the power needed to overcome the drag increases. How much? Well, it’s based on the cube of the cross-section. If you double it, you need 8 times as much power. If you triple it, you need 27 times as much power. And so on.
For the balloon, acceleration is down towards the ground and the cross-section is the diameter of the balloon. Doubling the diameter of the balloon means you would need 8 times the power to make it fall at the same speed as the smaller balloon. Since gravity (roughly) stays the same, that means you would expect to see it fall 8 times as slowly.
So, getting back to the wish to “fly into the sky”…
Sure. See, to make a balloon fly, you need something less dense than room-temperature air. That’s why hydrogen and helium are so popular. They’re gaseous at “room temperature”, and they weigh far, far less. Hydrogen weighs 0.000089 grams per cubic centimeter, and helium weighs 0.00018 grams per cubic centimeter. So, looking at the two balloons from the earlier example, we get the following information:
- The 1 foot balloon weighs 3 grams if you fill it with hydrogen, and 4.4 grams if you fill it with helium. It displaces 17.8 grams of air.
- The 2 foot balloon weighs 12 grams if you fill it with hydrogen, and 23 grams if you fill it with helium. It displaces 139.9 grams of air.
Regardless of which gas you fill the balloon with, it weighs less than the gas it displaces. So it has positive lift and it goes up. In fact, it could even lift additional weight – the 2 foot balloon filled with hydrogen would have neutral buoyancy with a 127.9 gram weight attached to it, so you could attach two Hershey’s chocolate bars (1.55 oz, or 44 grams each) to the balloon and still watch it go skyward.
Heating the air will also work, as gasses become less dense with heat. Sadly, I don’t have a good equation (that I understand) to show how much you’d have to heat the air to make it lift.
How many balloons would I need to fly to the sky, then?
Well, that’s more or less easy. How much do you weigh, and what gas are you using? I’ll illustrate with my son. He weighs around 60 pounds right now. That’s 27.2155 kilograms, or 27,215.5 grams. Looking at the two foot balloons, the lift for the hydrogen balloon is 127.9 grams per balloon and the lift for the helium balloon is 116.9 grams. So, it would take 27,215.5/127.9 = 213 2 foot hydrogen balloons to give him neutral buoyancy. 233 2 foot helium balloons would be required to achieve the same effect. You’d need another 18 hydrogen (20 helium) balloons to offset the weight of his clothes (maybe more if he’s planning on flying high). And I have no idea how many balloons would be required to offset the weight of the lines he’s holding on to or the harnesses to keep the balloons attached to him. And, of course, he’d need more to actually go up.
By contrast, I weight 316 pounds. So I’d need 1,121 hydrogen balloons or 1,227 helium balloons to achieve the same effect. That’s 37,553.5 cubic feet of hydrogen balloons, or a sphere roughly 42 feet in diameter. Oh, and it could explode.
Don’t do this at home.
You’d think so, but at least one person did. Larry Waters used 45 8-foot weather balloons filled with helium to lift himself, his lawn chair, his parachute, his pellet gun (so he could pop balloons and descend), his CB radio, his camera, and sandwiches and beer to a height of 16,000 feet. He flew for 45 minutes, and got fined $1,500 by the FAA after an appeal. But, because he was a trained pilot and lucky, he didn’t die.