physics, short, silly, thought experiment

Late for Work

I work a pretty standard 9-to-5 job. Now I know 9 to 5 is actually pretty cushy hours. I’ve got friends whose hours are more like 6 AM to whenever-it’s-done. But my lizard brain won’t get the message that 9 AM isn’t that early a start. Apparently, my brain thinks that getting up at 8 AM is the same as getting up at 3:30 and having to walk ten miles to work (in the snow, uphill both ways).

Luckily, I really don’t like being late, so I manage to be on time by pure stubbornness. But sometimes, it’s a pretty close shave. And while I was driving to work the other day, I got to wondering just how late I could leave the house and have any chance of getting to work on time.

My commute to work is 23.1 miles (37.2 kilometers). According to Google Maps, it should take about 39 minutes, which seems about right. That means an average speed of 35.5 miles per hour (57.2 kilometers per hour). Considering at least half that distance is on the highway at 70 miles per hour (113 km/h), that seems a little slow, but to be honest, there are a lot of traffic lights and weird intersections in the non-highway section, so it probably works out.

But the question remains: how quickly could I possibly get to work? And, therefore, how late could I leave the house and still get to work on time?

The most obvious solution is to convert myself into a beam of light (for certain definitions of “most obvious”). Since there are no vacuum tunnels between here and work, I can’t travel at the full 299,793 kilometers per second that light travels in vacuum. I can only go 299,705. Tragic. Either way, by turning myself into a beam of light, I can get to work in 0.124 milliseconds. So as long as I’m dressed and ready by 8:59:59.999876 AM, I’ll be fine.

Of course, there’d be machinery involved in converting me to light and then back into matter again, and considering what a decent internet connection costs around here, it ain’t gonna be cheap to send that much data. So I should probably travel there as matter.

It’d make sense to fire myself out of some sort of cannon, or maybe catch a ride on an ICBM. The trouble is that I am more or less human, and even most trained humans can’t accelerate faster than 98.1 m/s^2 (10 g) for very long without becoming dead humans. I am not what you’d call a well-trained human. Sadly, I don’t have easy access to a centrifuge, so I don’t know my actual acceleration tolerance, but I’d put it in the region of 3 to 5 g: 29.43 to 49.05 m/s^2.

Figuring out how long it’ll take me to get to work with a constant acceleration is pretty simple. We’ll assume I hop in my ridiculous rocket, accelerate at 3 to 5 g until I reach the halfway point, then flip the rocket around and decelerate at the same pace until I arrive. And since the math for constant acceleration is fairly simple, we know that

distance traveled = (1/2) * acceleration * [duration of acceleration]^2

A little calculus tells us that

duration of acceleration = square root[(2 * distance traveled) / (acceleration)]

Of course, I have to divide distance traveled by two, since I’m only accelerating to the halfway point. And then double the result, because decelerating takes the same amount of time, at constant acceleration. So, at 3 g, I can get to work in 71.2 seconds (reaching a maximum speed of 1,048 meters per second, which is about the speed of a high-powered rifle bullet). So, as long as I’m inside my rocket and have the engines running by 8:58:48.8 AM, I’ll be at work exactly on time. Though after struggling with triple my usual body weight for a minute and twelve seconds, I’ll probably be even groggier than I usually am.

I have no idea if I can even physically tolerate 5 g of acceleration. I mean, I’m hardly in prime physical condition, but I’m not knocking on death’s door either. But I’m gonna venture to guess that anything above 5 g would probably kill me, or at least leave me needing a sick day by the time I actually got to work, which would defeat the whole point. At 5 g, I only need 55.06 seconds to get to work, reaching a maximum 1,350 m/s. So, if I’m in my rocket by 8:59:04.94, I’m golden!

Of course, that was assuming that, for some reason, I do all my accelerating along my usual route. And frankly, if you’ve got a rocket that can do 5 g for over a minute, and you’re not flying, you’re doing it wrong. According to an online calculator, the straight-line distance between home and work is 13.33 miles (21.46 km). Re-doing the math, at 3 g, I can make it to work in 38.18 seconds (meaning I can leave at 8:59:21.82 AM, and will reach 568.1 m/s). At 5 g, I’ll be there in 29.58 seconds (leaving at 8:59:30.42, reaching 936.4 meters per second).

And yet, no matter how quickly I can get to work, I’m still gonna wish I could’ve slept in.

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Supersonic Submarine

I seem to be obsessed with submarines, which isn’t something I realized until now. There’s nothing worse than a stealth obsession.

Anyway: sound has a speed. That speed depends on the properties of the medium. The speed of sound in air is about 330 meters per second. The speed of sound in olive oil is 1430 meters per second (Yes, somebody measured that, and here’s the proof, along with some other handy tables of speeds of sound in other materials). The speed of sound in aluminum is 6320 m/s. The speed of sound in beryllium is an amazing 12,900 m/s, which is not only faster than the International Space Station’s orbital velocity, it’s actually faster than Earth escape velocity.

The speed of sound in seawater is a much tamer 1500 m/s (the exact speed depends on depth (meaning pressure), temperature, and salinity). That got me thinking that, since I’ve abandoned the submarine car in favor of an actual submarine, why not make it a supersonic submarine?

There’s nothing in the laws of physics to stop me. There’s no physical reason that makes it impossible to move through water faster than the speed of sound in water. There are plenty of engineering reasons, but we’ll get to those in a second.

