# What is the maximum viable speed for a projectile within earth's atmosphere?

Assume a projectile,

made of a very heat-resistant and durable material (something like tungsten), formed like a modern saboted tank cannon shot.

If tungsten, like suggested, is not ideal, you can choose whatever material you think would be ideal to fulfill the mission as defined below.

If the mechanism of accelaration is taken care of by my science-fantasy means - and this mechanism and aiming etc. is not part of the question - what would be the maximum viable velocity within our atmosphere?

At what velocities would the (aerodynamically shaped) projectile just burn up? If somehow the burn-up-problem would be solved (handwaved), what would be the next threshold? Is there a velocity that would just make matter disintegrate?

I was thinking about having it fly at around 75000 km/s (~0.25c), but somehow I got a feeling this would not be entirely viable :)

Definition of viable for this question: "Emerge" already at full velocity within earth's atmosphere, travel for 100-10000 m in a straight line, hit a target while still being a solid lump of matter.

If you think about telling me what a impact will look like, please wait a bit, I will ask that question next.

Image from Wikipedia

• Comments are not for extended discussion; this conversation has been moved to chat. Sep 16 '19 at 20:00
• I'm not saying this would change any answer, but I want to mention that you probably don't really need a solid lump of matter. What on earth are you hitting with this, that could stand a hit from liquid/gas/plasma at 0.25c but not from a solid slug at 0.25c? Sep 18 '19 at 15:42
• @Blueriver I was more concerned about accuracy and predictability
– user6415
Sep 18 '19 at 18:52
• @openend that's understandable. But I don't think they would be impacted either, at least if your projectile is solid at launch (which it is, if you're accelerating it with a handwaived gun). Storage is also important, and also solved in your question. You could look into existing railguns to see if they have any accuracy or predictability issues that would be solved with a solid projectile. Anyway, it was just food for thought Sep 18 '19 at 19:32

The answer depends on a mighty host of factors, including how pointy your projectile needs to be.

However, we can put together a pretty reasonable upper bound by looking at reentry vehicles. They are pretty much the fastest manmade things in the atmosphere. Apollo 10 came in at roughly 11km/s. The record fastest reentry vehicle was Stardust, which came in at 12.9km/s. These vehicles all have a very blunt front end covered in ablative material. Heating due to compression of the air at supersonic speeds is a really big deal at these speeds, and blunt shapes do a better job of keeping the hot shock wave further from the body of the craft.

It will be harder to make a small projectile fast. Heating is much more of an issue when you're small. Navy railguns under test throw projectiles at 2.5km/s. Of course, you're handwaving a lot of difficult things, so you should be able to beat them.

The fastest objects in the air are meteors. They can hit 72km/s. However, most of them don't stay together under that strain, so it's not clear whether they count.

Given that I have been talking about 2-72km/s, it should be rather clear that 75,000km/s is a whole different world. I hate to say never, because human innovation is an incredible thing, but I don't think we'll ever be able to travel at those speeds in the atmosphere. If you did, it would certainly be quite the spectacle. The energy of a small projectile in that range is on par with that of atomic bombs, and I would expect the compression heating in front of the shock wave to be hotter than the sun.

And, of course, I would be remiss if I didn't point out that there's an XKCD on the topic. As always, Randall Munroe has a way with words and pictures:

• "A careful reading of official Major League Baseball Rule 6.08(b) suggests that in this situation, the batter would be considered "hit by pitch", and would be eligible to advance to first base." I love XKCD. Sep 16 '19 at 5:14
• Thanks. That xkcd links even tells me how it would look like.
– user6415
Sep 16 '19 at 8:47
• Yes, well its a story, so 'developing fireball' *plasma front' and 'devastation in the vicinity' plus crater will suffice :)
– user6415
Sep 16 '19 at 8:53
• The reentry speeds you quote are at the top of the atmosphere, where the air density is many orders of magnitude lower than at sea level. Once they reach the thick layers of the atmosphere, reentry vehicles travel at subsonic speeds. Sep 16 '19 at 15:30
• @FlyingLemmingSoup Love XKCD and Randall Munroe, but in this case I think that since the batter, the pinch runner, the umpire, both teams, the stadium and everyone in it, as well as a substantial portion of the city it resides in would be vaporized, the correct ruling would be "Game called on account of Inclement Weather" (or merely suspended if it is after the fifth inning). Sep 17 '19 at 12:55

At what velocities would the (aerodynamically shaped) projectile just burn up?

Only a few km/s. Read up on the Sprint missile, which could reach Mach 10 in 5 seconds (which would be about 3.5km/s, though that slightly depends on the altitude it had reached at that point) which resulted in skin temperatures of 3400 degrees C and needed an ablative heat shield. The Sprint also went up pretty quickly to try and get out of the densest part of the atmosphere ASAP... shooting horizontally at sea level, you'd have much greater heating to deal with.

