Machine Gun Jetpack
The principle here is pretty simple. If you fire a bullet forward, the recoil pushes you back. So if you fire downward, the recoil should push you up.
The first question we have to answer is “can a gun even lift its own weight?” If a machine gun weighs ten pounds but only produces eight pounds of recoil when firing, it won’t be able to lift itself off the ground, let alone lift itself plus a person.
In the engineering world, the ratio between a craft’s thrust and the weight is called, appropriately, thrust-to-weight ratio. If it’s less than 1, the vehicle can’t lift off. The Saturn V had a takeoff thrust-to-weight ratio of about 1.5.
Despite growing up in the South, I'm not really a firearms expert, so to help answer this question, I got in touch with an acquaintence in Texas. (Judging by the amount of ammunition they had lying around their house ready to measure and weigh for me, Texas has apparently become some kind of Mad Max-esque post-apocalyptic war zone.)
As it turns out, the AK-47 has a thrust-to-weight ratio of around two. This means if you stood it on end and somehow taped down the trigger (Note: Please, PLEASE do not try this at home) it would rise into the air while firing.
This isn’t true of all machine guns. The M60, for example, probably can’t produce enough recoil to lift itself off the ground.
The amount of thrust created by a rocket (or firing machine gun) depends on (1) how much mass it’s throwing out behind it, and (2) how fast it’s throwing it. Thrust is the product of these two amounts:
Thrust=Mass ejection rate×Speed of ejection
If an AK-47 fires ten 8g bullets per second at 715 meters per second, its thrust is:
=57.2 N≈13 pounds of force
Since the AK-47 weighs only 10.5 pounds when loaded, it will be able to take off and accelerate upward.
In practice, the actual thrust turns out to be up to around 30% higher. The reason for this is that the gun isn’t just spitting out bullets—it’s also spitting out hot gas and explosive debris. The amount of extra force this adds varies by gun and cartridge.
The overall efficiency also depends on whether you eject the shell casings out of the vehicle or carry them with you. I asked my Texan acquaintences if they could weigh some shell casings for my calculations, but for a while they couldn't find a scale anywhere in the house. I helpfully suggested that given the size of their arsenal, really they just need to find someone else who owned a scale. (Ideally someone with less ammo.)
So what does all this mean for our jetpack?
Well, the AK-47 can take off, but it clearly doesn’t have the thrust to spare to lift anything weighing much more than a squirrel.