How deep is the hole to Sombra's dark magic door? A rough calculation.

[Music Chart Fan 819]

While watching Season 3 Episode 2, when Twilight is descending down the steps leading to Sombra's dark magic door (the one which leads to your worst fear), she breaks a crystal off the wall and drops it down the hole to see how deep it is. According to my timing, Twilight hears the crystal hitting the bottom about 9 seconds after she dropped it. I thought that with this time being so long, the hole should have been MUCH deeper than it was depicted.

 

Using this time, can we make a rough calculation of how deep the hole is? That is what I'll try to do. If you're not interested in the details, skip to the FINAL RESULT below. If you're up to it, you can check my work; making this rough calculation involved a fair number of assumptions and equations, and I very well might have made a bad assumption or miscalculation somewhere.

 

We can use the following kinematic equation to calculate distance traveled by a falling object.

 

x = x₀ + v₀*t + (1/2)*g*t²

 

We can assume that x₀ = 0 (initial height of the crystal when dropped) and v₀ = 0 (initial velocity of the crystal when dropped). I assume that gravitational acceleration is the same as that of Earth: 9.8 m/s².

With our time of 9 seconds, can we just plug and chug to obtain the depth of the hole?

 

Maybe not - what if the crystal reaches free-fall terminal velocity before hitting the bottom? Terminal velocity can be calculated using the following equation:

 

v_t = sqrt( (2*m*g)/(rho*A*C_D) ) = 60.624 m/s

 

g = 9.8 m/s2 (gravitational acceleration)

rho = 1.225 kg/m³ (density of air at sea level and at 15 ^oC)

C_D = 1.15 (drag coefficient of a short cylinder, assumed shape of the falling crystal)

A = 0.020645 m² (projected area of the falling crystal, assumed 8 in long x 4 in wide)

m = 5.453614 kg (mass of crystal, assumed density of ~2.6 g/cm³, same as average rock, and volume of 128 in³, an 8 in x 4 in x 4 in rectangular solid)

 

We can use another kinematic equation to calculate the time to terminal velocity.

 

v_t = v₀ + g*t

(60.624 m/s) = (0 m/s) + (9.8 m/s2)*t

t = 6.186 s

 

So, the falling crystal would reach terminal velocity in 6.186 s. We can use the following kinematic equation to calculate the distance the crystal falls before reaching terminal velocity:

 

x = x₀ + v₀*t + (1/2)*g*t²

x = (0 m) + (0 m/s)*(6.186 s) + (1/2)*(9.8 m/s²)*(6.186 s)²

x = 187.5 m

 

Now bear in mind that our time of 9 seconds includes the time taken for the crystal to fall to the bottom AND the time taken for the sound of the crystal hitting the bottom to travel back up the hole to Twilight's ear. We can use the following equations to calculate the times:

 

187.5 m + (60.624 m/s)*t₁ = (340 m/s)*t₂

 

9 s = 6.186 s + t₁ + t₂

 

The first term on the left-hand-side of the first equation - 187.5 m - is the distance traveled before reaching terminal velocity. The second term - (60.624 m/s)*t₁ - is the distance traveled after reaching terminal velocity; 60.624 m/s is the terminal velocity, and t₁ is the time traveled at terminal velocity before hitting the bottom. The right-hand-side of the first equation - (340 m/s)*t₂ - is the distance from the bottom of the hole to Twilight's ear; 340 m/s is the speed of sound in air at 15 ^oC, and t₂ is the time taken for the sound to travel to Twilight's ear. The left-hand-side - total distance the crystal fell to the bottom - and the right-hand-side - distance the sound traveled to Twilight's ear - should be about equal.

 

The second equation tells us that the 9 seconds elapsed between Twilight dropping the crystal and Twilight hearing the crystal hit the bottom consist of (1) the time taken to reach terminal velocity, (2) the time traveled at terminal velocity before hitting the bottom, and (3) the time taken for the sound to travel to Twilight's ear.

 

Solving these two equations, we obtain:

 

t₁ = 1.92 s

t₂ = 0.894 s

 

Plug the appropriate time into either the left-hand-side or right-hand-side of the first of the previous two equations to obtain the depth of the hole.

 

FINAL RESULT:

 

The depth of the hole from Twilight's location on the stairs to the bottom is about 304 m, or 997 ft.

 

Assuming a story is about 12 ft, this is a depth of 83 stories!

