• The Physics Behind the Sonic Rainboom

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    Some crazy math brony did a bunch of extremely intimidating looking calculations to find out just how amazing Rainbow Dash is. 

    I'm pretty sure we can safely assume that Dashie will be the next princess of Equestria, replacing tyrant Celestia of course.
    Thursday, April 28, 2011 op
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    71 comments:

    1. IntrovertedApril 28, 2011 at 7:39 PM

      Saw this weeks ago on Reddit. Still quite mind-bogglingly awesome nonetheless.

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    2. ZAquanimusApril 28, 2011 at 7:40 PM

      but what about the allmighty princess luna!? D:

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    3. SystemdertotenApril 28, 2011 at 7:44 PM

      My Brain...
      ...is full of fuck...

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    4. Prince ShadariApril 28, 2011 at 7:44 PM

      I read all of that, and understood it. I'm frightened, my education seems to be teaching me things.

      Also who needs Physics? Apparently not Rainbow Dash.

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    5. SandmanApril 28, 2011 at 7:44 PM

      I suppose that in addition to allowing them to fly and manipulate clouds, Pegasus magic creates a naturally occuring inertial dampening field around the pegasus. It is this field that allows them to pull these otherwise suicidal maneuvers without their internal organs turning into so much paste.

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    6. AluzirdApril 28, 2011 at 7:45 PM

      @Sandman
      That or their bodies are hard as hell.

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    7. AnonymousApril 28, 2011 at 7:45 PM

      A little interesting perspective: if Dash could sustain this sort of acceleration for ~.7 seconds, she could achieve (Earth's) escape velocity.

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    8. AnonymousApril 28, 2011 at 7:47 PM

      I've thought about this before and I am pretty sure this is wrong. You need to calculate centripetal acceleration, not the change in velocity in the y direction. If she came to a complete stop, this would be correct, but she just changed direction.

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    9. SandmanApril 28, 2011 at 7:48 PM

      @Aluzird If their bodies were "hard as hell" then they wouldn't be able to bend the ways they do. A dampening field allows for such movement. Also, a "hard as hell" body would only apply to the skeletal structure, not the soft, squishy organs like the brain, eyes, etc...

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    10. KurogoApril 28, 2011 at 7:53 PM

      All that math made me go derp-eyed.

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    11. AnonymousApril 28, 2011 at 7:53 PM

      Dash is tapping the speed force. Not to the lengths her red costumed brothers can but, enough to do stuff like this.

      This is good news because Dash seems like the type to not care and go too fast and be reabsorbed into it.

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    12. TecknoJockApril 28, 2011 at 8:03 PM

      This comment has been removed by the author.

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    13. AnonymousApril 28, 2011 at 8:07 PM

      @TecknoJock
      Radius is 5m. I got about 2357 G's.

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    14. GentApril 28, 2011 at 8:14 PM

      Oh geez, in that case I wonder how many G's filly Dash hit when she did the sonic rainboom at an immediate 90 degree angle? O.o

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    15. AnonymousApril 28, 2011 at 8:14 PM

      @TecknoJock
      This thing keeps eating my comments..
      ANYWAYS, the radius is 5m, dunno what you were doing there. I got about 2357 G's.

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    16. TecknoJockApril 28, 2011 at 8:20 PM

      This comment has been removed by the author.

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    17. AnonymousApril 28, 2011 at 8:22 PM

      There is actually a much easier and more accurate way to calculate how many Gs Dash had to deal with.

      First, her velocity as she enters the turn is equal to her velocity as she exits the turn. From this we can assume that the magnitude of her velocity through the turn is constant, and therefore that she is not experiencing any tangential acceleration. Her normal acceleration is given by v²/r. We assume that v = 340 m/s and r = 5m, as in the example above. 340²/5 = 23120m/s². Converting into Gs we get 23120/9.81 = 2356.8 Gs of normal acceleration. This is assuming a perfect quarter-circle turn. If you have the turn varying in radius, then you will see higher Gs in some spots and lower Gs in others. The margin for error here is rather high though, due to the rather lose estimates given. A more proper number (assuming the 5m figure is accurate to only one decimal place) would be 2000 Gs, due to significant figures rules.

