Dot Physics

Flying R2-D2, you are doing it wrong

You know I can’t help but like Star Wars. Even with the new stuff, I watch it. Recently, I was watching the Clone Wars cartoon and noticed something odd about the way R2-D2 flies. I know what you are saying….”the odd thing is that he flies at all. Why didn’t he fly in episodes 4-6?” Who knows. Here is the best image I could get of R2-D2 flying (from wookieepedia).

i-ecec3843338e4cd73595064e4fdfaa1a-2010-01-20_150px_r2_flying_ep_2_model.jpg

What is wrong? Well, maybe you can’t tell from the image I posted. Here is a diagram of flying R2-D2.

i-c03597afbe4eb2ffe9cffc6848d0f941-2010-01-20_untitled_1.jpg

If R2 (I can call him that because we are good friends) was flying like that, why would that be a problem? That would be a problem if he was going at a constant speed. I re watched the end of Clone Wars (the movie) and it seems like R2 is flying at a constant speed. Why is that problem? Let me draw a free body diagram.

i-2463fa3a30f798474b22c90b18ec9910-2010-01-20_untitled_3.jpg

If I assume that R2 is flying horizontally at a constant velocity, and the the “thrust” is at an angle theta from the vertical then:

i-9b251c857c502450a1931fd904eeda5f-2010-01-20_la_te_xi_t_1_3.jpg

I guess I should say I am using g as the local gravitational field for whatever planet he (R2) is on. Is R2 even a he? Maybe R2 is a she. I don’t know. The point is that the vertical component of the thrust must equal the vertical component of the gravitational force. So what you say? Well, that leaves the horizontal component of the thrust so that the net force in the horizontal direction is:

i-c27206d84bda66edea27f1a6fca65937-2010-01-20_la_te_xi_t_1_4.jpg

Obviously I have missed the air resistance force on R2 while he is flying. That is what makes the net horizontal force zero, right? Well, let us calculate this. First, some assumptions (or maybe starting declarations):

  • I would assume the world is like Earth because they seem to jump like a person on Earth. Needless, I will call the local gravitational field g.
  • Also, I will assume other things are like the Earth. Most importantly, the density of air (I will call rho) – if it is even air.
  • I am not sure what the air drag coefficient for R2 is, but I will call this C. I assume this value will be somewhere between a brick and a sphere.
  • Let me call R2′s linear speed v. This will need to be estimated later.
  • I will need to estimate three things about R2, its mass ( m ) volume ( V ) and cross section area in the direction of motion ( A ).
  • Finally, I need to know R2′s thruster angle. I will call this angle theta from the vertical.

If R2 is moving at a constant velocity, then the forces in the x (horizontal) and y (vertical) directions will be zero. I will assume there is air resistance in the horizontal direction, and I will use the following model for the magnitude of air resistance:

i-b3cc17f57f64051ec7a0960669bb653e-2010-01-21_la_te_xi_t_1_5.jpg

So, the x- and y-direction equations for force are:

i-1a7da52a0594284545747834f9ed1499-2010-01-21_la_te_xi_t_1_6.jpg

I will solve the y-equation for Fthrust and put that into the x-equation. Doing this, I get:

i-e867f3f10432de623722da10b2a37f7b-2010-01-21_la_te_xi_t_1_7.jpg

Let me tell you where I am going. If R2 is firing is thrusters a little forward and moving at a constant speed then I suspect R2 has a very low density and the thrusters are not pushing very hard. So, from the above stuff, I will solve for the mass.

i-d0ed8818bee4dd366a1b07c868d6a947-2010-01-21_la_te_xi_t_1_8.jpg

Here are my estimations (you are welcome to come up with your own):

  • rho = 1.2 kg/m3
  • Area: Wookieepedia says that R2 is 0.96 meters tall. Using tracker video on an image of R2, I am going to approximate it as a rectangle that is 0.42 meters by 0.62 meters for an area of 0.26 m2
  • Wikipedia lists the drag coefficient for a smooth sphere as 0.1. It has a smooth brick with a coefficient of 2.1. A skier has a coefficient of 1.0. Wikipedia does not list the drag coefficient for R2, but a value of around 1.0 seems reasonable.
  • For the velocity, I took it a little far. I was just going to ballpark guess at his speed, but I didn’t. I used Tracker to look at R2′s motion in Clone Wars where he flies to rescue Padme. From this, I get a speed of 2.3 m/s.
  • I already said I would assume Earth-like gravity. So, g will be 9.8 N/kg
  • Theta is about 35 degrees (although it could be as high as 45 degrees).

