In both the movie version and book version of The Hobbit, the dragon (Smaug) is defeated by an arrow - the Black Arrow. There are some differences between the book and movie versions of this arrow so I will just be using the movie version from now on.
You can see in this clip from The Hobbit: The Desolation of Smaug trailer, there is some machine like a ballista that shoots the black arrow (in the book version, it was shot with a normal bow). How fast would this arrow have to shoot? What kind of force would this ballista need to exert on the arrow? Let's get started.
Dimensions of the Black Arrow
Clearly this isn't a normal arrow. It's much bigger and maybe it's made of iron or steel (can't tell for sure). The first thing I need to do is to estimate the size. In one scene, Bard (the defender of Lake Town) takes the last remaining Black Arrow in an attempt to hide it. This shot shows the full length of the arrow.
Here I am using the approximate height of the actor that plays Bard - Luke Evans who is listed at 1.83 m. From this, I am going to say the arrow is 1.9 meters long. Maybe it's as short as 1.7 and as long as 2.1 meters. Still, I have a range of values.
Finding the length of the arrow wasn't too bad. What about the mass? Let me just approximate the arrow as a cylinder with a radius of 1 cm +/- 0.3 cm. If I use the length as 1.9 +/- .2 meters, this would give a volume of 5.96 x 10-4 +/- 4.26 x 10-4 m3 (using the crank three times method of uncertainty calculations). Yes, this uncertainty is huge compared to the value of volume. That's what you get when you just guess at stuff.
The mass of the arrow can be found by assuming it is forged from iron (or steel - whatever makes you happy). Using a density of 7870 kg/m3, this gives an arrow mass of 4.698 +/- 3.357 kg. That's a weight between 2.7 and 17 pounds. The 17 pound arrow seems a bit much, but the mean value of 10 pounds seems ok.
Shooting the Arrow
Fortunately, we have a quick video showing a dwarf shooting a different Black Arrow when Smaug originally attacked the Lonely Mountain. It shows just enough frames with the arrow in it for me to get a plot of the position. Here is a plot showing both the horizontal and vertical position of the end of the arrow as it's shot from the ballista.
From these graphs I get an x-velocity of 12.4 m/s and a y-velocity of 6.66 m/s. The magnitude of this velocity is 14.08 m/s (31.5 mph) and at an angle of 28.2° above the horizontal. This is not very fast for an arrow. If I include the uncertainty from the length estimate, I get a launch speed of 14.08 +/- 1.48 m/s. Really, this is more like the speed of a nerf dart (yes, I measured the speed of a nerf dart) than a actual weapon that could kill an actual dragon. Maybe this is why Smaug destroyed the dwarves - their weapons sucked.
Actually, I don't think this weapon could harm anyone very far away. Lets say I shoot this arrow at 14 m/s at an angle of 28 degrees. Here is the plot of it's trajectory.
This is an arrow shooting from the top of a 15 meter tower (with out air resistance). The arrow rises about 2 meters above the starting point and travels just over 30 meters horizontally. This is a terribile arrow. You will never kill a dragon with this arrow unless you are at point blank range.
What About Real Arrows?
Just for completeness, let's look at a real arrow. Actually, it might be a stretch to call this a real arrow, it's my kids' bow. I guess it's still a legitimate bow. Here is slow motion video (240 fps) showing me shooting an arrow. By the way, I love the 240 fps video from the iPhone 6+ (my wife's phone).
Honestly - it was just a coincidence that I was wearing a Tree of Gondor shirt. Here is a plot of the horizontal motion of the arrow from Tracker Video Analysis.
From this, the launch speed of an arrow (from a simple bow - not even a fancy one) is 30 m/s. That's about twice the speed of the black arrow shot from the ballista. Sure, I don't think this arrow could kill a dragon either - especially since it has a rubber tip. Really, a more high quality bow (and higher quality archer) could probably shoot an arrow at twice this speed.
Just for fun, here is one more archer, an 8 year old archer.