 

IMPLICATIONS IN THE EPISODE:

 

This would mean that when Twilight trips and falls down the steps, the jump cut between Twilight beginning to fall down the steps and Twilight reaching the bottom is probably a LOT longer than you might expect. Also, Twilight would NOT be unscathed after falling down 83 stories of steps. I have no idea if she would even survive that fall or what the extent of the damage would be.

 

When the camera shows the view from the bottom of the hole to the top, I'm not sure that the depiction of the depth of the hole is accurate.

 

Finally, in the episode, roughly 45 seconds elapse between the time that Twilight enters the door and the time that Twilight first hears Spike calling to her in her "hypnotized" state (not counting the time for the commercial break). Somehow I think it unlikely that Spike descended down 83+ stories of steps in such a short time.

 

In fact, I timed the rate at which Spike was going down the steps as roughly 5 steps/second. Assuming a step height of 8 in, it should take Spike about 300 seconds (5 minutes) to go down the steps, not including any time that Spike waited before starting the descent. Normally I would think that Spike would be incapable of keeping up this high rate of going down the steps for 5 solid minutes, but apparently Spike IS physically capable of doing this, since he shows no signs of physical exertion (sweating, heavy breathing, etc.) after reaching the bottom. I guess Spike is in better physical shape than he looks!

 

Are my assumptions and calculations okay? Can you think of other implications of the depth of the hole in the episode? I hope you enjoyed my analysis, and thanks for reading!

[Crispy 5,563]

Finally, in the episode, roughly 45 seconds elapse between the time that Twilight enters the door and the time that Twilight first hears Spike calling to her in her "hypnotized" state (not counting the time for the commercial break). Somehow I think it unlikely that Spike descended down 83+ stories of steps in such a short time.

 

I think your calculations are valid in terms of depth, however, time is subjective in a memory. We've all been caught off guard by how long we've been daydreaming, say nothing for being under the sway of dark magic-induced fear. Spike could have easily taken as much time as he needed to properly get down those stairs before he got through to Twi.

 

Edit:

 

This would mean that when Twilight trips and falls down the steps, the jump cut between Twilight beginning to fall down the steps and Twilight reaching the bottom is probably a LOT longer than you might expect. Also, Twilight would NOT be unscathed after falling down 83 stories of steps. I have no idea if she would even survive that fall or what the extent of the damage would be.

 

Cartoon damage will be cartoon damage. I direct you to this.

 

Edited November 14, 2012 by Crispy

[CloudFyre 848]

And this, ladies and gentlecolts, is why I love this fandom. Is it relevant? Nope. Is it useful? Probably not. Is it hilarious and great fun to read? All the way.

 

Seriously, I wouldn't have even thought to try calculating something like that. Kudos to you!

[Shankveld 4,951]

I came.

I saw math.

I left.

 

 

my applause to you for doing all this work,

maybe you should give the cure for cancer a crack next haha ~

[Titan Rising 2,156]

I think you have made a grave error in assuming the acceleration due to gravity is 9.8. According to my calculations from several different points in the show I would place the acceleration due to gravity at around 2.375, meaning the hole would be significantly less deep, around 19.732 stories deep to be precise.

[Music Chart Fan 819]

I think you have made a grave error in assuming the acceleration due to gravity is 9.8. According to my calculations from several different points in the show I would place the acceleration due to gravity at around 2.375, meaning the hole would be significantly less deep, around 19.732 stories deep to be precise.

 

Would you mind explaining how you got that number? I feel like it would be a big paradigm shift if that number were true, since on the surface everything looks very Earth-like. Would all the physics of the characters' normal actions be explainable in light of the lower gravity? Would the plants and animals we see be as Earth-like as they are if they developed in a low-gravity environment?

[Nothing&NoOne 695]

I just..I don't..

Wat.

 

....What about the staircase that leads back up to where the crystal heart is? You'd have to factor in how fast she's going.

 

Nice work, though. I suck at math.

[Shanks 10,816]

I can't really even begin to say whether any of that is anywhere even close to accurate as I understood pretty much none of it, I am just going to chalk it up to "it's a cartoon" and stop thinking about it before I get a permanent migraine.

[Music Chart Fan 819]

....What about the staircase that leads back up to where the crystal heart is? You'd have to factor in how fast she's going.

Unfortunately I don't think we have enough information to make that calculation. We aren't given any idea how many steps there are or how long Twilight and Spike slide up the underside of the steps, given all the jump cuts, not to mention trying to figure the speed at which they are sliding. We lucked out with the calculation I made because we were given a pretty clear time measurement and the physics isn't too difficult to calculate the distance traveled by a falling object.

