Thank you, gentlemen; good answer(s).
Yes, the needed speed would be less than escape speed, but not by enough to worry about. So no, not with any man-portable weapon, but if you were custom-designing a gun to do it, it’d be a lot easier than shooting from Earth (even before you consider the atmosphere).
No, but it’s worse than that. You’ll notice in the simple formula I cited that the force of gravity (g) is assumed to be constant, and is therefore only valid as an approximation of low-velocity near-surface ballistics, such as the behaviour of a baseball. You can see that the constant “g” makes this equation ignorant of the existence of escape velocity at all! I just used it as a quick approximation but the fairly high velocity relative to the moon’s gravity means it was likely very much on the low side.
To properly model the behaviour of high-speed projectiles where diminishing gravity with distance becomes a factor, you need to get into differential calculus. The problem is even more complicated when you’re approaching the L1 Lagrange point with a much larger body, as you suggest.
The short answer is no, not with any gunpowder projectile. The theoretical muzzle velocity limit obtainable with gunpowder is around 5,000 fps. I agree with wolfpup, the best you can do with this from the moon is 480 km by my pre-coffee calculation, so you wont hit L1 as well.
However! It may possible to make a Coilgun from commercially available components. A multi-stage, 22% efficient coilgun would need about 7000 joules of energy to propel a 2.4 gram bullet to 2400 m/s. I think that’s doable, make the barrel however long you need for the number of stages required, and supply the energy with a capacitor bank/battery combo. 7000 joules (87.5 millifarads capacitor bank) at 400 volts might do it.
For the first few shots, anyway. While some of the heat created by firing will escape in the gasses and ejected cartridges, the rest will be trapped in the gun. Your barrel will probably melt before you empty your magazine.
In fact, I think we proved something very much like that. Over 50 years ago now
Is it? If you tip something into Earth’s gravity well, wouldn’t it just spiral it’s way into the center, or is there a chance it’s going to achieve a sustainable orbit?
My view is that the OP’s goal was to get the shot into Earth’s atmosphere, and ideally the surface, some time soon, not multiple millenia from now. Avoiding having it take centuries probably requires you get the orbital mechanics close to right.
Orbits are not my area of expertise though. I await someone who really knos what they’re talking about.
Your best bet would be to fire the gun towards the horizon, in the direction that goes along with the moon’s rotation, at the moment when this points against the moon’s orbit around the Earth.
So if you stand on the moon’s equator facing east, and fire your gun, its velocity is added to the surface velocity to make it easier to escape the moon’s gravity.
If you wait to do this until the Earth is overhead on yhe exact opposite side of the moon, then as soon as you escape the moon’s gravity you will find that all of the velocity you used to do so is now subtracted from your orbit around the Earth.
If your orbit decays enough because your bullet went fast enough, then the bullet will enter the atmosphere and be captured by it.
As for how the numbers actually work out, I will leave that for people whose expertise in orbital mechanics doesn’t come from Kerbal Space Program. I’d suspect a powerful conventional firearm could launch a bullet into orbit, but not fast enough to overcome the moon’s orbital velocity.
OK, looking at the numbers a little bit, it seems like the moon’s surface rotates very slowly (because it’s tidally locked) so you only get a few km/h by launching from the equator.
Gerald Bull enters the chat…
What about Phobos and Mars? Whilst standing on Phobos, could I shoot Mars?
While the convection provided by the atmosphere will cool a gun somewhat faster than radiation alone in a vacuum it isn’t very much over the timespan of repeatedly firing the weapon, and firearms are designed to fire continuously without coefficient of thermal expansion (CTE) or thermal stress issues through multiple magazines of ammunition. Some belt-fed machineguns were designed with water jackets to cool the barrel but even modern squad automatic weapons will fire several hundred rounds without requiring cooling.
The bigger problem would be lubrication and ‘vacuum welding’—most firearms are lubricated with oils that have some volatile components that will rapidly vaporize in vacuum. There are dry lubricants that don’t suffer this issue that could be substituted but an unprepared gun might well lose all lubrication, and then even a minimal buildup of carbon residue from the propellants may freeze the action solid in vacuum. Also, the lunar regolith is a really fine, electrostatically charged dust which acts kind of like graphite except that instead of being slippery it sticks to everything, causing anything with mechanical joints or action to get jammed up.
It is also possible that if the gun were fired in the dark (approximately -130 °C, -210 °F) that the metal components (particularly hardened steel) might be so brittle that they could fracture under the shock loading. Frames (particularly those made of polymer-based composites) might also warp significantly in the Lunar day with temperatures around +250 °C (+480 °F), so the kinds of problems experienced with the H&K G36 might be manifest in all types of weapons for which those effects have not been considered.
As already noted by several posters, no gunpowder-propelled gun will be able to fire a projectile that will escape the Lunar gravity field, even if fired through the Lagrange points. However, a rocket-propelled projectile capable of achieving the 2.38 km/sec escape speed could feasibly be small enough to be fired from a man-portable firing unit. Actually hitting the Earth, on the other hand, is a significant guidance challenge; it only takes less than 2% greater speed to escape from the Earth-Moon system entirely above achieving Lunar escape speed, and while the Earth looks pretty stationary from the Moon’s surface it is actually speeding along at a fast clip. Figuring the ‘windage’ of such a shot is no mean feat, and it is far more likely that your projectile will end up wandering around in a chaotic solar orbit, leading or trailing the Earth for many millennia before being thrown out into interplanetary space.
Well, you could. But should you?
Stranger
Escape velocity of less than 30mph? Yeah, that wouldn’t be nearly as difficult.
It’s not exactly a sphere but it’s less than 20 miles across no matter where you measure. The earth’s moon is relatively unusual in the solar system for its large size compared to the planet it’s orbiting (with the possible exception of Charon/Pluto if you’re counting it). Phobos is a flyspeck by comparison.
Mars being ~42 degrees in diameter when viewed from Phobos makes aiming at it a bit easier too. Earth viewed from the Moon is ~2 degrees in diameter. You still need to do some orbital mechanic planning to make sure your projectile hits Mars, not enters orbit around it.
XKCD can probably answer this entire thread with a single image:
The images shows the hypothetical depth of each body’s gravity well, rescaled in such a way that its depth indicates how far you’d have to climb, fighting earth’s surface-level gravity the whole way, to obtain the same gravitational potential energy as you would for any given body.
It shows that the moon’s gravity-well depth by this math is 288 km. According to Wikipedia, the 0.220 Swift is a commercial cartridge with the highest available muzzle velocity, 1,422 m/s. Here in Earth’s surface gravity, if you fired that straight up and somehow neglected aero drag, you’d get one of those bullets up to an altitude of 102 km. That’s only about 1/3 of the energy you’d need to escape the moon all by itself. The L1 Lagrange point is about 57,610 miles away from the moon too; even knowing that the moon’s gravity fades as you get to these kinds of lunar altitudes, I doubt that there’s nearly enough energy to get there.
As for Phobos and Deimos, Munroe says you could escape the former with a bike and a ramp, and you could throw a baseball free of the latter.