Using these values, the mass of R2 is 0.1 kg. Yes, 100 grams. How do I know I am correct? I know because Wookieepedia doesn’t list R2′s mass or weight. They know it is silly, so they left it off.

If this mass is so low, I think R2 doesn’t even need thrusters. He would just float (which would actually change my calculations above – I left off the buoyancy force). By my estimations, R2 is about .42 meters in diameter. This would put its volume at about 0.1 m3 and R2′s density would be:

i-bcd6a3fc48c5b69e039324c900dfbc67-2010-01-21_la_te_xi_t_1_9.jpg

I was originally thinking that maybe R2 was made of styrofoam – but that has a density of about 40 kg/m3. So there.

Pre-emptive comment

I know someone is going to say “hey, chill man! It is just a movie. Don’t ruin it by bringing in all your physics stuff.” My reply, someone has already ruined the Star Wars movies, his name is George. Just kidding, I still like Star Wars.

Update:

There were two questions in the comments that I addressed in a second post. The first question: Was R2 flying at a constant speed – I included the data. Second: what was the angle of the thrusters.

Flying R2-D2 Part II

Comments

  1. #1 Neil
    January 21, 2010

    I just want to say: R2D2 shouldn’t fly, and George stomped all over my childhood memories.

  2. #2 matt
    January 21, 2010

    I think this is better explained by a simple anti-gravity device. It’s either that or a wizard did it.

  3. #3 william dyer
    January 21, 2010

    This leads me to remember the scene at a Comic/Sci-fi convention from an older Simpsons Treehouse of Horror episode .

    Professor Frink: Yes, over here, n’hey, n’hey. In episode BF12, you were battling barbarians while riding a winged Appaloosa, yet in the very next scene, my dear, you’re clearly atop a winged Arabian. Please do explain it.

    Lucy Lawless: Ah, yeah, well, whenever you notice something like that, a wizard did it.

    Professor Frink: I see, all right, yes, but in episode AG4 –

    Lucy Lawless: Wizard.

  4. #4 Hamish Grant
    January 21, 2010

    Look. R2D2′s been closely associated with the Jedi for so long (probably from his very creation date, I don’t know the full canon, don’t kill me!) that he is probably qualified to use a little Force-fu now and then if the need presents itself. I’m thinking he can balance the physics of his ‘flight’ using a bit of Force-push-and-pull. That’s your answer.

  5. #5 Luci
    January 21, 2010

    This was great! You had me at the force diagram. So you’re basically saying R2 has the density of a dust bunny?

  6. #6 Chris
    January 21, 2010

    It seems to me your main hypothesis (“If R2 is moving at a constant velocity”) doesn’t hold. I admit I haven’t seen the movie in a long time, but I don’t recall much in the way of extended level flight. I doubt R2′s flight is very well modeled in the movie, but I don’t think this analysis is applicable either…

  7. #7 EurekaBizB
    January 21, 2010

    Have you seen this TED Talks video?
    A New Beautiful Theory of Everything
    http://www.ted.com/talks/garrett_lisi_on_his_theory_of_everything.html

    I don’t understand much about Physics or Quantum Physics, but maybe R2D2 and you can find a (or THE) missing puzzle piece in Garrett Lisi’s beautiful 8-dimensional model of elementary particles and forces.

    May The Force be with you!

  8. #8 Dave
    January 21, 2010

    I agree with the antigravity device. Based on your calculations, there is no theta that gives a reasonable answer for his mass. Obviously, he would need to turn the a-g device on and off with his thrusters, since having such a low mass would cause no end of problems as he rolls along.

