Let’s Unpack the Pendulum Rocket Fallacy

It's difficult to make a rocket if you are one of the first rocket makers. Sometimes, you simply don't know the best way to design a rocket so you just have to pick something. This is exactly what happened with the early rockets. Robert Goddard's early design put the thruster at the very top of the rocket with the fuel tank at the bottom. The idea was that a top-mounted rocket would make the whole thing more stable. If the vehicle deviated from a perfect vertical motion, its lower center of mass would just make it swing back to the vertical position—just like a pendulum (a mass swinging on a string).

That's not what happens. A top-mounted thruster doesn't inherently make the rocket more stable and a bottom mounted rocket works just fine. Of course, you already know that you can put a rocket engine at the bottom of a rocket since just about every modern rocket does it this way. This idea that a top-mounted rocket makes the vehicle swing back and forth vertically is known as the pendulum rocket fallacy. Let's play around with some physics models so we can figure out exactly what's going on here.

A Rocket Pendulum

I want to build a model that is as basic as possible—but can still show the main properties of a real rocket. It might seem silly, but here is my rocket design.

Illustration: Rhett Allain

Yes, this rocket is just three masses connected by springs. Why? It's because this is the simplest design that would be a "mostly" rigid object but could still be modeled by calculating the forces on individual point masses. For each mass, there are three forces acting on it—the gravitational force and then the two spring forces. If I know the stiffness of the springs and the locations of all the masses, I can find these spring forces. Once I have the net force on each mass, I can use that to update the momentum and position of each mass after some very short time interval—then I can just keep repeating this process to get the motion of the whole rocket.

The cool thing about this method is that I only use simple forces on individual points, but from that I can get the motion of the entire rocket. Oh, I'm also going to plot the center of mass of this thing. OK, let's do this. I'm going to start off with my rocket as an actual pendulum. That means I will fix the position of the top mass and then just let the two bottom masses do whatever the forces tell them to do. But wait! Since this is a rocket, I need to hold it in place with some type of rocket engine. In this case, that rocket will push on the top mass in whatever direction (and magnitude) it needs to keep the thing still so it can act like a pendulum. It will essentially put the rocket into a hovering mode instead of taking-off mode. Here's what I get. Note: this is ACTUAL code. You can see and edit the code by clicking the pencil icon.

Yes, it does indeed swing back and forth—but what about that force? Since the body of the rocket is changing momentum while swinging, this causes the connecting springs to stretch and exert a changing force on the top mass. In order to keep that top mass stationary (for the whole rocket pendulum thing) the thrust force must also change both in magnitude and in direction of the thrust. So, this is not a normal rocket—but it is a fairly normal pendulum. It looks like things are working. That means we can go to the next version of the rocket.

Rocket With a Thruster at the Top

What if we replace the thruster in the previous example with a more realistic rocket? This means that it will have a constant magnitude thrust and always point directly away from the center of mass for the rocket. So, if the rocket rotates, the thrust force will also point in a different direction—you know, like a real rocket engine.