      Of course, this is from a rusty memory of Dynamics, and is probably not perfectly accurate.

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    18. Thomas CoolApril 28, 2011 at 8:26 PM

      @Anonymous You are right. That's what I got, but this thing keeps destroying my comments..

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    19. AnonymousApril 28, 2011 at 8:27 PM

      Ah man, I was going to use the sonic rainboom in a physics class presentation on physics impossibilities in shows/movies, I'll have to find another scene from MLP

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    20. Thomas CoolApril 28, 2011 at 8:29 PM

      @Post-Above-This Do it! You can calculate the G's and show how it would crush any normal being into a creamy paste.

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    21. AnonymousApril 28, 2011 at 8:29 PM

      How do we know that it's a 5 meter turn? Did someone measure the turn radius?

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    22. Thomas CoolApril 28, 2011 at 8:31 PM

      @The-Post-Above-This-Because-It-Won't-Let-Me-Reply
      Guesstimated from perspective and the trees, I believe.

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    23. AnonymousApril 28, 2011 at 8:33 PM

      @Thomas Cool But now it may be seen as plagerism in the off chance that the teacher or a student is a closet brony

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    24. AnonymousApril 28, 2011 at 8:37 PM

      In comparison, a turn at the maximum Gs a human can withstand (9) and at the same velocity would have a radius of about 1310m, or 1.31 km. That's 0.81 miles, for those who don't do metric.

      I remember reading somewhere that the maximum that a unmanned plane could handle before being ripped apart was 27Gs... Which would need a 436m (0.27 mile) radius to survive that turn.

      So Rainbow Dash (and all the ponies holding on to her) are either tougher than anything the universe should be able to make... Or the laws of physics are entirely different in Equestria. Most likely the second.

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    25. AnonymousApril 28, 2011 at 8:37 PM

      I've said it before and I'll say it again. There's no way a pegasus could fly with wings that small in an atmosphere as thin as Earth's. There is clearly a much thicker atmosphere, and therefore a slower speed of sound. For reference, Peregrine Falcons frequently dive and catch prey at speeds of up to 89 meters per second with no ill effects.

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    26. DeenrekaApril 28, 2011 at 8:37 PM

      I'm getting 2356.778797 m/s^2 using centripetal acceleration, however that doesn't take into account the fact that at some point she begins fighting gravity while turning adding to the total centripetal acceleration needed. Ponies help me study for AP Physics C exam, hooray!

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    27. AnonymousApril 28, 2011 at 8:40 PM

      Also for reference, since somepony was mentioning aircraft, the Hercules X-265/271 accelerated at 100g, reaching mach 10 in 5 seconds. (much better than 27 g)

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    28. AnonymousApril 28, 2011 at 8:43 PM

      Ugh, that made my brain hurt a little. Still awesome, though :)

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    29. AnonymousApril 28, 2011 at 8:45 PM

      @Anonymous
      In straight-line acceleration, not turning flight. No aircraft could survive a turn at that many Gs and keep it's structure intact.

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    30. AnonymousApril 28, 2011 at 8:46 PM

      I'm glad that had a TL;DR

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    31. VenbrouApril 28, 2011 at 8:47 PM

      Damn...
      At 100Gs, death is almost certain. Dash pulled almost twenty times that amount in this instance. Thing is, this was only a 90 degree turn within a five meter radius. A screenshot of the one she did in Cutie Mark Chronicles shows a 110 degree turn in a pinpoint turn. That's zero meter diameter. To calculate that, one would have to divide by that zero, which isn't possible.

      TL;DR: Rainbow Dash theoretically pulled an infinite number of Gs as a filly.

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    32. VenbrouApril 28, 2011 at 8:51 PM

      @Anonymous And there's actually a way to figure out just how thick the atmosphere on Equestria is... Too bad I don't know how.

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    33. NinestempestApril 28, 2011 at 8:55 PM

      Friendship goes in (SAVE RARITY!) and magic comes out. Never a miscommunication.

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    34. VenbrouApril 28, 2011 at 8:55 PM

      @Gent
      That would be (340^2/0)/9.8. In other words, it's impossible to calculate.