  9. #9 Dan
    January 21, 2010

    I love this. Two questions (and feel free to mock me as I was never any good at physics):

    - Is theta the angle between R2′s leg and F(g)? It’s not labeled in your diagram.

    - What would be the correct theta angle for R2′s thruster leg to be firing at, presuming a realistic mass for R2? He does seem quite heavy when he gets electrocuted and falls to the ground (in Ep 4 I believe?)

  10. #10 Mario
    January 21, 2010

    IN this photo in wookieepedia
    http://starwars.wikia.com/wiki/File:R2_flames.JPG
    you can see also R2D2 flying and I would tell its theta angle is not greater than 10º or even 5º which would lead in a still sissy 8 times factor correction.

    Also, his forward lean possition could make a glider effect (like long jump skiers) that could help reducing the necessary vertical thrust. That force will end in the denominator of the final formule substracting to your denominator, that is, increasing the possible mass of R2.

  11. #11 felix
    January 21, 2010

    since you don’t actually know he is travelling at a constant speed, or what the drag and gravity are, it’s a bit pointless to say he’s doing it wrong

  12. #12 Mario
    January 21, 2010

    #8, of course there is a theta that will give a reasonable mass: 0,5º will make it weight 80 kg. Quite a bit for a computer but still a small weight for that lot of servos and strong metal.
    And also it makes sense, if you need a lot of power to keep in the air a big mass, you only need a bit portion of it (0,5º!!!) to push it through a quite soft medium as Earth air.

  13. #13 Rhett Allain
    January 21, 2010

    @felix,

    Actually, I did measure R2′s velocity. It was constant. I will reproduce that graph and post it if I get a chance.

  14. #14 Chris Lee
    January 21, 2010

    I hate the fact they made him fly at all. Stupid CGI.

    My R2 (which does not fly, and is built to R2 Builders club spec) has 18.25″ diameter (body cylinder only, not counting width of legs, feet, etc.), stands roughly 42″ tall (in 2-leg mode), and weighs approx 250 lbs. with batteries.

    If you need other dimensions for R2, got to http://astromech.net

  15. #15 Stuart
    January 21, 2010

    You are assuming that the visible thrusters are the only forces being applied by R2. There might be some sort of technology akin to that of landspeeders at work as well… possibly accounting for extra lift.

  16. #16 GERSH
    January 21, 2010

    oh ahh no sorry, you’re not factoring in all anti -gravity components, nor the calculations of that galaxies anti magnetic repulsion, in his formula therefore you’re actually quite wrong. The spatial indents at the front and back of d2 on both sides are instant-’nute anti gravity wells. The propulsion from the “legs” are solely for minimal direction at best. Earth like gravity being a factor only as much as down is up in relativity. R2D2 is therefore able to fly by it’s AI input; the same input that enables it to help a pilot navigate their galaxy “far far away” at close to speed of light “warps” we still to this day can only “fantasize” of (with our calculations) in the hope of developing safe holding containers for anti-matter. :)

    http://www.amazon.com/Art-Amos-Star-Sketch-Book/dp/0975927302 –filled with delicious star nutrients…ahhh! yum!

  17. #17 NelC
    January 21, 2010

    It seems to me that it should be possible to estimate R2′s mass by the way he moves in the first trilogy of movies (Episodes 4-6). When R2′s being played by Ken Baker inside the R2 suit, the wobbling makes it seem about the order of 100kg, to my eye. The radio-controlled R2 seems lighter, maybe half that or less. But that’s just my instinct, I wouldn’t know how to quantify that.

  18. #18 rikchik
    January 21, 2010

    Agreed with other people that R2 is probably using anti-gravity of some sort. We see that kind of tech in use on similarly small items constantly throughout the series, from Luke’s landspeeder on Tatooine, to frozen Han being floated through Cloud City, to the imperial speeder bikes on Endor. Perhaps R2 has an underpowered floating unit, which only neutralizes some of its mass – thus the jets are needed.

