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#The Laws of Physics Do Not Apply to Legolas

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Author: Rhett Allain. Rhett Allain Science Date of Publication: 04.08.15. 04.08.15 Time of Publication: 12:28 pm. 12:28 pm

SPOILER ALERT. If you haven t seen the final Hobbit movie (The Hobbit: The Battle of the Five Armies ), then maybe you should just walk away. In fact, if you haven t seen the other movies with Legolas (which is all of them including Lord of the Rings) then you might see some spoilers. Oh wait, you read the book? You will still see spoilers because Legolas wasn t in the book version of The Hobbit (which was just one book).

Ok, now that the warning is out of the way let s get back to Legolas. Of course you know Legolas. He is the elf in the Fellowship of the Ring and he likes to shoot his bow while doing crazy stuff. It s just what he does. Here is a small sample of his moves in the movies:

• He surfs on a shield sliding down stairs. During this time he is shooting his bow.
• He likes to surf. He also surfs (or slides) down the trunk of a dying elephant (giant elephant).
• Oh no! Tauriel is over there and she needs help. Legolas jumps on a troll and stabs two daggers into the troll s brain. He then proceeds to use these two daggers as a sort of remote control to steer the troll where he wants. Yes, you probably missed this part go back and watch The Battle of the Five Armies.
• It isn t enough to make an RC troll. He drives the troll into a tower and knocks it over. This knocked over tower then becomes a bridge across a chasm. Yes.

Up to this point, I am accepting of the awesomeness of Legolas. I mean, it s just a movie, right? But my acceptance can only go so far. This next trick is just too much for me. Let me describe what Legolas does.

Legolas is fighting an orc on the bridge that was once a tower (remember the RC troll). The orc tosses a large rock at Legolas, but he rolls out of the way so that the rock lands on the bridge . Oh darn. Now the bridge is falling apart with Legolas on it. Don t worry, this is Legolas. He won t die (because he has to appear in the Lord of Rings movies). To save himself, Legolas just runs up the blocks of the bridge as they fall. Simple right? Well, I hardly ever say anything during a movie but in this case I blurted out what the what?

Video Analysis

There is one part of the falling bridge with a camera off to the side and mostly stationary. This seems to be almost perfect for video analysis. I need a couple of assumptions.

• Middle-earth is like Earth in that it has a similar gravitational field (9.8 N/kg). Really, this isn t an assumption. Tolkien even wrote that Middle-earth is located on Earth.
• Legolas is 1.8 meters tall (because Orlando Bloom is 1.8 meters tall).
• The camera is far enough away to ignore perspective (not technically true) and the camera looks perpendicular at the bridge (also not true).

Ok, that s it. Now I just need to scale the video based on the height of Legolas and then start tracking objects using Tracker Video Analysis. Here are three of the bridge blocks falling.

Although these look fairly linear, I can still fit a quadratic function to the data. Doing that, I get the following vertical accelerations: -1.122 m/s 2. -2.792 m/s 2. and -2.46 m/s 2. Clearly, those are not the same accelerations we would expect for a falling object on the Earth (it should be around -9.8 m/s 2 ). So really, there are two options to fix this. First, my scale could be wrong. If Legolas were MUCH bigger, then the size of the blocks would also be bigger making a greater acceleration. Second, this could be played back in slow motion. I am going to go with option 2.

Let me assume a block acceleration of -1.2 m/ss 2 (I am using units ss because these are slow motion seconds). If I make this equal to -9.8 m/s 2. then 1 ss would be 0.35 seconds. This will be useful when I look at the motion of Legolas.

Ok, one more thing. Let me just make a quick model of a falling bridge. Here I have several blocks. Each one starts to fall after a short time interval. This is what that would look like (with Earth-like gravity).

Yes, I know my bridge is backwards. You still get to see something. Notice that at the end of the fall, the falling blocks form a diagonal? That s just like in the clip. So, I think the blocks are indeed accelerating as they fall, just not at the correct value.

How do you run on a falling block?

Yes, this move is theoretically possible. Actually, this is very similar to a question posted on Twitter.

another oral exam week is upon me. potential ?s: if you had a bucket of pebbles, how fast and often would you throw them to hover?

— Andy Rundquist (@arundquist) April 6, 2015

If I answer the question for Legolas, you should be able to do the pebble version as homework. Let s start with a diagram showing Legolas pushing on a falling block. There s a lot going on in this diagram so let s just look at one thing at a time. First, there are the two red arrows. Legolas pushes on the block (downward). Since forces are an interaction between two objects, the block pushes back on him (in the opposite direction). Now, if Legolas is to stay stationary (which he doesn t), then this upward force that the block pushes on him would be equal to his weight. Really, the block will have to push greater than the weight so that he can get a little hop to the next block. What do forces do? A net force on an object (like Legolas or the block) changes the momentum of the object where the momentum is the product of the object s mass and velocity. When Legolas pushes on the block, the push (along with gravity) will change its momentum. In the case of a block, that s not too big of a deal. The thing probably has a mass around 1000 kg (just a guess) so that this change in momentum won t mean such a large change in velocity. Of course this is not true if you are throwing a pebble down. However, the problem is the time of the push. If the block is falling, Legolas won t have much time to push on this block to push himself up. Let s make an approximation. Suppose that Legolas can push over a distance of 0.2 meters in order to jump. How long would it take a falling block to move this distance? The answer depends on how long the block has been falling (the longer it falls, the faster it goes). Here is a plot of the time for a block to move 0.2 meters as a function of falling distance (assuming the block has a constant acceleration).

%img src="https://plot.ly/

RhettAllain/444.png" /%

You can see that by the time a block has fallen just half a meter, Legolas could only push on it for 0.05 seconds. Now for a quick calculation. Suppose that he wants to push on the block such that he jumps 0.5 meters up to the next block. He would have to push in a way to give himself a final speed of about 3 m/s. Suppose that Legolas has a mass of 60 kg I can now calculate the force he would need to push with in order to make this jump. (this equation is just in the y-direction):

Putting in my values, (his initial velocity would be the initial speed of the block about -3 m/s) I get a force of 7788 Newtons (1750 pounds). I guess that isn t too crazy but I think that would be hard enough to push on a stationary block much less one that is moving down. But wait. During this time, Legolas also has to move in the horizontal direction to get to the next block. If the blocks have a length of 0.5 meters (that is my guess) then his horizontal velocity would be (0.5 meters)/(0.05 seconds) = 10 m/s (22 mph). Again, not too crazy.

Ok, let s look at the actual motion of Legolas during this move. In this plot, I am using the corrected time to make the blocks have the Earth-like free fall acceleration. Here is the vertical motion of Legolas.

The speeds look ok but if you look at the position of the blocks he is pushing on, they are about 1 meter below the bridge. That s not good. Also, if you look at one push he pushes on a block for about 0.1 seconds and the block falls a distance of about 1 meter. The time is ok, but if the block falls a meter then Legolas would have an initial downward velocity of about 4.5 m/s. This means that he also has to jump higher and push even harder.

In the end, it s just darn hard to push on something that s falling. Even the MythBusters tried something like this with a collapsing suspension bridge. Check out the awesome footage.

Of course there is one difference in the Legolas jump (ok two differences he is an elf and not a human). The MythBuster bridge has a very low mass. When you push on it to jump, it increases its downward speed. Since the blocks have a much larger mass, a person pushing wouldn t change the speed so much. You still have to push in a very short time. That s just plain tough.

Really, I like to remind myself of the role Legolas had in the book version of the Hobbit. Yes, he had no role. I think the movie version of The Hobbit (all three movies) would have been better without Legolas.