In any case, range at low altitude and high speeds is clearly limited by the lifetime of your ablative heat shield. On the bright side, it'll look awesome as it'll form a plasma trail. Hope you weren't intending to be at all stealthy!

If somehow the burn-up-problem would be solved (handwaved), what would be the next threshold? Is there a velocity that would just make matter disintegrate?

When the stagnation pressure behind your bow shock exceeds the yield strength of your projectile. Or to put it another way: the force of the air ramming into the front of your projectile is greater than the strength of the bonds holding your material together, and it will simply break up like a jet of fluid, starting at the tip. It'll be eroded away all down its length til nothing is left but an expanding cloud of fragments, which will probably resemble an explosion.

Due to the lack of studies on hypervelocity projectile weapons (along with the lack of such weapons), I'll turn to papers on shaped charges. The impact pressure of a shaped charge jet with a solid object exceeds the yield strengths of any material, so both can be treated as incompressible fluids splashing against each other. Obviously there are differences between a soft shaped-charge jet penetrating solid armour, and a hard projectile penetrating air (for a start, air is definitely not an incompressible fluid), but the basic idea is the same so this figure probably isn't too wrong. Take the figures with a small pinch of salt, but they're a good first guess.

Looking at studies of shaped-charge jets, you get handy formulas like $$P = L\sqrt{\frac{\rho_j}{\rho_t}}$$ where $$P$$ is the penetration depth, $$\rho_j$$ and $$\rho_t$$ are the densities of the jet and target respectively and $$L$$ is the length of your projectile. In this case, the "target" is the atmosphere itself. A half-metre long tungsten rod will therefore travel $$0.5\sqrt{19300 / 1.225}$$ or $$62m$$ max through the air once it had exceeded this critical velocity, disintegrating as it went, regardless of its temperature.

What would be that velocity threshold, you ask? Well, uh, I haven't the faintest idea (aside from "higher than re-entry speeds"). Compressible aerodynamics turns out to be Quite Hard, and my previous efforts to wrestle with it came to nothing. But now you know the magic words to search for, so maybe you'll have more luck than me.

If I had to hazard a guess though, I'd look at shaped charges again. The tip of shaped charge jet travels at about 10km/s. Armour is about 10000 times more dense than air. The force exerted by a fluid jet is $$F \propto {\rho}v^2$$, so you'll need your projectile to travel about $$10km/s * \sqrt{10000} = 1000km/s$$ to develop the same forces from impacting air. This is a very, very loose approximation and shouldn't be taken too seriously. It does seem plausible though, give or take an order of magnitude.

I was thinking about having it fly at around 75000 km/s (~0.25c)

This is more than 10 times faster than the 1000km/s threshold I invented above, so I'm pretty certain that there's no way that a projectile travelling this fast could stay intact even if you did handwave all the heating issues away.

Definition of viable for this question: "Emerge" already at full velocity within earth's atmosphere, travel for 100-10000 m in a straight line, hit a target while still being a solid lump of matter.

Well, if your projectile was long enough, some of it might survive to hit the target. You'd need to fire it perfectly straight though... any deflection or manufacturing defects will cause bits of it other than the very tip to be blasted off, and that will a) ruin your accuracy and b) ruin your range. You're clearly limited in how long you can realistically make your projectile, and given the speeds you want it'll almost certainly have to be too long.

• "Hope you weren't intending to be at all stealthy!" I can't imagine much of a reason to be stealthy at 0.25c within an atmosphere; you reach your target within half a second, max, or you leave the atmosphere and the scope of this question. Sep 16 '19 at 13:59
• @Skyler yeah, but the shot might end up being visible to the naked eye from space, even if you miss the target. Sep 16 '19 at 14:15
• @Skyler Stealth mode: everything that was looking at it is dead within seconds =p Sep 16 '19 at 14:38
• Interesting the square root -- a Newtonian Penetrator has the same equation, without the square root.
– Yakk
Sep 17 '19 at 17:28
• @Yakk looks like it comes out of fluid mechanics, and the force exerted by a jet against a moving target: $F = \rho_jA(v-u)^2$ where A is the cross sectional area of the jet. When you model this as two jets moving against each other, the point at which the jet stops is $\rho_j(v-u)^2={\rho_t}u^2$ where $v$ is the jet velocity and $u$ is the target velocity (given a stationary coordinate system centred on the tip of the jet) and the areas are the same and so cancel out. Given a jet of length $l$, $P = {lu}/(v-u)$, and from the last equation, $u/(v-u) = \sqrt{\rho_j / \rho_t}$. Easy as that! Sep 18 '19 at 8:28

Just as food for thought: Some fraction of c (light speed).

First, you fire a laser or something that will ionize the air between yourself and the target (= create a plasma). Think big: Something that ionizes a channel that is between 100 and 1 km wide. Bigger is better.