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    35. AnonymousApril 28, 2011 at 8:59 PM

      Cool. Now somebody do the math explaining how Dash caught the Wonderbolts & Rarity at the speed of sound without turning them into pony paste.

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    36. AnonymousApril 28, 2011 at 8:59 PM

      The explosion you see is actually the effect of Rainbow Dash tearing a hole in the space time continuum, where she makes a gentle 2g turn through a 7 dimensional wormhole, then re-appears an instant later going in a different vector.

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    37. AnonymousApril 28, 2011 at 9:00 PM

      @Venbrou
      Well, a turn radius of one centimeter would have been impossible to notice on the show, so if we take that as an assumption for the turn radius...

      1,180,000 Gs.

      Ok, Rainbow Dash is invincible.

      Also, this conversation has made me realize that I would have done MUCH better in dynamics if everything was done in terms of ponies.

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    38. AnonymousApril 28, 2011 at 9:58 PM

      Nothing like applying real life Physics to cartoons....

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    39. AnonymousApril 28, 2011 at 10:13 PM

      As a fellow physicist, I contend this info graphic. In the initial setup, it is assumed that RD turns a radius of 5 meters HOWEVER, it is over 90 degrees (vertical to horizontal). The initial graphic is wrong. If the arclength (R*theta) of her turn was 5m, then the radius of her turn was actually ~0.06m. However, if the radius was indeed 5m, then her turning actually took 450m.

      If her arclength was in fact 450m (as the initial graphic implies) then she actually pulled ~17.7 gs (still quite a lot), not the bone-smashingly impossible 1670 gs.

      As we can clearly see, < 20 gs is far more reasonable and therefore correct.

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    40. AnonymousApril 28, 2011 at 10:16 PM

      Honestly, this math isn't anything more difficult than introductory physics plus a force diagram. Trivial really.

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    41. AnonymousApril 28, 2011 at 10:18 PM

      Must take into account the resistance and viscosity of air.

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    42. AnonymousApril 28, 2011 at 10:22 PM

      @Anonymous

      That would just be wind resistance. It would just be the cross-sectional area of Rainbow Dash (with wings outstretched if I remember correctly) times something (I forget). More importantly is the friction on the hair and feathers of an animal at those speeds. It might not be unlike receiving a noogie with a power sander.

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    43. AnonymousApril 28, 2011 at 10:29 PM

      9:13 PM, Arc length has nothing to do with centripetal acceleration!

      Average centripetal acceleration = (v_tangential)^2 / average radius of turn

      (340 m/s)^2 / 5 m * 1 g/9.81 m/s^2 = 2356 g, rounding to 2000 for significant figures.

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    44. AnonymousApril 28, 2011 at 10:46 PM

      Shouldn't there be some way of figuring out the relative weight/thickness of wingbones that a pony would have in comparison to a bird's body frame? In other words, if the atmosphere is thicker, then the winged pony body frame (which probably would have a somewhat lighter hollow bone structure similar to a bird) would not have to compromise as much as it would in our atmosphere to fly?

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    45. AnonymousApril 28, 2011 at 11:16 PM

      @Anonymous

      It seems, sir, that you are indeed correct. And rounding it 2000 is more accurate as Mach speed is actually around 315 m/s not 340.

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    46. AnonymousApril 28, 2011 at 11:19 PM

      By all accounts, Rainbow Dash should have torn her hair out, liquefied her organs, and pulverized her bones.

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    47. AnonymousApril 29, 2011 at 1:06 AM

      So physics guys, how about you explain how she can BOUNCE OFF the sound barrier and get thrown the complete opposite direction.

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    48. AnonymousApril 29, 2011 at 1:35 AM

      Yay, an aerodynamics question!

      If we remember the episode, we can see a bow shock form around Dash; she's already flying past Mach 1 if a bow shock is forming. Bow shocks only occur if the turning angle of the flow exceeds the flow's maximum turning angle for the given Mach number. Notice that as she continues to attempt to make the rainboom, the bow shock becomes an oblique shock and attaches to Dash, signifying that she's focusing on streamlining her body to reduce the drag. An oblique shock produces a lot less drag than a bow shock, but if Dash doesn't stay focused, the oblique shock will detach, becoming a bow shock again, greatly increasing the drag, and thus the forces acting against her.

