Ich schaute mir „Die zwei Türme“ noch einmal an und war neugierig, wie weit Gandalf im Kampf gegen diesen verdammten Balrog gefallen ist, also habe ich die Rechnung mit einem lustigen kleinen Kritzelbild gemacht.
76.616 Fuß
Von WeHaveToEatHim
Ich schaute mir „Die zwei Türme“ noch einmal an und war neugierig, wie weit Gandalf im Kampf gegen diesen verdammten Balrog gefallen ist, also habe ich die Rechnung mit einem lustigen kleinen Kritzelbild gemacht.
76.616 Fuß
Von WeHaveToEatHim
26 Kommentare
Ooh, you could also work out how tall Saruman’s tower is!
Then how many stairs did he climb after to get to the top of the mountain?
But what is the terminal velocity of a balrog?
What’s the terminal velocity of a balrog of morgoth?
Im watching The Two Towers (Non-Extended Version) and clocked the time stamp of the moment Gandalf falls (2:36) to the moment he hits the water (3:45) and used that to calculate how far he fell assuming Earth gravity.
I used my Remarkable Paper Pro to draw the image.
Just a fun data doodle.
Edit: Guys Im so fucking wrong. Listen to the people much smarter than me in the comments.
You are assuming an infinity velocity is possible, which is not in the presence of air. (EDIT: Also in the vacuum, but this isn’t a relativistic scenario)
EDIT From my other post, the balrog terminal velocity could be around:
Vt = sqrt(2mg/dAC)
Where m = 1276kg balrog mass
g= 9.81m/s gravity
d=1.255kg/m3 (sea level for simplicity) density of the medium/air
D= 4.5m balrog approximate diameter, so assuming circular area is around 15 sqm
C= 0.73 (assuming elephant drag coeficient)
That gives a Vt of approx 41 m/s, 148kmh.
Numbers from google and AI, take it as you wish!
And I am too out of practice to make an integral to get the appropriate depth. A physics/engineering student can pick it up from here!
This assumes unlimited velocity, when air resistance (especially in a confined space) would seriously limit speed
All I know is that he arrived exactly when he intended to.
This is Middle-Earth. It may not have had the same mass as the current Earth and would have a different gravitational constant.
Could you calculate Middle Earth’s mass/diameter based on the fall time and Gandalf’s estimated mass?
There is no evidence that, in the world of Middle-earth, the acceleration due to gravity is a constant and equals 9.81 m/s²
In Felix Baumgartners famous free fall from space he fell from 39 km altitude for 36.5 km in 4m 20s… Just for reference, the Balrog wasn’t instrumented
You need to account for air-beard resistance.
This is a common fallacy. A human has a rule-of-thumb terminal velocity of 120 miles per hour (53.6448 m/s), which is hit in ~12 seconds after falling 450 meters: [https://www.thoughtco.com/terminal-velocity-free-fall-4132455](https://www.thoughtco.com/terminal-velocity-free-fall-4132455)
After which, the remaining ~57 seconds would be Gandalf falling at terminal velocity, which at that point is as such: 53.6448*57=3057.7536
Add that to the first 450 meters and we got a fall distance of 3507.7536 meters. This is assuming whatever planet Tolkien’s works is set on has the same gravity and atmospheric composition as Earth.
Cool doodle though.
I’m the book they fell for days, prehaps 8 days, so the distance is affected by old magic’s.
Are we gonna account for the flowy wizard robes?
bro definitely hit terminal velocity…
The teacher in me wants to tell you to always use units in order to sanity check your work!
I’ve always wondered how big the lake is, given how slow it looks like they’re falling even though it’s terminal velocity.
African or European Gandalf?
Reminds me of the time years ago where I tried to calculate the size of the planet in WOW.
Middle Earth has a lower gravity than earth, about 5ms². You can clearly see that by how high the elves can jump.
That seems like a bit much! But the crust of the Himalayas is estimated to be over 70 km thick, and those dwarves were busy for generations.
Nice illustration, but you’re ignoring air resistance. This [page](https://www.gigacalculator.com/calculators/terminal-velocity-calculator.php) says average terminal velocity for a human is 66 m/s, and is often reached after accelerating for ~12 seconds. Wikipedia says it’s 55 m/s for a „belly to earth“ position, but Gandalf is in much more of a head first position, so I think 66 m/s is fair.
So using your formula d = 0.5(9.81)(12^2) = 706 m during acceleration. That leaves 57 seconds of falling at terminal velocity, covering 3762 meters. 706 + 3762 = 4468 meters, or 14,659 feet.
Of course it’s possible that Gandalf is not of the same density as a human, so who knows what his terminal velocity would be.
Imagine knowing you’re going to die in 4 minutes and still being bored on the way down
Off topic but your hand writing is so nice
Air resistance is important. You should cap the speed at 50 m/s. Plus the Balrog’s flames would greatly increase the drag.