Hot Air Balloons, Algebra, & the Creator

hot air balloons

Hot air balloons have always fascinated me, and this last weekend I got a chance to see some up close. They’re huge—and it’s incredible to watch them float up and down by increasing or decreasing a flame of fire at the base.

But you know what’s even more amazing? The fact that day in and day out, the air in these balloons responds consistently to the heat from the flame used to control their movement. Let’s use math for a minute to look at some consistencies God both created and sustains in the atmosphere—consistencies that make hot air balloons possible.

Density & Temperature

For starters, there’s the Ideal Gas Law. This law is simply a way of describing the consistent relationship between the atmospheric pressure (P), the density of the air (ρ), the gas constant (R), and the temperature (T). Using letters to stand for each of these, we can describe the consistent relationship between them like this:*

P = ρRT

Now, using algebra (which is based on creation operating so consistently that we can multiply, divide, etc., by equal quantities on both sides of the equation, and know it will work even if we don’t know the actual values), we can rearrange this equation like this:


Now we know that the density of air (ρ)—how tightly packed the molecules are—depends on the atmospheric pressure (P), the gas constant (R), and the temperature (T). Since the atmospheric pressure and gas constant are constant for a specific area on earth, we can view them as constant values and realize that as the temperature (T) changes, the density (ρ) will too!

In other words, if we change the temperature inside the hot air balloon, it will change the density of the air inside that balloon (increasing temperature decreases the density, and vice versa). Why do we care? Well, let’s look at another consistency…

Buoyancy, Density, & Volume

The buoyancy force (the force that makes the balloon float) changes as the density of the air inside the balloon changes…meaning that we can use temperature to change a balloon’s ability to float! The buoyancy force (FB) of a hot air balloon equals the difference between the density of the air outside of the balloon (ρa) and that inside (ρ), times the volume of the balloon (V), times the local acceleration due to gravity (g).

FB=(ρa – ρ)Vg

Look at that equation and think about what would get a higher buoyancy force. The lower the density inside the balloon (ρ), the greater the force will be, as when we subtract it from the outside density (where the air is not being heated but staying relatively the same temperature), we’ll get a greater number to multiply by the volume and the acceleration due to gravity. Thus getting a lower density (which we do by heating up the air inside the balloon) gives the balloon a greater buoyancy force (which helps it float).

Now, there’s another variable that affects the balloon’s ability to float.  Look back at the buoyancy equation and notice the V, which stands for volume.

FB=(ρa – ρ)Vg

The larger the volume (V), the greater the buoyancy force will be! So large balloons will float better than small ones (assuming you can evenly heat all the air inside to a temperature warmer than the outside air). And now you know why hot air balloons are so large!

(You might wonder how the g in the equation influences the buoyancy. The g in the equations depends on the consistent way God causes gravity to operate, so we can’t change it. It’s a fixed value—about 9 m/s^2 close to the surface of the earth.)

Getting the Balloon to Float

Besides the buoyancy force, the only other force acting on a balloon is the weight of the balloon, the basket, and anyone or anything in the basket. We can describe this force algebraically like this:

where m is the total mass of the balloon, basket, and everything in the basket and g is again the local gravitational constant.

Once the air in the balloon has reached a temperature such that the buoyancy force is greater than the gravitational force, or FB > Fg, the net force on the balloon will be upward and the balloon will start to rise. That’s why when we heat up the air in the balloon and it becomes less dense, making FB > F g, it rises in the air!

The Consistency of Creation & the Creator

Hot air balloons are one example of how men, using their God-given abilities to explore God’s creation—have utilized the consistencies God created and sustains around us to develop a useful device (in this case, a balloon that floats). But don’t miss the miracle of our ability to use hot air balloons. We can only get in a hot air balloon with confidence because atmospheric pressure and buoyancy operates in a consistent way, day after day, year after year. While individual balloons may have different volumes, densities, masses, and forces, the relationship between them stays the same no matter what the individual values. Without the consistencies of creation, we would be unable to use hot air balloons. It’s this consistency of creation that makes modern science (and hot air balloons)–as well as algebra–useful.

Yet why is creation so consistent that we can describe how it will operate with letters and know that relationship will hold true, no matter the actual values we plug in?

The Bible gives us an answer: because of the biblical, consistent, faithful God. Jesus is faithfully upholding all things. We have a faithful Creator God.

He is the radiance of the glory of God and the exact imprint of his nature, and he upholds the universe by the word of his power. After making purification for sins, he sat down at the right hand of the Majesty on high,” Hebrews 1:3 (esv)

Transform Your How Your Students See Pre-Algebra and Algebra!

Want to teach algebra from this perspective? Check out our pre-algebra curriculum, and stay tuned for Algebra 2 next year! In fact, Algebra 2 students will get to explore hot air balloons in more depth as they explore God’s creation using math.

*Note: You may have seen this equation before written PV=nRT, where V is volume and n the number or mass of gas molecules. Since ρ = n/V, they’re actually the same equation.

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Math, Man on the Moon, & the Creator

Photo Credit: NASAThis Saturday, July 20, 2019, marks the 50th anniversary of the Apollo 11 landing on the moon.

This achievement would never have been possible apart from the Creator’s faithful sustaining hand, and men using the ability He gave them to use math to explore His creation.

  • Day after day, God holds creation together in such a consistent way that we can use math to describe that consistency. For example, we can describe the force due to gravity as gravitywhere G is a constant value, the ms are 2 masses, and the r is the distance between those masses. There are many, many formulas used in exploration of space—each one is a way of describing a consistency God created and faithfully sustains.
  • By describing the consistencies around us mathematically, we can use math to figure out how to send a spacecraft into space. For example, using algebra we can calculate the acceleration due to gravity that the spacecraft has to overcome. Using more math, we can figure out how fast the spacecraft has to go to escape from the pull of gravity into space, which can be described like this:Escape Velocity(see “Gravitational Escape Velocity with Saturn V Rocket” for more information). Then we can use math to figure out how to design that spacecraft to do that!

Whole books could be spent describing the math behind getting man to the moon. The point here is that modern science (including the space program) rests entirely on there being consistencies in creation (which enable us to design a spacecraft so it can escape the earth’s pull—if creation weren’t consistent, we wouldn’t know ahead of time if the spacecraft would really make it). And those consistencies in turn exist because a faithful, consistent Creator is holding all things together.

Jeremiah 33:25-26a (esv) says, “Thus says the Lord: If I have not established my covenant with day and night and the fixed order of heaven and earth, then I will reject the offspring of Jacob and David my servant.” God has established His covenant with the “fixed order” of heaven and earth—with the consistencies all around us. Here He points to that very consistency as a reminder that He is a God who keeps His covenants. He will do all He’s said in His Word—saving all who believe upon Jesus, and punishing those who reject His gift of salvation.

As you remember the landing on the moon, lift your eyes higher to the Creator of it all. Truly, creation declares His praises and reminds us to take head to His Word.

“The heavens declare the glory of God; and the firmament sheweth his handywork. Day unto day uttereth speech, and night unto night sheweth knowledge. There is no speech nor language, where their voice is not heard.” Psalm 19:1–3 (kjv)

Note: Our upcoming Algebra 2 book will give students the opportunity to explore consistencies such as  and see more up closely how math helps us describe God’s creation, pointing us to the Creator. Be sure to check out the math curriculum we offer for other grades too!