SPOILER ALERT. In the movie The Hobbit: Desolation of Smaug, Legolas fights some orcs. Yes, Legolas isn't supposed to be in this movie, but he is. At one point, Legolas takes a breather from fighting and some orc thinks this would be a perfect time to take him out with an arrow. The orc's arrow flies towards Legolas, but BOOM. The arrow is shot out of it's trajectory as it is hit from another arrow fired from the wood elf Tauriel.

That's your plot. Now for the physics.

How Fast Is Tauriel's Arrow? —————————-

Looking at the video, there are only a few details I can get. First, it takes about 0.1 seconds for the arrow to go from the orc almost all the way to Legolas (but not all the way). If I know how fast an arrow is fired from a bow, I can use this to get the distance from the orc to Legolas. It seems a decent arrow speed (shot from a modern bow) would be about 100 m/s. Of course the question is - are orc bows as good as a modern bow? Let's say no. Let me put the arrow speed at 60 m/s. That would put the orc-Legolas distance at just 6 meters.

What about Tauriel? I'm just going to estimate the position of the wood elf. Here's a diagram showing the locations. Note: this is just an estimate.

With this estimate, Tauriel is just 5 meters from the orc and about 9 meters from Legolas. In order to hit the arrow fired from the orc, I would estimate that Tauriel would have to fire AFTER the orc fired. Suppose she has lightning fast reaction time and can aim and fire in just 0.05 seconds. This would mean that her arrow has to travel 9 meters in just 0.05 seconds giving it a speed of 180 m/s. Yes, that's fast. Right now, real humans are trying to break the 400 fps arrow speed (121 m/s). Ok, but she's an elf. I guess she's better than modern humans.

How difficult would it be to hit that orc arrow? ————————————————

I will assume the orc arrow is traveling in a horizontal motion. Tauriel has to aim two ways. First, she has to aim vertically in order to hit the arrow. Let me assume the orc arrow has a diameter of 1 cm (and so does the elf arrow). In order to obtain an arrow-arrow hit, the elf arrow has to be in a certain position.

From this, the elf arrow has to be in a 1 cm vertical range to get a clean hit. If the arrow is 9 meters away, this is an angular size of just 0.06 degrees. Is that a small target? Let's say you wanted to hit a 10 cm apple. In order to have the same angular size of the target, you would have to be 90 meters away. So, that seems like a tough (but maybe not impossible) target.

What about the timing of the shot? If you shoot the arrow too early, it will pass in front of the orc arrow. Too late, and it goes behind it. If I take one point in space (the point where the arrows might meet), I can find the time difference between the front of the orc arrow passing until the back of the arrow passes. Using an arrow speed of 60 m/s and a length of 0.6 meters, this give a passing time of 0.01 seconds. Tauriel must release her arrow in this small time window in order to hit the orc arrow. But wait! It's even more difficult than this. I assumed that the orc arrow and the elf arrow ere perpendicular to each other - they are not. Tauriel shot from behind the orc making the target appear even smaller.

Needless to say, this is a tough shot.

What Was Tauriel Thinking? ————————–

We have missed the obvious question. Why didn't Tauriel just shoot the orc? The orc is closer than the arrow, the orc is bigger than the arrow. Shooting the orc would be way easier than shooting the arrow. Oh sure, you could say she just arrived on the scene and didn't have time to shoot the orc. However, there was only a 0.1 second window after the orc shot the arrow.

If I asked Tauriel what she was thinking, she might say this.

"Could I have just killed the orc right away? Sure. That would be simple. I mean, have you seen the other stuff I have done? Killing one orc is just too easy. It's good to challenge yourself. Do that which is difficult, not that which is easy." "Ok, that's not the real answer. The real answer is that I was tired of Legolas and his father. They think I am this lowly wood elf and they are all uppity high elves. I'm not good for anything important am I? Well, what if I show you how awesome I am with a bow? Maybe then you will notice. Yes, Legolas, you could have died if it weren't for me. If I would have just shot that orc, you wouldn't have even noticed that I saved your life." "Try to ignore me again. Let's see what happens. Oh, don't worry. I'm not attracted to you, I just want respect. Anyway, you aren't my type. I like tall dwarves."

See. This scene says something about Tauriel. You just have to think about it a little bit.