Although he doesn't pull the string back all the way, he still gets an arrow speed of 13 m/s.
How to Kill a Dragon, Dwarf-Style
Maybe you are thinking the following:
I hear you, and that makes sense. However, how much damage could you do to a dragon if you don't even hit it? The answer is no damage.
Well, then how fast do you have to shoot an arrow to have a halfway decent chance of hitting it? If the arrow is shot with an initial speed of 50 m/s and at an angle of 45° it will rise about 60 meters. This seems at least a little bit reasonable. The dragon might make a swoop down to within range. Really, a speed of 100 m/s might give the people fighting the dragon a reasonable chance. Let's go with that value.
Now you want to launch this black arrow with a speed of 100 m/s. How much kinetic energy does that arrow have? Of course this depends on the mass of the arrow - and I just have a range of masses. The kinetic energy is calculated as:
Using the crank three times method for uncertainty, I get a KE of 23.5 +/- 16.8 kJoules. A normal arrow might have a mass of around 20 grams. If fired at 50 m/s, it would have a kinetic energy of just 25 Joules. Doubling the speed would still only give 100 Joules of kinetic energy. So, it's no where near the black arrow.
What about launching this Black Arrow? Here is a diagram of the arrow while it's being launched.
Since I know the final kinetic energy of the arrow (the initial KE is zero), I could find the average force on the arrow if I just know the distance over which this force pushes on the arrow. Looking at the Hobbit trailer, the ballista pushes on the Black Arrow over a distance of about 0.95 meters. This puts the average force at 24,726 +/- 17,668 Newtons. Bam. Oh, I left off the energy that goes into the change in gravitational potential energy when the arrow is launched at an angle. Why? Becuase it's tiny compared to the kinetic energy.
Ok, here's why you don't want to hold this ballista and instead shoot it from some mounted location. Since I know the mass (at least a guess) and the velocity of this Black Arrow, I can find the momentum.
Using the values from above, this would put the momentum of the Black Arrow (after being launched) at 469.8 +/- 335.7 kg*m/s. That might not seem so impressive. But wait! The momentum principle says that in order to change the momentum of an object, you need to exert a net force. If you just had a person holding onto the ballista, then this force on the arrow would have to be the same as the arrow pushing back on the ballista (due to the nature of forces). The result is that an increase in momentum of 469 kg*m/s for the arrow would mean that the person would have to have the same increase in momentum (in the other direction).
Let's say that a dwarf plus the ballista has a mass of 125 kg. In order to have the same momentum, the dwarf would have a recoil speed of 3.76 +/- 2.69 m/s. Yes, that's pretty fast for a recoiling dwarf. Of course the recoil speed would actually be lower since the dwarf has other forces acting on him other than the arrow pushing back. There is also a frictional force. Still, the recoil would be significant - and it would probably hurt.
Why Did They Make the Movie That Way?
So, in the movie, the arrow is going too slow. Also, in the book version of The Hobbit, there isn't a ballista and instead Bard shoots a normal bow. Why the changes?
First, let me address the ballista. I suspect this is a plot point. There is only one Black Arrow, so having that is essential. However, if Bard has the arrow and his bow he can shoot the dragon from anywhere in the town. That makes things too simple. If they have a ballista, then Bard must both get the arrow AND get to the ballista. This can make for a more complicated action in the movie. Well, that's just my guess.
What about the arrow speed? Suppose they made the arrow shoot at 100 m/s. If it launched over a 0.95 meter distance (like in the movie), then it would have an average speed of 50 m/s (since it started at rest). With this average speed, it would only take 0.019 seconds for the arrow to launch. Most movies are recorded at 30 frames per second which gives each frame only 0.033 seconds. With a launch speed of 100 m/s, you wouldn't even see the arrow being fired. Making a slower arrow actually makes a more dramatic scene. The exact same thing happens in Star Wars. The speeds of blasters in Star Wars is also incredibly slow (yes, I did an analysis).
In the end, it's all about making a better movie (I guess).