Then you use electronic and magnetic fields to move as much plasma out of the way as possible. A good shape would be a cylindrical field which moves the ionized gas away from the center. Of course, air will try to get back in but that will take a moment (a lot of molecules will try to get away and the cold molecules need to wrestle their way in).

Getting the plasma to move towards the target could be a bonus.

Use the channel with reduced air pressure to fire your projectile. You might be able to achieve a temporary near-vacuum this way which would allow you to fire with the speed of light - if you can accelerate your projectile to those speeds.

Notes:

• Everyone living within 1000 miles of the combat zone will hate you - this approach will fry electric devices for miles around.
• Creating the channel will probably cost more than just firing several (slower) slugs.
• The initial blast with the laser will cause a lot of damage at the target; you most likely will not need the projectile ...
• This will disrupt radio waves for days, maybe weeks. Say goodbye to mobile phone service during that time. Everyone on the planet will hate you. Pitchfork & torch kind of hate, not just Facebook cybermobbing.
• The collateral damage will make this a hard sell to even the military.

My conclusion: Just drop tungsten rods from low orbit (a.k.a "Hammer of God"). Takes a minimum of acceleration (or they would just stay in orbit), cheap to make, easy to scale. The tip will be vaporized by the heat of reentry so you probably need some kind of ablative heat shielding which pushes the air around the body so it flies/falls in a cavity.

• Regarding the cheapness of rod from god... a 6*0.3m tungsten cylinder isn't that cheap to make, and is Quite Expensive to put in orbit. Its terminal velocity will be about Mach 10. You'll probably find a boring old railgun or even a scramjet to be a more convenient alternative, and both can be reasonably fired from the surface, too. Sep 16 '19 at 14:23
• The plasma will move out all by itself: it will expand because of its high temperature. If it is created very hot and very fast, this will be an explosive expansion that should leave a very low density gas on the laser axis... for a fraction of a second. Sep 16 '19 at 15:17

## Real-world example: Operation Plumbbob

Cort's answer to this question is pretty great, but makes two faulty assumptions. One, the reentry vehicles aren't the fastest objects humans have ever made, and two, they're interacting largely with the top of the atmosphere. As evidenced by the host of other answers, the theoretical side of things is difficult at best to figure out, so we'll be best off going with real-world data.

Operation Plumbbob was the codename for a series of US nuclear tests. Pascal B is the one we're interested in - in which a steel cap was welded onto the top of the pressure release. This steel cap was then launched by the nuclear blast into the atmosphere at estimated speeds of >60km/s. It's assumed that this hunk of metal made it into space, but given that we only managed to capture a single frame of its existence, we may never know.

Assuming you have the ability to produce such an acceleration, the main concern is remaining "solid". However, at such high speeds, literally everything acts like a fluid and you're essentially "spraying" molten metal through the air as a series of droplets - which is really up to you as to whether you consider such a "shotgun" effect still a solid object.

• Considering that Escape Velocity from the Solar System is 42km/s, someone (somewhere, somewhen) is going to have a very bad day... I wonder if this test was proposed by Serviceman Chung? Sep 16 '19 at 23:10
• A truly informative answer. Basically eclipses everything else on this page. Well done. Sep 18 '19 at 2:53

For a given value of "projectile" and an arbitrary distance...

Anatoli Bugorski was hit by a particle accelerator beam in the face. The accelerator in this case was the U-70, which can impart 76GeV on a proton beam. At those energies each proton would be travelling at close to 99.99% of the speed of light. You don't need to worry about the projectile disintegrating at short distances, but then again this is a weapon that weights 200 metric tons and can only fire at targets up to a couple meters away from the muzzle. You might wish to use it only for executions.

• Yep, the answer is c. I thought of the Oh-My-God particle. With a small enough projectile, its stability will be retained even while moving at relativistic speeds. Sep 17 '19 at 4:30

The jet from a shaped charge is one of the fastest objects on earth: formed from a flat metal liner blasted into a narrow line by a conical explosive charge, it is still essentially solid (many erroneously describe it as a liquid), but at that speed it does not behave like one.

Shaped charge jets are most effective when dense metals are used (eg tungsten) but lighter metals are generally more practical (eg copper). They are used to penetrate thick tank armor, and the fact that you can do this with copper tells you something about the odd physics at these speeds. Pushing stuff (even air) out of the way is a challenge, and because the jet moves faster than a shockwave through it, it effectively behaves as a stream of particles rather than a single object.

So your projectile will 'hang together' but will ablate rapidly depending on how much heat it absorbs. When penetrating armor, a hydrodynamic process operates: it erodes the armor at the same rate is the jet is eroded. If the same applies with air - and I don't think anyone has fired anything fast enough to confirm this! - then 1 kg of penetrator will be eaten up by 1 Kg of air. So a 10-ton block of steel (1 metre cube) would get through 10,000 metres of air before being entirely vaporized.

This is why long, narrow jets work better than aerodynamic blobs for armor penetration at such speeds.