  19. #19 Billy Brooks
    January 21, 2010

    I have to admit, I kinda suck at physics…but I had fun making him fly. What did you all think of how he walked up stairs earlier in the movie? Did I get any closer with that one? :)

  20. #20 Eric Lund
    January 21, 2010

    R2D2′s been closely associated with the Jedi for so long (probably from his very creation date, I don’t know the full canon, don’t kill me!)

    R2D2 was first built by young Anakin Skywalker (as was C3PO) some unknown interval before we meet them in Phantom Menace. They end up in Padme Amidala’s custody after Anakin goes off for his Jedi training, and after she dies her daughter Leia inherits them. It’s easy to explain why R2D2 flew in episodes 1-3 but not 4-6: whatever mechanism allowed him to fly broke down sometime in the ~20 year gap between episodes 3 and 4, and nobody knew how to fix it.

  21. #21 Poe
    January 21, 2010

    Who said those were rockets? When I saw those flames I thought.. “cool exhaust”.

    Isn’t that the exhaust to the well established ‘device’ that allows all craft in the movie to fly? Why would he use rockets when such wonderful alternate technology exists?

    That’s like writing about how the spaceships couldn’t be made of cast iron.

  22. #22 DG Lewis
    January 21, 2010

    No, the flaw in your model is due to the fact that during the time of Episodes 1-3, the Force is out of balance…

  23. #23 Anonymous Coward
    January 21, 2010

    I’m with #1 and #14.

    But now this is all starting to make sense! The reason why the flying was wrong was that R2-D2 DOESN’T fly: that part of the movie was fake!

  24. #24 Jellodyne
    January 21, 2010

    Midi-chlorians in the hydraulic fluid. Fixed! Good day, sir.

  25. #25 Jared
    January 21, 2010

    #20 actually on C3po was built by Anakin. RD-D2 worked on the Queens ship before her or the Jedi meet Anakin.

    As to why he didn’t fly in episodes 4-6, it is very obvious. His memories were wiped at the end of episode 3 and he forgot how to fly.

    Also, my theory is not that he has an anti gravity device but rather a more complicated warp type bubble device that allows him to reduce his mass. Thus the mass calculated in this theory is correct inside the reduced mass bubble.

  26. #26 tenenbaum
    January 21, 2010

    What about dynamically controlling the amount of thrust produced by the thrusters and adjust their angle accordingly to produce constant speed? I’m surprised the people at Lucasfilm didn’t actually write a control system for the thing and simulate it out in a physics engine. I’ve seen kids program control systems for dynamically stabilized rocket hovercraft things in physics engines, it’s not that hard really.

  27. #27 Jon H
    January 21, 2010

    Obvs, the mass is all in the hidden, weakly interacting “dark midget”.

  28. #28 SMurph
    January 21, 2010

    Hell, X-Wings are capable of VTOL flight without any apparently downward thrusters. I don’t have as much of a problem with “how” R2 flies as I do with the fact that it just looks stupid.

  29. #29 Mikee
    January 21, 2010

    How much thrust do his boosters produce?

    What is the gravity on the various planets/ships that R2D2 has used the boosters?

    How can you solve a math equation when the two most important variables are unknown?

  30. #30 MattF
    January 21, 2010

    Note that air has a density of about 1.3 kg/m^3– so R2 is really just floating, and the thrusters are just for show.

  31. #31 hanov3r
    January 21, 2010

    #25 – actually, there’s nothing in the dialog for Ep3 showing that R2′s memory gets wiped. Which, of course, leads (in part) to the greatest conspiracy theory ever.

  32. #32 Neal Freeland
    January 21, 2010

    So they just should have shown R2D2 pointing his feet straight down to counteract the gravity force, and then back a little to accelerate or forward to decelerate.