      Notice also that when she stopped in midair during the first rainboom attempt, she flared her wings slightly right before she flew off. This would rip the wings off any lesser creature or machine, but not Dash. In this case, instead of her wings being torn away, she simply gets sent into a very fast, very high G pullup that sends her into the opposite direction.

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    49. AnonymousApril 29, 2011 at 1:37 AM

      Also, forgot to mention, the maximum turning angle of an oblique shock increases with Mach number. Dash is either streamlining her body to reduce the turning angle of the flow, or increasing her speed, or both.

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    50. QuicksilverApril 29, 2011 at 1:47 AM

      I love this fandom. :D

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    51. Parallel LogicApril 29, 2011 at 3:20 AM

      Hah, you bronies are silly.

      @Anonymous
      Yeah, I was going to say an acceleration related to radial velocity approach would be simpler than breaking down the event into components, gravity's pretty much negligible at this point anyway.

      I'd be curious to see the original work, I'd like to see where the 5m comes from.

      @Anonymous
      While the point you make is true, your calculations appear to be off.
      L=5=(90/180)*pi*r
      http://en.wikipedia.org/wiki/Arc_%28geometry%29#Arc_Length
      r=3.18m
      I think this answer makes more sense just from a sanity check, the trace along a quarter of a circle shouldn't be two orders of magnitude different than the radius.

      ~< 20 gs is far more reasonable and therefore correct
      >>Flying at super sonic speeds and making a turn on a dime does not really sound like 20gs would be the expected solution.

      I think of all the dynamics to look at, the number of g's pulled seems most trivial. She shouldn't be able to breathe at those speeds. She'd need machine-level precision on her wings to ensure she neither flies off in an arcing path nor crashes into the ground (jets that travel at these speeds have intricate control systems that fly the plane, it's not as simple as taking joystick input and adjusting the wings a proportionate amount). I'd have to check the math, but I'm not even certain this could be accomplished since Dash's wings have a specific area and air can only provide so much lift, meaning there's a lower bound on the radius she could turn where her wings could provide the necessary upward force to help her turn. Of course, that calculation would assume her wings were fully extended, when in fact they were crumpled at or near her side when she surpasses the speed of sound.

      @Anonymous
      ~how about you explain how she can BOUNCE OFF the sound barrier
      >>Another good cartoon absurdity

      @Anonymous
      ~Must take into account the resistance and viscosity of air.
      >>With a jet this would be perfectly counteracted (assuming a constant velocity) by the thrust provided by the engines since they act in opposite directions, so the net acceleration would be zero. Presumably Dash has some sort of similar thrust since she maintains her speed and even gains altitude without falling back below the speed of sound.

      ~Engineering brony

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    52. JeqstremApril 29, 2011 at 3:23 AM

      2356 g is the correct figure, based on the 5m radius and constant 340m/s. Seeing these calcs makes Twilight Sparkle cry...

      You don't really need to account for gravity. Dash would just need to adjust her thrust such that the resultant vector is the centripetal force required, but pulling a turn like this it's pretty much negligible.

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    53. Mather MaddoxApril 29, 2011 at 1:28 PM

      How do we even know that Equestria has the same gravitational force as Earth does? Hey, maybe it's a way bigger planet, since, y'know, its star ORBITS IT. Or maybe it's way smaller.

      Point is, we don't have all the data and most likely never will.

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    54. AnonymousApril 29, 2011 at 1:40 PM

      Now, thats some heavy free time right there.

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    55. AnonymousApril 29, 2011 at 2:34 PM

      Not really. Nothing quite like studying for aerodynamics final like applying it to ponies.

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    56. RainbowfinchbirdApril 29, 2011 at 4:27 PM

      oh man i can't read it!!!!!!!

