Category: Physics

The Physics of Getting Hit By An Arrow

February 17, 2014 / Ryan / 4 Comments

So, while I was reading “Genghis: Birth of an Empire” I noticed that there are quite a few instances when the book describes a man being knocked down by an arrow. As someone who reads and watches a lot of fantasy and historical fiction, this was far from the first time I’ve seen such a thing, but as a physics person I wondered whether an arrow would really hit someone with enough force to knock them down.

Now, I’m not disputing that getting hit by an arrow might cause someone to lose their balance and fall off of a horse, or off of a castle wall or the like. The question here is how hard the arrow actually would hit. For anyone who remembers their high school physics, that means we’re talking about a momentum problem. We need to figure out how much momentum a typical arrow would have, and figure out what that amount of momentum would do when transferred to a human. Specifically, this type of problem is an “inelastic collision”, where two bodies collide and then stick together. If the arrow somehow bounced off the person, that would be an elastic collision, but we won’t worry about that.

We are going to assume that our hapless victim is standing still and the problem starts with an arrow speeding toward them. To find the arrow’s momentum, we need to know its mass and velocity. For an estimate of the arrow’s mass I will go with 65 g based on this source. The velocity depends on the draw weight of the bow that fired the arrow, but generally I am seeing arrow velocities of around 200 feet per second (61 m/s) for longbows. Momentum is just mass x velocity, so the momentum of a 65 g arrow going 61 m/s is 3.965 kg*m/s.

I don’t know about you, but that doesn’t mean very much to me, so let’s figure out how fast some more familiar objects would have to be going to have the same momentum, starting with a baseball. A standard baseball has a mass of ~145 g. So a baseball with the same momentum as our speeding arrow would be going 27.3 m/s, or 61 miles per hour. We can do the same calculation with a soccer ball, which has a typical mass of 450 g. A soccer ball with the same momentum as our speeding arrow would have to be going 8.8 m/s, or ~20 miles per hour.

A 60 mile per hour baseball or a 20 mile per hour soccer ball don’t really have enough oomph to knock a person over unless they’re already off balance. So, a typical arrow would not knock a man down, or stop him in midair, or the like. He might fall down because he suddenly has an arrow sticking in him, but it’s the damage done by the arrow, not its momentum that would make him fall.

Jupiter as the Moon

July 9, 2013 / Ryan / 1 Comment

Have you seen this awesome set of pictures depicting what it would look like if you replaced our moon with each of the planets? I particularly like the images for Jupiter and Saturn – they absolutely dominate the night sky:

Jupiter, as it would look if it were our moon.

I always enjoy pictures like this: they are a great way to illustrate the relative sizes of the planets. But the following tweet made my skeptic senses tingle:

Now, Ezra Klein is a smart guy, and I have followed his blog for a while for his excellent political coverage, which is nice and wonkish, with lots of graphs and charts. I’m a scientist, I like me some graphs and charts. But as much as his blog loves to factcheck political stuff, he failed to fact check this fact check.

The tweet in question says that the Earth and Jupiter would collide if they were the same distance apart as the Earth and the Moon. it’s not that this is wrong, but that it doesn’t state its assumptions and so it is misleading. I was tempted to reply on Twitter, but this is the sort of thing that, alas, involves lots of caveats so I thought it would be better to tackle it here.

The main question is what the motion of the two planets is when you suddenly replace the moon with Jupiter. If both planets are stationary relative to their center of mass, then yeah, they’re going to collide. But the same would be true of the Earth and the Moon. If the two planets have any angular momentum at all, they will orbit their center of mass rather than simply falling toward each other. So that means that in all cases except that in which the planets start at rest, they won’t hit each other.

But not quite. Because planets have some size. If they don’t have enough angular momentum, they will follow a very narrow elliptical orbit and will crash into each other. To avoid collision, there has to be enough angular momentum for the periapsis of the orbit to be greater than half the sum of the planets’ radii. I tried switching Jupiter for the moon, but otherwise keeping the angular momentum and orbital eccentricity of the system the same and I found that the periapsis is about 4400 km (I did this calculation quickly – my numbers might be off so feel free to check me!). This is too small to keep Earth and Jupiter from colliding, so the tweet is correct, given the assumption that the angular momentum and eccentricity of the system stays the same when you swap the moon for Jupiter.

But what bothered me was that the tweet could be read to imply that an Earth-sized object and a Jupiter-sized object could never comfortably orbit each other, and that’s just not true. With enough angular momentum, an Earth-sized object could be in a stable orbit around Jupiter.

So, let’s say the system has enough angular momentum for the Earth to stay in a stable circular orbit at the specified distance. What other misfortune might befall us? One possibility is that we would be inside the Roche limit and the powerful tides from Jupiter would tear us apart. I calculate the rigid-body Roche limit for Jupiter and Earth to be ~108,900 km, which is safely less than the Earth-Moon distance.

