Today we find our feline travelers, Henri and Cassie, visiting the famous hot-air balloon festival in Château-d’Oex, Switzerland. Certainly not the kind of educational event Henri prefers for her young companion, but when Cassie saw the brochure with its pictures of giant balloons floating through the mountains, the begging began. Against her better judgement, the older cat finally agreed.

All the way there, Cassie talked about how exciting it would be to go up in one of the balloons. However, after their experiences at the Leaning Tower of Pisa and the Eiffel Tower, Henri was determined to make sure that this time, their paws remained firmly on the ground.

Having no desire to deal with crowds of people who might step on their tails, the two cats took the early morning bus to the festival grounds. Upon their arrival, Cassie watched in amazement as giant fans and gas burners were attached to the baskets and the balloons slowly took shape. Cassie tugged on Henri’s arm. “Why doesn’t the air come back out of the opening on the bottom of the balloon?”

Henri smiled; maybe this would be educational after all. “It’s because hot air wants to go up, causing it to be trapped inside the balloon.”

The young kitten nodded, not entirely understanding what Henri meant, but was glad the older cat hadn’t answered, “it’s physics!” as she often did.

The balloons were of every imaginable shape and color and it soon looked as if they were standing in a massive flower garden. Cassie happily ran around trying to see each one. Whenever she got too close, Henri, anxious to prevent any misadventures, quickly shooed her away.

Seeing a stand selling small helium balloons decorated to look like the larger ones, Cassie rushed over to Henri. “Can I have a balloon?” she pleaded.

“I don’t see why not,” replied Henri.

Cassie picked out a balloon with pink stripes, and Henri tied it to one of her paws so she wouldn’t lose it. Unfortunately, being a cat (curse you lack of opposable thumbs!), Henri hadn’t done such a good job at tying the string. It wasn’t long before the balloon came free and began drifting upward.

“My balloon!” exclaimed Cassie running after it and leaping into the air. Sadly, it was too high for her to reach.

“We’ll get you another …” began Henri, but Cassie was already out of earshot.

“I could have been at home sleeping on a nice comfortable couch,” muttered Henri as she hurried after the kitten. Seeing Cassie leap into an unoccupied balloon basket (I think you can see where this is going), Henri cried out "No!" Running faster than she had in years, the older cat was just able to scamper into the basket as the balloon left the ground.

Cassie was perched on the edge of the basket pointing forward. “Follow that balloon,” she called out excitedly.

It took several minutes for Henri to catch her breath. By the time she did, they were nearly fifty feet in the air. “Cassie!” she shouted. The little kitten looked over at the older cat and ducked her face under a paw.

“I’m sorry,” she mewed softly. “I just wanted to catch my balloon.”

Henri looked at her sternly. “We could have gotten you another balloon.” Seeing the kitten’s crestfallen face, Henri sighed and patted her on the head. “Let’s see about getting ourselves down.” Pointing at the two ropes hanging down from the balloon, she continued. “I think one of these ropes opens the top of the balloon to let air out, but we have to be careful because one of …”

“I’ll get it,” cried Cassie, leaping upward and grabbing one of the ropes before Henri could stop her. There was a loud roar from above as flames shot into the balloon. The balloon rose rapidly, throwing Henri to the floor. Cassie let out little fearful gasps as she held tightly to the wildly swinging rope.

“Hold on,” shouted Henri, struggling to get to her feet. It took several minutes, but at last she was able to grab the rope so Cassie could climb down. Unfortunately, the burner had been firing the entire time and the festival grounds were now well behind them and the ground below was covered in trees.

Henri gave the kitten a dour look. “No place to land here. Looks like we’ll be taking a little trip.”

Cassie was about to clap her paws, but seeing the look on Henri’s face, restrained herself. However, as they floated onward, the older cat’s mood softened. After all, there are few places more peaceful than a hot air balloon. She and Cassie sat there in (mostly) silence watching the scenery below.

After nearly two hours they finally reached an area free of trees near a small village. Henri grabbed the other rope and tugged on it. A welcome whir could be heard coming from the top of the balloon and they started slowly drifting towards the ground. When they reached the ground, the basket bounced a few times and came to a stop.

Our two feline friends jumped out and found themselves in a quiet mountain meadow with a small stream passing through it. Tired and thirsty, our travelers enjoyed a refreshing drink and rested on a small grassy hill.

“I’m sorry for all the trouble I caused,” apologized Cassie in a small voice.

Henri looked over at the kitten. “You certainly are a pawful.” Breaking into a smile and gesturing around them, she continued, “but how else would we have found this wonderful place?”

