One of the big problems with rockets is their size. They need to be truly humungous to get anything into orbit. Interestingly, the reason for this was explained back before modern day rockets were even invented. A Russian scientist named Konstantin Tsiolkovsky described why rockets need to be really big way back around the turn of the last century (like 1900).

The graphic below helps to understand what is going on. Let’s say you want to lift a blue cube into space. The blue cube has some mass to it. In order to accelerate the blue cube up to some speed it takes a total of two bricks of red fuel. Let’s put some pretend numbers to this to make it a bit easier to understand. Let’s say that you want to reach a speed of 4, and using two bricks of red fuel will give you a speed of 1. That is too slow. So, we need more fuel.

Now the problem is that we have the blue cube plus two red bricks of fuel, which is more massive than just a blue cube. So, we will need even more fuel to accelerate this. In order to accelerate the blue cube plus two red bricks by 1, we will need four red fuel bricks:

We can keep going on this. Now we have a blue cube and 6 red bricks. In order to accelerate all of them by a speed of 1, you need 8 red bricks of fuel.

After that, we will be going at a speed of 3. We are very close to 4! To accelerate all of those red bricks (we have 14 now!) plus the blue cube by another 1, it takes 16 red bricks of fuel! Yikes! This is growing out of control!

For a rocket, what would happen is that the 16 red bricks would burn to allow the rest of the fuel plus the blue cube to be accelerated by 1. Then the 8 red bricks would fire, accelerating the 6 red bricks plus the blue cube by 1, giving a total speed of 2. Then the four red bricks would burn and accelerate the two red bricks and blue cube by 1, giving a total speed of 3. Finally, the last two red bricks would burn, accelerating the blue cube by 1 and giving a total speed of 4.

The point of all of the above is that the amount of fuel that you need grows very quickly, since you have to have more fuel to lift the other fuel that lifts the other fuel which lifts the other fuel, etc. Tsiolkovsky realized this more than 110 years ago and came up with a formula that describes this phenomena (of course, he knew calculus, which helps to explain things a bit). There are two forms of his equation:

They are exactly the same equation (but probably don’t look like it because of the “e” and the “ln”), but just re-arranged to allow two different questions to be answered:

- If we need the rocket to change speeds by a certain amount (V), and the empty rocket has a given mass (Mempty), how much mass does the rocket have to have at the start (Mfull)?
- If we have a given amount of fuel and a rocket that has a given mass (Mempty and Mfull), how much change in velocity (V) can we get out of the rocket?

One detail that I left out, which was talked about in the last post about chemistry, is that there is a term in the equation that represents the exhaust velocity of the rocket (Ve). If we take the top equation above, there are simplistically two terms of the right hand side: the exhaust velocity and the ratio of the full mass of the rocket to the empty rocket. What this multiplication means is that if you want to reach a given speed (V), you can use less fuel (smaller ratio of masses) if you have a larger exhaust velocity (Ve). The amount of fuel still exponentially increases (this is sort of what the “ln” means), but if you use a fuel with a higher exhaust velocity, you can use significantly less of that fuel. So, you want to get a fuel that will really leave the rocket with as much speed as possible. Then you can use less of it!

You can also use these equations to prove that a rocket with stages is much more efficient that a single-staged rocket. I won’t do this here, but you can think of it conceptually given the diagrams above. Let’s pretend that the black boxes around the fuel and blue cube are different stages of the rocket and that they have mass, which is pretty much exactly how it works. For the biggest rocket (with the blue cube and the 30 red fuel bricks), the rocket will be quite heavy and it will really be hard to get the fuel and everything up into the air. When we burn the 16 red bricks, we then get to drop the gigantic storage tank and some motors and plumbing and all sorts of stuff. The rocket then has significantly less mass. The next 8 red cubes have a MUCH easier job to do in this case, and they can accelerate the rocket much faster. The same is true when the 8 red cubes are done burning and the rocket drops the second stage with the motors and plumbing and stuff for that.

Rockets typically have three or four stages, each with smaller motors (or less motors) and smaller fuel tanks, just as illustrated above. The most efficient rocket in the world would destroy itself as it burned, having an infinite number of stages. That is quite difficult to engineer, though.

Chemical rockets that use fuel like this are about the only thing that we have ever used to get something off the ground. But, there are other methods. Some of them are just scary, and could get you killed by the CIA. Let’s talk about that next time.

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