That’s not to say the tides wouldn’t do some weird things of course. Jupiter’s tides would still be huge and so it would raise enormous tides in our oceans and in the solid rock of the Earth itself. The friction generated by these tides would heat up the Earth’s interior (and probably trigger volcanoes and earthquakes in the process) and gradually slow the Earth’s rotation until one side of the earth was permanently locked facing Jupiter (the same thing has already happened to the moon, which is why we don’t see the far side except from spacecraft).

We would also be pounded by the intense radiation of Jupiter’s magnetosphere. Our magnetic field might protect us for a while, but as our spin rate slowed, our magnetic field would die, leaving us exposed. It’s possible that the radiation would then strip away our atmosphere, leaving the earth a desiccated and dead world. I’ll leave that calculation as an exercise for the reader. Correction: our orbital period around Jupiter would actually not be much more than a day, so our magnetic filed would probably survive.

So, bottom line: the tweet in question is only sort of right. Earth could stably orbit Jupiter, just not with the same orbital parameters as the current Earth-Moon system. If we were in a stable orbit, there would be some… interesting side effects, which might prove to be deadly, but nothing as dramatic as crashing into Jupiter.

How do Zerg Fly in Space? Part 1: Mutalisks

October 18, 2010 / Ryan / 41 Comments

Ok folks, it’s time. We’ve all asked this question, but I’ve been putting off answering it because we all actually know the answer.

How do the Zerg fly in space?

Magic.

There’s a certain point in sci-fi or fantasy where you have to just suspend disbelief and go along for the ride, and I think that the Zerg ability to travel through space is a good example of this. That said, I’d like to take a look at one of the most common explanations that people give to justify the mutalisk’s ability to flap its wings and propel itself through space. As you’ll see, it’s completely implausible.

Now, we know that flapping doesn’t do anything in a vacuum: there’s nothing to provide any resistance, so you can flap all you want and it won’t move you forward. But what if mutalisks used their dragon-like wings as solar sails, catching the photons from a nearby star to cruise through interplanetary space? That might not explain the flapping, but it could explain how they can move, so let’s take a closer look.

The idea behind solar sails is the conservation of momentum. Even though photons of sunlight have no mass, they do have momentum. High school physics tells us that momentum is conserved, so if you have a bunch of photons with momentum being absorbed by a solar sail (or a mutalisk’s leathery wing) then their momentum must be transferred to the thing they’re hitting, exerting a force on it and causing it to move through space. So, how large would a mutalisk’s wings have to be to let it accelerate at a reasonable speed? To figure this out, we need to do a back of the envelope calculation, making some assumptions about how big mutalisks are.

In general, the sizes of units in the game are not reliable: I prefer to consider the cinematics as the authoritative source. So let’s take a look at this cinematic showing Jim Raynor’s battlecruiser being attacked by a swarm of mutalisks.
[youtube=http://www.youtube.com/watch?v=pOh1YoaPbi0&w=640&h=385]

At about six seconds, one of the mutalisks flies over the right-hand side of the battlecruiser and crosses near a long row of windows. Based on its size compared to the set of windows, I would say that the creatures have a wingspan of around 100 meters and that their tube-like body is about the same length and maybe 5 meters across. If mutalisks are like most earth life, then they are mostly water. to get a rough idea of their weight we can calculate their volume and then multiply by the density of water. A cylinder 5 meters by 100 meters has a volume of about 2000 cubic meters. That would correspond to a whopping 2,000,000 kg or 2000 metric tons!

Now, obviously that’s too big for something to fly in at atmosphere, let alone in space by flapping its wings. But lets be generous and say that maybe mutalisks are made of some very low-density material, and maybe I overestimated their size. What if they were closer to 20 tons? How much oomph would their wings give them if they used them as solar sails. Again I’ll be generous and treat the wings as a square of material 100 meters on a side.

The momentum of a photon is given by: Momentum = Energy/Speed of Light, so near a sun-like star, where most photons are in the visible range, they have an energy of 2-3 electron volts, yielding a momentum of 1.6×10^-27 kg m/s per photon. That’s not much, but a star puts out a lot of photons, so let’s see if that balances it out and gives us a decent thrust.

Let’s say our theoretical mutalisk is orbiting the sun at the same distance as Earth, 150,000,000,000 m from the sun. The sun puts out 3.8×10^26 watts or roughly 8×10^44 photons per second, but that power is spread out in all directions. To figure out how much hits our solar-sailing mutalisk, we have to imagine spreading that power out over a sphere the size of our mutalisk’s orbit with a surface area of 4*pi*R^2 where R is the radius of the orbit. That gives 2.25×10^20 photons per square meter per second.

Solar sails don't make for the most maneuverable spacecraft, and they would be sitting ducks in any sort of space battle.

With a wing area of 100 m x 100 m (10,000 square meters), our mutalisk would intercept around 10^24 photons per second, corresponding to a whopping force of 0.0004 newtons! That’s enough force to accelerate a 20 ton mutalisk up to about 14 miles per hour in a year.