The two cats spent the rest of the morning enjoying the beauty of their surroundings. However, as lunchtime neared and their stomachs began to gurgle, Henri stood up and stretched. “We should be going. I hope that village has a place where we can have lunch.”

“Me too,” agreed Cassie as the two cats happily went on their way.

It’s hard to disagree with Henri about Cassie being a pawful. She means well, but sometimes just gets a bit carried away (in this case, literally). However, she did bring up a good question about the hot air balloon, so let’s see if we can find an answer.

Henri was right about hot air rising, but why is that? To discover the answer to that question we must travel to Ancient Greece where the famous philosopher Archimedes is about to take a bath. Lowering himself into the tub, he notices the water level in the tub increasing. He’s perplexed for a moment, but a sudden flash of insight causes him to leap from the tub crying out that now famous phrase, “Eureka!”

Yes, most of us know the story of Archimedes leaping from the tub crying “Eureka!” It’s a common misconception that this is how he discovered how objects float (buoyancy for all you science types), but all it really showed is that the volume of a fluid (such as water) displaced by a submerged object is equal to the volume of the object.

Now Archimedes did actually discover the principle (which is named after him) we’re interested in. It’s included in his book On Floating Bodies which was written in 250 B.C. Archimedes’ Principle states:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

So what does this mean? Think back to the last time you went swimming. Didn’t you feel lighter when you were in the water? That’s Archimedes’ Principle at work. Your body is filling up a certain volume in the water and water, being selfish, wants that volume for itself. So the water tries to push you out of the way, making you feel lighter.

Now why doesn’t the water shoot you into the air? Because, and don’t take this personally, you’re denser than water. That is, a cubic foot of you weighs more than a cubic foot of water. As Archimedes showed us in his tub, the volume of water you displace equals your own volume. Therefore, the weight of the water you’re displacing (and thus the force acting on you) is less than what you weigh, so the water can’t get rid of you.

But what about something less dense than water? As a kid, did you ever force a beach ball under water and let it go? What happened? It shot out of the water into the air. Since air is less dense than water, the force of the water acting on the ball is greater than the ball’s weight ejecting it from the water.

You’re probably thinking, “Hey, what’s water got to do with hot air balloons.” Actually, in physics the term fluid refers not only to liquids, but to gases as well (confusing I know). Therefore, everything we said about water above is also true for air. Even now, air is trying to push us out of the way, but its density (0.000044 pounds per cubic inch) is so low compared to water, we don’t even feel it.

So what is it about hot air that makes the air around it want to push it out of the way? For the answer to that, we must turn to (cue dramatic music) the Ideal Gas Law. It provides an idealized (physics term for simplified) representation of the behavior of a gas as a function of temperature and pressure. A common form of the Ideal Gas Law is:

We can use this equation to figure out how the density of air changes when it's heated. Now when the air is heated, the temperature goes up (T increases), but what happens to the air pressure (P)? Ah, remember Cassie’s question about the bottom of the balloon being open? Because the bottom of a hot air balloon is open, excess hot air escapes and the pressure inside the balloon is only slightly higher than the outside air. Since R is constant for air, it doesn’t change.

Now in the above equation, what happens to the air density if P and R don’t change, but T increases? Through the magic of graphs, we present:

From the graph we see that as the temperature of the air increases, the air density decreases. As we saw with the beach ball pushed underwater, the cooler air surrounding the balloon will try to push it away generating an upward force on the balloon.

Speaking of that ball, if the hot air in the balloon is less dense that the air outside, why doesn’t the balloon shoot up like the ball did? Well, the balloon has extra weight (the basket, burner, and cats) that has to be lifted as well. To achieve this, the density of the air in the balloon has to be significantly less than the surrounding air. In fact, for a typical hot air balloon, the air must be heated to about 210 degrees Fahrenheit to generate enough lift to get the balloon off the ground.

Seems like a lot of work to get a hot air balloon off the ground, doesn’t it?. However, based on Henri and Cassie’s experience, it certainly seems worth the effort. Things were certainly a lot easier with Cassie’s helium balloon. There’s no heating or anything, just fill the balloon and it floats. Why is that?

It’s because helium is naturally lighter (less dense) than air. In fact, helium is the second lightest element after hydrogen. Unfortunately, hydrogen is extremely flammable and unsafe to use. The blimps you see around sporting events all use helium. It’s easy to use, abundant and safe. If helium works so well, why even use hot air balloons? Why for the experience of course! While I can’t say for certain, I think Henri and Cassie would heartily agree.