Over on Wired, Rhett Allain from Southeastern Louisiana University has been crunching some numbers on an interesting problem — all in aid of encouraging people to think about maths. The problem in question? How long was the Balrog’s whip in Peter Jackson’s Lord of the Rings?


His answer, in part:

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/s2.
  • 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.

Allain calculates, with a few caveats about air resistance, that in that short space of time, the Balrog tumbles a good couple hundred metres. You can head over to Wired to read the rest of Allain’s answer and check out his figures.

As an interesting addendum, DwellerinDale — one of the smart cookies who inhabits the TORN Reading Room — performed a similar analysis about 18 months ago, which was published as part of the very first TORn Amateur Symposium.

His numbers come out a little differently (possibly due to a different accounting of air resistance and Balrog mass.

We find the distance the Balrog falls before catching Gandalf with his whip … 437.5 m … which is very close to a quarter of a mile. The best possible scenario with air resistance gives a distance 270 meters less than free fall, but it still can’t make the Balrog’s whip catching Gandalf physically possible.

If you’d like to check Dweller’s numbers for comparison, you can download a PDF of his full paper here to check out the working.

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