  33. #33 amphiox
    January 21, 2010

    I agree with #21. Those aren’t thrusters. It’s some kind of repulsor-lift anti-grav tech. What looks like thrusters aren’t thrusters at all, just the exhaust of whatever fancy power source they have for the anti-grav device venting its waste, or maybe some kind of weird spacetime fabric distorting effect associated with anti-grav that just looks like thrusters to our ignorant 20th Century eyes.

    And those point-blank galleon-style capital ship space battles? That’s because both sides have some kind of unmentioned super stealth/jamming technology that makes all long distance targeting futile. Yep. Something like that.

  34. #34 Justa Busta
    January 21, 2010

    Unless I missed it, you have also made a pretty big assumption in that the thrust is always equal. R2-D2 could be using a “thrust variator” or some such thing that could easily explain horizontal flight* especially if he had the nozzles that could also rotate.

    *Please don’t come back by saying the flame is almost always the same size. That is easily explained through luxury and the desire to have such a design feature of stabilized flame.

  35. #35 Justa Busta
    January 21, 2010

    Oops, I did miss it, but it is in the comments. See #26 by tenenbaum.

  36. #36 Fran
    January 21, 2010

    Sooooo many comments! All of you should make sure to sign up for Geek/Nerd Pride Day.
    http://www.facebook.com/event.php?eid=231355998949&ref=nf
    http://en.wikipedia.org/wiki/Nerd_Pride_Day

  37. #37 Rhett Allain
    January 21, 2010

    @Billy,

    That is ok, it R2 still looks cool. I checked out your site and your sample video has a way better quality shot of flying R2 than I got from using my flip camera on the TV.

    Oh – and cloudy with a chance of meatballs was awesome.

  38. #38 Billy Brooks
    January 21, 2010

    @Rhett,

    Thank you so much! I love that you did this. It’s really fun to see the discussions. I’m as big a R2 fan as anyone and it made my day. And it’s on my other favorite blogs too (Boing Boing, io9, etc…) Nicely done!

  39. #39 Paul Camp
    January 21, 2010

    They do it with wires.

  40. #40 Sean Meaney
    January 21, 2010

    So what? R2-D2 is made entirely from Spidersilk fibers.

    The new R2 Astromech is made from Light weight spidersilk and has a fun personality to keep the kids entertained. Now made from the same materials as your light weight millitary A, B, X, and Y wing Aerospace fighter. Buy two and get one free.

  41. #41 nacho
    January 21, 2010

    what planet is he on the gravity might just be different

  42. #42 Loves To Spooge
    January 21, 2010

    I can’t believe this. How much time did you waste with this crap? Star Wars is shit full stop. The most overrated film in history.

    And to those losers who go ‘George stomped on my childhood’, he didn’t do the prequels for you idiots, he did it to make money of more kids who are as stupid to think it is any good.

  43. #43 esfas82
    January 21, 2010

    Along the same lines, maybe you could ask why the spaceships have to bank in order to turn. The thrust seems to be along the fore-aft axis of the ship. Must be the air in space which transmits the noise of the ship going by and the explosions. (It even transmits the sound faster than sound)

  44. #44 Justin
    January 22, 2010

    @43: Banking makes turns more comfortable for the people inside them. Look at tilting trains and cambered/banked roads.

  45. #45 Jameson
    January 22, 2010

    Wookieepedia also says that R2s gender is “Masculine programming” :D

  46. #46 Ian Kemmish
    January 22, 2010

    R2D2 can understand English, which is far harder than speaking it, but chooses to whistle instead. He’s just toying with us primitive carbon-based life forms.

  47. #47 SDGlyph
    January 22, 2010

    Oh, OK… this explains so much. Not only do we now know that R2 had a portable suspensor unit (he has to, the physics proves it!), we also have an explanation for why he suddenly doesn’t fly in the original trilogy: the suspensor broke and was never fixed / replaced.

    Thus are the teetering towers of fanon built. ;-)

  48. #48 Poe
    January 22, 2010

    If anyone was wondering why the new Star Wars sucks:
    http://www.youtube.com/user/RedLetterMedia#p/u/7/FxKtZmQgxrI

  49. #49 Flipside
    January 22, 2010

    Hmmm very interesting this is…

    And for the record R2 weighs about 200Kg. (I should know I have one!) Maybe 210Kg post Christmas ;)

    Cheers.

