Machine Gun Jetpack: The Real Physics of Improbable Flight
Explore the contents of this article with a free Wolfram SystemModeler trial. Could you fly using machine guns as the upward driving force? That’s the question asked in Randall Munroe’s What if? article, “Machine Gun Jetpack.” It turns out you could, because some machine guns have enough thrust to lift their own weight, and then some. In this post, I’ll explore the dynamics of shooting machine guns downward and study the actual forces, velocities, and heights that could be achieved. I’ll also repeat the warning from the What if? post: Please do not try this at home. That’s what we have modeling software for.
Machine gun with a squirrel on top
Let’s start smaller than a human, with a gray squirrel from the original story. Put this squirrel on a machine gun, fire it downward at the full automatic setting, and see what happens. I’ll be using Wolfram SystemModeler to model the dynamics of this system.
Model of a machine gun
The image above shows the model of a machine gun. It contains bullet and gun components that are masses that are influenced by gravity. They are easily constructed by combining built-in mechanical components:
Mass influenced by the Earth’s gravitational force
The magazine component is a little more advanced because it ejects the mass of the bullet and the bullet casing as each shot is fired. It does this by taking the initial mass of the full magazine and subtracting the mass of a cartridge multiplied by the number of shots fired, which is given by the shot counter component.
Combining this together with a simple model of a squirrel, a sensor for the position above ground, and a crash detector that stops the simulation when everything crashes on the ground, I now have a complete model.
To get a good simulation, I need to populate the model with parameters for the different components. I will use a gray squirrel, which typically weighs around 0.5 kg (around 1.1 pounds).
Then I need some data for our machine gun. I’ll use the ubiquitous AK-47 assault rifle. Here is some basic data about this rifle:
The thrust generated by the gun can be calculated from the mass of the bullet, the velocity of the bullet when leaving the muzzle, and how often the gun is fired:
I can then estimate the percentage of each firing interval that is used to actually propel the bullet through the barrel. I am going to make the assumption that the average speed in the barrel is equal to half the final speed:
The force during this short time can then be calculated using the thrust:
Now I have all the parameters I need to make our squirrel fly on a machine gun:
Now we simulate the squirrel on the machine gun with a single bullet in the gun:
Seeing the height over time, I conclude that the squirrel reached a height of about 9 centimeters (3.5 inches) and experienced a flight time of only 0.27 seconds.
To put it another way:
That didn’t get the squirrel very far above the ground. The obvious solution to this? Fire more bullets from the gun. A standard magazine has 30 rounds:
This gives a flight time of almost 5.8 seconds, and the squirrel reached the dizzying height of 17.6 meters (58 feet). Well, it would be dizzying for humans; for squirrels, it’s probably not so scary.
Now we’re getting somewhere:
I have shown that a squirrel can fly on a machine gun. Let’s move on to a human, going directly for the standard magazine size with 30 bullets:
One gun is not enough to lift a human very far. I need more guns. Let’s do a parameter sweep with the number of guns varying from 1 to 80:
This shows some interesting patterns. The effect from 50 guns and above can be easily explained. More guns means more power, which means higher flight. The simulations with 15 and 32 guns are a little more interesting, though. Let’s look a little closer at the 15 guns scenario. The red dots show the firing interval, meaning the guns shoot one bullet each every 0.1 seconds:
You can see that the craft manages to take off slightly, starts to fall down again, gets off another shot, but then falls farther than the height it had gained. You can also look at the velocity over time:
For the first shot, the craft starts at a zero velocity standing still on the ground. It gains velocity sharply, but before getting off the next shot, the velocity falls below zero. This means that during one firing cycle, there is a net loss in velocity, resulting in the eventual falling down, even though there are bullets left in the gun. It could then start over from standstill on the ground, doing tiny jumps up and down.
The scenario with 32 guns exhibits yet another behavior. The start looks similar to the behavior with 15 guns, where it gains some altitude, but then falls back down because it loses net velocity during each firing cycle. But then at around 2.5 seconds it starts to gain altitude, until all the ammunition is spent at 3 seconds.
This can be explained if you look at the mass of the magazine over time:
You can see that at each shot, the magazine loses weight because it ejects a bullet and a bullet casing. After a while, this makes the whole craft light enough to gain altitude. This indicates there is some limit to how many bullets you can carry for each machine gun and still be able to fly, which is another interesting parameter you can vary. Let’s try to fly with the following magazine sizes for an AK-47, assuming I create my own custom magazines:
Because more guns means more power, I will use a large number of guns, 1,000:
When using 1,000 guns, it turns out it is not a good idea to bring 165 bullets for each gun:
This is because if you bring too many bullets, the craft becomes too heavy to gain any altitude. Now that I have found a reasonable (if there can be anything reasonable about trying to fly with machine guns) number of bullets to bring along, let’s see the achieved heights when varying the number of guns. I would expect that with more guns, we will gain more height and flight time.
Here is the maximum height achieved with the different number of guns:
It turns out that increasing the number of guns drastically (from 1,500 to 50 million) only gives a marginal increase in the top height achieved. This is because as the number of guns increases, the part of the human carried by each gun decreases, until each gun only carries its own weight plus very little additional mass. This makes the total craft approach the same maximum height as a single gun without any extra weight, and adding more guns will give no more advantage.
In closing, the best machine gun jetpack you can build with AK-47s consists of at least around 5,000 machine guns loaded with 145 bullets each.
How high you can fly using machine guns
Download this post, as a Computable Document Format (CDF) file, and its accompanying models.
This post brought to mind Jerry Pournelle’s 1980 science fiction novel, King David’s Spaceship, in which a backward planet sends a human into orbit in a capsule propelled by rapid-fire canon. Perhaps the techniques discussed in this blog post could be applied to determine the characteristics required of a one or more rapid-cannon capable of accomplishing what Pournelle describes.
You may also search for “Nuclear pulse propulsion” and project orion or daedalus.
that’s the idea behind Project Orion (nuclear propulsion): http://en.wikipedia.org/wiki/Project_Orion_%28nuclear_propulsion%29
Can you model the tank flying scene from the movie “A-Team”.
More than application iself , I find the use of modelica what is really awesome. Simple and elegant way to solve the problem
This whole idea is a ripoff of one of the “What If?” from Randall (XKCD), just with worse drawings and more graphs.
As I state in the beginning of the post, the idea comes from Munroe’s “What If” post. As a big fan of Munroe’s work, I wanted to try expanding on his post with a different approach, namely modeling and simulation software. I think our graphics nicely illustrate the various ideas and results.
Interestingly, this is precisely the technology originally under development by Robert H. Goddard in the early 20th century. Indeed, he stuck to this approach until finally pressured by his funding source (the Smithsonian Institution) to actually show some progress — at which point he cobbled together a liquid fuel rocket (using pressure feed) just to get something successfully off the ground. He ultimately realized that liquids were far more practical and that pumps were the key to success.
I agree with Malte
I encourage you to try it with the model attached at the end of the post! It should be pretty easy to vary with WSMParameterValues, as the emptyMagazineMass is a parameter in the model already.