 

I think your calculations are valid in terms of depth, however, time is subjective in a memory. We've all been caught off guard by how long we've been daydreaming, say nothing for being under the sway of dark magic-induced fear. Spike could have easily taken as much time as he needed to properly get down those stairs before he got through to Twi.

I understand your point here, and it is well-taken. As I was thinking about this, I came up with some interesting observations from the episode to consider.

 

When Spike looks into the dark magic door, the events of his dream begin almost immediately and look like they are unfolding in real time. If the door's dark magic acted the same way on Twilight, we might think that her dream also started immediately and the events of her dream also unfolded in real time.

 

However, the dark magic door seems to be inconsistent in its effects. As Spike's dream is occurring, he looks like he is doing and saying the same things in the real world as he is in the dream world (talking, stepping back, eyes tearing up). Conversely, when Spike calls out to Twilight and jerks her out of her dream, Twilight is laying on the floor sobbing with her head in her legs in her dream, whereas in the real world she is simply sitting and staring into the door. Spike also gives no indication that he heard or saw Twilight doing anything even as her dream was occurring.

 

In conclusion, we can't say with certainty whether the events of Twilight's dream unfolded in real time immediately after she went through the door. This did happen with Spike, but the dark magic of the door may have affected Spike and Twilight in different ways.

[Crispy 5,563]

When Spike looks into the dark magic door, the events of his dream begin almost immediately and look like they are unfolding in real time. If the door's dark magic acted the same way on Twilight, we might think that her dream also started immediately and the events of her dream also unfolded in real time.

 

However, the dark magic door seems to be inconsistent in its effects.

 

I would guess it's because of the difference in activation. The dark door worked on Spike with no input of magic at all, probably still feeding off of the dark energy Twilight gave it. I believe Twilight was more directly under it's thrall, and as such less subject to actions in real time. As you pointed out above, there's no way Spike could have reached her so quickly, save maybe falling and somehow slowing himself.

[Doctor XFizzle 8,670]

I'm just here because I saw "how deep is the hole" and it made me giggle like a high schooler.

 

Yeah, that's too much math for me but nice work setting it all up. It's fun to calculate stuff in cartoons but by no means should it be taken seriously and as accurate. Now real life movies those things can be tested, in fact I think Mythbusters has a couple episodes where they test out movie myths and stuff.

[Zygen 6,066]

Oh my that is quite alot of math... IT BURNS!!!!! Ok not really, but being in 9th grade I don't really understand it that much(Actually to be totally honest I really didn't look at it)

ANYWAYS. I can't judge your measurements since I don't know enough about it. However the fact that she was not hurt in any way really is well because its a cartoon . I love thinking about theory's and the such, but there is some cases where the cartoon thing applys.

It was an interesting thing to know though .

[Music Chart Fan 819]

I'm just here because I saw "how deep is the hole" and it made me giggle like a high schooler.

 

Yeah, that's too much math for me but nice work setting it all up. It's fun to calculate stuff in cartoons but by no means should it be taken seriously and as accurate. Now real life movies those things can be tested, in fact I think Mythbusters has a couple episodes where they test out movie myths and stuff.

 

I must admit, the possible sexual joke of the title never crossed my mind; I guess you have a dirtier mind than I.

 

And yeah, I know that the crystal taking a long time to hit the bottom was really a one-time gag and the writers forgot all about it and treated the hole as being much shallower afterward. It was a funny gag, though; I laughed out loud at the look on Twilight's face when she finally heard the crystal hit the bottom of the hole.

[Party Cannon 876]

 

 

There was only one video that I could think of to add to this commendable yet hilariously unnecessary use of physical calculation.

Edited November 17, 2012 by Party Cannon

[BlinkZ 1,151]

Eays Spike has long fall boots in his legs   hehe well it is a cartoon so eh..spike puleld a gem out of his ass during the diamond dogs episode so anything possible.

"its bigger on the inside!"
I mean do the math for the staircase that went UPwards to the tower.. now that insanely tall

Edited October 6, 2016 by BlinkZ

[Castle Bleck 19,390]

It was all underneath a secret portal, and the door was both semi-sentient and yet another, multi-destination portal (to both the white void full of extra stairs and -- in the comics -- Sombra's private study).

 

Basically, Umbrum magic could rival Draconequus magic in terms of taking physics out to lunch.

Edited October 7, 2016 by A.V.