  50. #50 carolinahaze
    January 22, 2010

    I didn’t read any of the above comments, but I’m pretty sure that trying to apply modern physics to a supposed galaxy that has developed anti-gravity technology just fails. You did not take any of that into consideration in your numbers.

  51. #51 Tuttle
    January 22, 2010

    R2D2 was first built by young Anakin Skywalker (as was C3PO) some unknown interval before we meet them in Phantom Menace.

    For the same reason? ie, To “help out” his mother?

    So to help mom do the dishes he first designed a robot with no arms before designing one who has arms, but they don’t move. What, was Shmi in desperate need of a translator and an astromech? That’s all C3-P0 does and I didn’t see any starships in the backyard.

    And what was the point!? Look in Wato’s shop; there’s a freakin’ protocol droid just laying around! Full skin and everything. And why, if you’re building your own droid, would you make it look like just another mass produced unit? You can’t swing a dead womp rat in that universe without hitting a protocol droid.

    That kid had all the imagination of a rock.

  52. #52 jimmy from the farm
    January 22, 2010

    im on lsd and just fucking lost. i stumbledupon the wrong thing right now.

  53. #53 Mike
    January 23, 2010

    There is something very obvious which has been over looked relative to the total forward thrust necessary to propel a droid at constant speed on a planet similarly equivalent in gravitional force to planet Earth. Namely, you forgot to account for a wizard helping R2 fly.

  54. #54 Gaevs
    January 24, 2010

    Have you considered that R2 may have a repulsorlift installed?, that will lower it’s mass enough to use it’s trusters…

    I know that it’s stupid, but could be a explanation..

  55. #55 GAZU
    January 25, 2010

    If the force/s is/are balanced, then the speed os constant. If not, then there is an acceleration in some direction.

  56. #56 1d30
    January 25, 2010

    I love how so many people say your analysis isn’t applicable, for whatever reason, but don’t give any concrete reasons for that assertion. Or they mention some piece of technology that R2 must have that makes him lift like that – technology that is simply not present in the detailed specifications and diagrams floating around.

    Mainly I think they’re just unwilling to accept anyone saying anything negative about the fiction to which they are slavishly devoted. Fanatical, one might say.

  57. #57 Ganesh Nair
    January 27, 2010

    I was having a dumb facebook argument with someone about this article, who said you failed due to your assumption of Earth gravity. Whereas I thought it was a perfectly logical assumption, given the fact that nobody else on Geonosis was floating around like morons. But in the interest of completeness, I looked up the gravity of Geonosis, and came up with a value of 8.82, 90% of Earth gravity, which, from your equation, results in a mass of 0.13 kg. So, you weren’t far off.

    The internet is full of nerds.

  58. #58 Otto
    February 9, 2010

    Around 1979, Estes introduced an R2D2 model rocket kit. This flying can was “stabilized” by clear plastic fins on its robot legs and washers that the kit builder had to add to the dome based on how well it behaved when whirled around on a string.

    I recall that when launched, mine went about 20 feet straight up and seemed to hover briefly before it tipped over and began to circle, until the ejection charge blew the dome off.

    I took it as a lesson that flying R2D2′s are just wrong.

  59. #59 Wikiborg
    February 13, 2010

    R2D2 flew in episodes 1-3 but not in 4-6 because George Lucas had acquired the factors of production (CGI) using the money paid by the Star Wars franchise customers.

  60. #60 Paul
    April 17, 2010

    I have one of those Estes R2D2 rockets on my desk right here. I’ve flown it a number of times, and while it is prone to damage on landing, it is great fun. The key is I modified it to take a larger D engine, a C engine is too weak for a good flight.

    It’s retired from flying now, one of the fins broke and I doubt I’ll find a replacement.

  61. #61 Neil
    February 3, 2012

    Maybe R2 has a repulsorlift in his third leg?

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