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    57. CondorApril 29, 2011 at 8:41 PM

      Ahh, I see, they used V^2=2ad. That is an awesomeexample of using what you learned in school practically :D

      On a seperate note, I'd like to clarify how amazing that kind of G resistance is. Once you convert those G's back into an acceleration (16355.2 m/s/s) and then use the average weight of an adult pony (and let's call Rainbow Dash one strapping and fit pony and put her weight at 200kg, or 440 lbs (ponies are surprisingly weighty)). Now using 200 kg times 16355.2 m/s/s, through Newton's Second law, Force=Mass x Acceleration, you find that Rainbow Dash sustained a force of 3,271,040N, or a whopping 734,315.1 pounds of force! That's roughly equivalent to the weight of 2 Boeing 747's. And considering Rainbow Dash weighs 440 lbs, that means she sustained a force roughly 1669 times her own weight. This puts her strength on par with that of Mr. Incredible, and in the general ballpark of Superman. Now THAT is what I call a pony.

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    58. AbdulApril 30, 2011 at 3:19 PM

      I'm comin' up with 32.33... repeating of course... percentage of survival.

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    59. CondorApril 30, 2011 at 3:44 PM

      @Abdul
      Lol, I commend you for that reference.

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    60. Stormy BraveMay 1, 2011 at 6:39 AM

      In the words of Applejack, don't you use yer fancy mathematics to muddy the issue!

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    61. AnonymousMay 7, 2011 at 6:06 AM

      Well now, it looks like Rainbow Dash needs to be going up against Superman and Goku.

      I'd assume she's comprised of the same hard to destroy matter that those two are made of. Anyone have a physics sheet of calculating just what the heck superman is?

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    62. Brett TurnerMay 10, 2011 at 7:29 PM

      if this is the only thing school teaches me, then i'll consider that a success.

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    63. AnonymousMay 24, 2011 at 12:57 PM

      Assuming the calculations used are correct, I plugged in the updated speed from the MLP Physics Presentation (http://www.youtube.com/watch?v=muVfidujxRg) and come up with our little filly pulling a 38,552g turn. Miss Dash is constructed entirely of unobtainium.

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    64. AnonymousAugust 16, 2011 at 2:09 PM

      I beg to differ too. In this kind of problem,the use of the polar coordinate system is way easier than the cartesian coordinate system.
      2357 G is the correct answer

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    65. AnonymousJanuary 1, 2012 at 9:11 PM

      Blast from 2012! YES RAINBOW DASH IS NEXT PRINCESS, SHE IS BEST PONY.

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    66. PrinceWhatevererJanuary 11, 2012 at 4:29 PM

      currently using this to disprove some troll on youtube.. oh fandom, you so awesome!

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    67. TaintedFutureMarch 23, 2012 at 1:24 PM

      @Anonymous
      SO what is her max speed in MPH or mach???

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    68. UnknownMarch 24, 2013 at 5:50 AM

      I think, that Rainbow dash should have for her pet boeing 373 instead of turtle...

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    69. UnknownMarch 24, 2013 at 5:50 AM

      sorry, a boeing 737 :x

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    70. deathsheirJuly 1, 2013 at 7:28 PM

      Few problems with this. Mach 1 is Trans-Sonic. During Trans-Sonic Flight, parts of the body is super-sonic, while other parts are sub-sonic. This, creates instability, to where all manuverablity is gone, including a simple turn. So Rainbow Dash is crashing into the ground at Mach 1.

      Second the G-force on Rainbow Dash... in our world, 13g you're blackingout, no matter what kind of training you've gone through. I'll give leway for "Magic" but even magic has a set of laws that cannot be broken. From 13g - 1670g, that's an increase of 12846%...I'm kind of skeptical by that much of an increase. I'm willing to give 300% and add 1, allowing Rainbow Dash to pull 39g without blacking out, while 40g she's blacking out no matter what.

      How fast is she going? Faster than Mach 1, and if real life is any indication, the Mach Cone is broken at roughly Mach 1.2. When the Mach Cone breaks, that is when you get the Sonicboom, and in this case the rainbow along with it.

      (408.35^2)/(39*9.81)=435.84. Rainbow Dash can therefore make a 435.84 meter turn without blacking out.

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    71. deathsheirJuly 1, 2013 at 7:33 PM

      @Homfrog and it is that point that much math can go incorrect.

      And we don't even know the radius of the turn because the whole scene is completely disporportional, unless it is possible for trees in their world to be fully grown at 8 feet in height...

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