The interesting thing about moving supersonically in water is that water isn’t a gas. Air isn’t very dense, it’s compressible, and it doesn’t have many phase transitions readily available. It can liquefy if you compress it while keeping it cool, and it can turn to plasma if you compress it and let it heat up. But when you’re talking about supersonic vehicles, the air heats up rather than cooling down. It heats up a lot. The air around re-entering spacecraft turns into plasma.

Water, on the other hand, is much denser (pure water is about 1,000 kilograms per cubic meter), and compared to air, is almost incompressible. Water is about five orders of magnitude less compressible than air. This means that a whole slew of new phenomena happen in supersonic submarines that don’t happen in supersonic aircraft. The coolest one is cavitation.

Cavitation is what happens when, for one reason or another, the pressure on a volume of water drops below that water’s vapor pressure, or when something moves through the water so fast that the cavity in the water doesn’t have time to close around the object. There are all sorts of cool videos of cavitation on the Internet, but I think this is my favorite:

Ain’t that beautiful? Many thanks to The Slow Mo Guys and Smarter Every Day for filming that, and for doing exactly what I would have done if I had access to one of those slow-motion cameras.

Notice the large cavity that opens behind the bullet as it travels. The spherical cavity around the gun’s muzzle is from the blast of hot, escaping gas, but the sort of sausage-shaped bubble attached to the bullet is pure cavitation. The bullet slams the water aside so hard that, even though water is usually very good at closing voids within itself, it has no choice but to stand aside for a fraction of a second. For the brief period that it exists, that cavity is full of a little water vapor that evaporated from the surface and not much else, and as soon as the moving water has deposited its inertia in the stationary water around it, pressure wins out and makes the bubble collapse again.

But a cavitation bubble isn’t the same thing as a sonic boom. The bullet in that video was fired from a revolver. Since I don’t know the make of the revolver or what kind of ammunition it was using, I don’t know the muzzle velocity, but if we assume it was in the same class as a Ruger firing .357 Magnums, then the muzzle velocity would have been around 450 meters per second. Not faster than the speed of sound in water. Barely faster than the speed of sound in air.

Either way, we know that our supersonic submarine would cut quite a large hole in the water as it flew. (Flew? That doesn’t sound right. What is the right verb for a submarine’s movement? Somebody let me know. That’s gonna bother me now). It would also, true to acoustics, generate a sonic boom. I would guess that this sonic boom would be more than enough to rupture the eardrums of unlucky divers who happened to get in its way, and that the drop in pressure after the shock would probably create a whole swarm of smaller cavitation bubbles in its wake. And because the water that evaporated from the surface of the cavity would be moving roughly in the same direction as the cavity (relative to the submarine), the submarine would likely create a second, much slower-moving sonic boom in the water vapor. After the submarine passed, the cavity would expand to a maximum size, then slam closed, possibly heating the gases inside enough to glow. This is called sonoluminescence, and is very impressive:

After the collapse, you’d have a soup of very hot bubbles and very hot water vibrating and rising to the surface. The water would be hot from the collapse of the cavity. Here’s about what our supersonic sub would look like:

Supersonic Submarine

And, from a practical perspective, it would be hot for another reason. To break the speed of sound in water, you’d need the engine power of 4 Saturn V moon-rockets.

Yes, really. This comes from the basic drag formula I’ve been using all along:

drag force = (1/2) * (density of medium) * (velocity of object)^2 * (drag coefficient (depends on shape and texture of object)) * (projected or cross-sectional area of the object)

I have no idea where we’re going to get a rocket four times as powerful as a Saturn V. I guess we could just make the end of the submarine a parabolic reflector and drop antimatter out the back and ride the blast of steam, but I hear people get pretty upset if you go dumping antimatter in the ocean. Especially if they happen to be swimming behind you.

But that’s the least of our worries. At 1500 meters per second, the front of the submarine would be experiencing pressures ten times greater than at the bottom of the Mariana Trench. Not unsurvivable, but between the pressure of the water against the front of the hull and the cavitation going on around the back of the hull, the whole thing’s going to need to be a pressure vessel. That’s going to be one heavy submarine. While we’re pretending that a submarine-sized craft could produce 141 million Newtons of thrust for an extended period, why not just turn the bastard into a rocket? Besides, I’m afraid that if I tooled around underwater making watery sonic booms, I might upset an octopus, and I have a deep and inexplicable affection for octopuses.

But before we stop doing weird things underwater, there’s a question that demands to be answered: if our supersonic submarine would need four times the thrust of a Saturn V to travel through the water, how fast would the Saturn V itself be able to go underwater. Well, input some reasonable values into the drag equation, set the drag equation equal to 35 million newtons (the Saturn V’s first-stage thrust), and we have:

41.2 meters per second

or

92 miles per hour

or

148 kilometers per hour

The Saturn V is one of the most powerful rockets ever built. And, under ideal conditions, it could manage 92 miles an hour underwater. I have driven my car faster than that. A good baseball pitcher or cricket bowler can throw faster than that. I guess the people at NASA weren’t planning for the possibility that the space between the Earth and Moon might inexplicably be filled with seawater. The fools.

But, although 92 miles an hour is not a very impressive speed, especially by rocket standards, you have to admit, it’d be one hell of a sight to behold:

SaturnV

Now that‘s a fucking torpedo!

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