I don't think I am going to give any Lord of the Rings spoilers here. The books are over 50 years old and even the first movie (the Peter Jackson version) was released in 2001. So let me just say that at one point, Gandalf fights a monster and essentially dies. That sucks for Gandalf (although I think he leveled-up in the process) but it is a great opportunity for some physics homework.

Just so that we all start with the same material, I am going to use the Gandalf vs. Balrog scene from the Peter Jackson version of Fellowship of the Ring and The Two Towers. The fight actually happens during Fellowship of the Ring but there are pieces of it shown in The Two Towers. If you do a quick search for "Gandalf vs Balrog", you should be able to find the video of it without too much difficulty.

No one is happy if I just give out homework questions. What about an example from the textbook? Well, there isn't a textbook. This is real world stuff. But ok, I'll start off with an example.

How Long is the Balrog's Whip? ——————————

Here's the scene. Gandalf and the rest of the party are trying to escape from Moria. They cross a narrow bridge and then Gandalf makes a stand agains the Balrog. In order to prevent the Balrog from crossing, Gandalf breaks the bridge and the monster falls. But wait! At the last second, the Balrog's whip grabs a hold of Gandalf and pulls him into the chasm.

Let's get data from the video to estimate the length of the Balrog's whip. But first, a couple of assumptions.

Even though this is Middle Earth, I will assume that the local gravitational field is just like on Earth. That means that a free falling object will have an acceleration of 9.8 m/s 2 .

. During this initial falling motion of the Balrog, I am going to assume that the air resistance is negligible so it accelerates with a constant acceleration.

Now for the only measurement from the video - time of fall. It's difficult to pinpoint exactly when the Balrog was in free fall as well as the time the whip hits Gandalf. I can use Tracker Video Analysis and mark a beginning and ending frame for the fall (with a conservative estimate of the start and stop times). This gives a falling time of about 13.8 seconds.

How far did the Balrog fall during this time interval? Well, if I stick with my original assumption that there isn't any air resistance I can use the following kinematic equation:

I can set the initial position at 0 meters and if the Balrog starts from rest, the initial velocity is 0 m/s. Putting in a time of 13.8 seconds and a value of g at 9.8 m/s2, I get a final position of -933 meters. That's just crazy.

The real crazy thing is my assumption that there is no air resistance. Without air resistance the Balrog would have a constant acceleration during this time. I can find the vertical velocity at the end of the 13.8 seconds with the definition of acceleration.

If I again assume the Balrog starts from rest, I get a final velocity of 135 m/s (over 300 mph). This is a problem. If the Balrog was moving this fast, then there should be a significant air resistance force. Suppose that I use the following model for the air resistance:

In this expression, ρ is the density of air. A is the cross sectional area of the object and C is a drag coefficient that depends on the shape of the object. Of course I don't really know C or even A for a Balrog. However, I do think that the Balrog has a terminal speed similar to that of Gandalf (who is human-like). At terminal speed, the air resistance force is equal to the weight of the object such that the net force is zero. When the net force is zero, the object doesn't change speed - thus the term "terminal velocity". If I guess a terminal velocity of 54 m/s (approximately true for humans) then I can solve for the things I don't know. Since I am assuming the Balrog falls the same way a human would, I am going to use a human mass of 68 kg (otherwise I would have to guess at the area and mass of a Balrog).

All the stuff on the left side of the equation is mostly unknown (I know the value of 1/2 and I can guess at the density of air). Since this stuff doesn't change, I am just going to call it K. Putting in my values for the terminal velocity and the mass, I get a value of K = 2.24 kg/m.

What now? Now, I can make a numerical calculation. If look at a falling Balrog in small time steps, I can approximate the forces as constant. With constant forces, I can calculate the change in velocity and change in position during this time interval. A new velocity means a new air resistance forces for the next time interval. So, I just keep doing this stuff over and over. Here is an older post with more details of a numerical calculation.

I'll skip all the details in the calculation - but, it's here if you want it. Here is a plot of the vertical position of a Balrog falling for 13.8 seconds.

You can see that the Balrog reaches a terminal velocity fairly quickly (after just about 3 seconds). Also, the falling distance with air resistance is 215 meters (700 feet).

So, how long is the Balrog's whip? Maybe my time is off for the start and finish of this falling motion. Maybe my air resistance model is wrong - or maybe the Balrog actually flies a little bit. I feel comfortable saying that the whip is at least 100 meters long and maybe as long as 200 meters.

Let me just look in the back of the book and check my answer for this homework problem. Oh snap. Only the even number problems have answers. I guess we will never know if I am correct.

More Homework ————-

Now it is your turn to answer some questions.

If you look at the fight scene between Gandalf and the Balrog as they fall, it takes about 69 seconds for the two to fall from the bridge to the underground lake. Estimate the height of this fall.

Suppose that when the Balrog grabbed a hold of Gandalf, it temporarily stopped falling at a position of 200 meters below Gandalf. When they are both falling, it take Gandalf 15 seconds to catch up to the falling Balrog. Assuming the Balrog has a terminal speed of 54 m/s, what terminal speed would Gandalf need to catch the Balrog? Use this to estimate Gandalf's percent decrease in cross sectional area multiplied by drag coefficient (AC).

While falling, Gandalf catches up with his falling sword - Glamdring. Estimate the density of this sword (assumptions and estimations required).

Assume the Balrog has a terminal speed of 54 m/s. If the Balrog has the same proportions as a human (roughly true) but a height 3 times larger than a human, what is the density of a Balrog? (hint: you can estimate the increase in cross sectional area for the Balrog and assume the drag coefficient is the same for a human).

Near the end of the fight, Gandalf and the Balrog fall into an underground lake. I went ahead and used Tracker Video to mark the position of the falling duo as a function of time. Here is that data.

If you assume that Gandalf and the Balrog are falling at a terminal speed of 54 m/s, how tall is the ceiling in this cavern. The units for the distance in this plot are in units of cavern height (since there was nothing obvious to scale the video). What is the tallest underground cavern known to humans? How does this cavern compare to the known caverns?

That's the homework. None of these questions have answers in the back of the book (because there is no textbook).