Edit: An astute reader points out that the size of mutalisks is described in the Starcraft novels as being much smaller than I described. They apparently have a wingspan of 20 feet and are only 7 feet long and about a meter across. to me that seems shockingly small, especially compared to the cinematics, and it also seems quite stubby compared to all of the art depicting mutalisks as having a body that is a long tube. Still, we can scale the above results to fit these new dimensions. Given a cylinder 7 feet long and 3.3 feet across, and again assuming a density like water, I get a mass of 1.7 metric tons. If we treat the wings generously as a 20 foot square, then their surface area is 37 square meters, so the thrust on wings of that size as compared to our 10,000 square meter example above would be .0004 newtons x .0037 = 1.5×10^-6 newtons. That’s enough to accelerate our 1700 kg mutalisk all the way up to 0.06 miles per hour in a year! Even assuming a much lower density, it would only accelerate a 170 kg mutalisk up to 0.6 miles per hour in a year.

You might not have followed every step of that (admittedly very crude) calculation, but that final value should give you some idea of how ridiculous it is to say that a Mutalisk’s wings could work as solar sails. Even assuming very large wings and a small body, the acceleration that you get is miniscule. Plus, solar sails are really only good for accelerating away from the star, and Mutalisks are like fighter jets: they need to be able to dodge and weave in all directions very quickly.

Bottom line, I can’t explain how the mutalisks fly in space. Heck, I can’t explain how something that big flies in air! Solar sailing certainly doesn’t cut it, so we’re left where we began. It’s magic. This is a part of the Starcraft Universe that just doesn’t fit with the laws of physics in our own universe, and that’s fine. Mutalisks are still cool, and it’s not like I’m going to stop playing Starcraft because I don’t know how they can fly in space.

Stay tuned: next I’ll take a look at all the non-winged Zerg fliers!

What is a Supernova?

September 13, 2010 / Ryan / 16 Comments

The splash screen for the 'Supernova' mission in Starcraft 2.

Everyone has heard of supernovae, but it seems like there is some unwritten rule that they must never be depicted correctly in popular sci-fi. Starcraft 2 is, sadly, no exception. I don’t think it’s too much of a spoiler to say that there is a mission in the campaign where you have to battle your enemies to gain control of a relic on a planet that is about to be consumed by the fires of a huge blue star that is “going nova”. Or supernova. The words are used interchangeably.

As an astronomer, I always cringe when this happens, so I want to clear up what exactly supernovae and novae are.

The confusion is understandable: both are stellar explosions, and their names imply that a supernova is just a big nova. But in most cases this isn’t true. Let’s consider supernovae first, since by understanding them we will already be well on our way to understanding novae.

A supernova is when a very large star explodes. But that’s the end of a long process, so let’s rewind to the beginning: a star starts off as a ball of (mostly) hydrogen gas that is compacted into a dense sphere by its own gravity. At some point, the immense pressure of all that hydrogen is enough to force the atoms in the star’s core to collide with one another and merge into helium, releasing a huge amount of energy. This process is called fusion, and for most of a star’s life, it sits there turning hydrogen into helium and producing lots of energy. That energy makes the star so hot that it glows, and it also provides the pressure that keeps gravity from crushing the star any more. Most stars are poised in perfect balance with gravity trying to compress them and the nuclear reactions in their cores trying to expand them. Continue reading

What is a Rail Gun and How does it Work?

September 6, 2010 / Ryan / 21 Comments

A swarm of diamondbacks fire their "railguns" at a train.

In the Starcraft 2 campaign, there is a unit called the “diamondback”. It’s a sort of hovertank, and it is useful for hunting down other vehicles because it is armed with dual “rail guns” that can fire while it moves. In the game, the rail guns are depicted as some sort of energy weapon: they fire hot blue beams at the target. This disturbed me because rail guns actually do exist, and they most certainly don’t fire blue energy beams!

So what’s a real rail gun and how do they work? I’m so glad you asked!

Force Fields and Plasma Shields

July 30, 2010 / Ryan / 8 Comments

I was always a fan of the Protoss: super-advanced technology, powerful units, and those awesome plasma shields protecting everything from the lowliest probe to gigantic carriers. But I always wondered: could those force fields really work?

Well… Sort of.

It really depends on your definition of a force field and what you want to prevent from passing through. Electric and magnetic fields can exert forces on charged objects, so you could call those “force fields.” If you accept that definition, then there have actually been some NASA studies of how to use force fields on spacecraft and lunar bases.

One of the biggest problems facing human space exploration (aside from the politics) is that space is a dangerous place for humans. Even if we can launch people to Mars, there is a lot of radiation out there, both from the sun and from high-energy events out in the galaxy. That’s why two studies at NASA’s Institute for Advanced Concepts (which has, sadly, been canceled due to lack of funding) looked into deflecting the radiation with electromagnetic shields. Continue reading

The Real Science of Starcraft 2

July 27, 2010 / Ryan / 6 Comments

The Overmind

Well folks, it looks like I’m not the only one interested in the Science of Starcraft! The gaming site GamePro has a very interesting article taking a look at several aspects of StarCraft science, including teleportation, zerg in space, a hive mind, and hybrid species. They cover each topic pretty quickly but it’s still worth reading. But be sure to check back here! I’ll dig into those and other topics in a lot more depth.