The Real Physics of Roller Coaster Loops

A look at the physics principles and calculations that engineers use to design roller coaster loops.
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Video Attributions:
“Wildfire Front Seat POV - Silver Dollar City” by AmusementInsider
“(4K) Kraken Back Seat POV - SeaWorld Orlando” by AmusementInsider
“Holiwood Nights 2017 | Voyage and Thunderbird Behind-The-Scenes Tour! Holiday World” by OhioValleyCoasters
“Dragon Challenge Off-Ride POV | Universal Orlando Islands Of Adventure 20” by OhioValleyCoasters
“Batman The Ride Off Ride Cinematic Footage [Six Flags Over Georgia 2019]” by OhioValleyCoasters
“The Incredible Hulk Roller Coaster! AWESOME B&M at Universal Orlando! Off-Ride POV HD” by OhioValleyCoasters
“OzIris - Parc Astérix (4K 60FPS On-Ride POV NO COPYRIGHT)” by Coaster Hipster
“The Monster - Walygator (4K 60FPS On-Ride POV NO COPYRIGHT)” by Coaster Hipster
“First Person Riding Roller Coaster "Medusa" at Six Flags” by Brian Wilson
Still Image Attributions:
“Raptor at Cedar Point in motion” by Craig Lloyd
“Centrifugal Railway Detailed Sketch” by Unknown author
“Revolution” by Prayitno
“ASTM F24 Logo” by ASTM International
“Carolina Cyclone (Double Loop)” by Coasterman1234
“Euler spiral” by AdiJapan
“Europa-Park - Blue Fire Megacoaster” by Jérémy-Günther-Heinz Jähnick
Music from Epidemic Sound
#RollerCoasters #Themeparks #ArtofEngineering

[@AVdE10000]

When I clicked the video I thought this was just gonna be a brief explanation on why circular loops can cause whiplashes, and thus they use more egg-shaped designs. But mad respect for the incredibly detailed and in-depth explanation given!

[@AVdE10000]

Also this gives me flashbacks to the intense math and physics classes from the one year I studied industrial engineering, god if only I were smart enough to finish that education :/

[@prestonhall5171]

The trick to any kind of higher-level education is to use as many resources as possible. That way if you don’t get it in class you still have an alternative. I’m a CS major and I plan to graduate knowing more that I was taught in lectures, so to aid with that I use non-curriculum textbooks and the internet (such as KZbin videos). Yes that’s more expensive but SO worth it in the long run I imagine.

[@ItsNoelle.]

@@prestonhall5171 yeah seriously if you're not 'cheating' then you're just doing yourself dirty, its so hard to learn everything just from lectures especially in physics math and engineering unless you're insanely intelligent I guess

[@whannabi]

@@ItsNoelle. true and sometimes even the written lesson given online isn't explained well so that's another reason to cheat. Uni just provides you a structure, a routine because you'd probably never study it by yourself anyways and then it's your job to do the rest, that's how I see it. And the degree of course but that's obviously expected.

[@Smashbro2013]

I like how so many early engineering designs and inventions were basically just death traps

[@jil7379]

That’s how we learn around here

[@dylandowling8133]

A real world application of calculus?! I never thought I'd see the day...

[@advaitpathak4330]

Lol i am doing engineering right now and i never thought i would see calculus in real life............

[@advaitpathak4330]

Lol i am doing engineering right now and i never thought i would see calculus in real life............

[@williamlee7119]

There are easier ways to do this where you can have more freedom of design. In college I was able to do this in excel, fairly similar to your approach but with more freedom in regards to design and g forces desired

[@KuraIthys]

Well, depends on what you mean by 'real world'. I've studied computer game development for most of my adult life (in an amateur context), and there's a lot of things that come up very frequently. Physics engines are, unsurprisingly, full of calculus. But, a 3d rendering engine will be full of geometry, vectors, matrix multiplication, and a whole bunch of methods of creating discrete plots of various functions. I hope you know your line and circle equations... And all their various forms. XD - because different forms are useful for different things. Especially when trying to optimise. Parametric equations are especially useful. Ever wondered how collision detection works? There are simpler forms that are easier to calculate, but the most general form is based around the intersection of two line segments (vector dot product), with the parametric form giving you the actual point of intersection. Alternatively, in 3d space it's more likely to be the parametric value of a line intersecting a plane. (even just being able to calculate the intersection of a line segment with an infinite plane is extremely useful.) Yeah... I guess calling that 'real world' is arguable, but I learnt way more mathematics from trying to write actual code to do these things than I ever did from formal mathematics or physics classes...

[@khyoman]

If it wasnt for calculus you wouldnt be able to share your stupid opinion

[@ArtofEngineering]

Quick note! If you are going through the math in this video yourself, please note that the derivation at 13:03 contains an error. The first equation for "G" has the "g" term in the wrong place, and the correct equation for "G" is shown at 3:50. The resulting equation for "r" should have g*[G - cos(theta)] as the denominator. I hope this helps if you're trying to replicate the plots and it doesn't work properly.

[@williamlee7119]

back in college I made a excel spreadsheet similar to your approach, I used a workaround for the calculus, using extremely small time increments (for graphing purposes on excel). The result is you can input your desired g forces as a function of time and excel will output the shape as a graph. (also works for hills or anything really) you can come up with some pretty interesting track elements

[@willdavis752]

Hi, what computer software do you use to plot these?

[@ArtofEngineering]

There are many options available. I used Microsoft Excel but you could also use something like MATLAB or Mathematica.

[@willdavis752]

@@ArtofEngineering Is Microsoft Excel able to give you the equation of the relation generated? or just an image of it?

[@ArtofEngineering]

It's not possible to get an equation because the differential equations cannot be solved analytically. Excel or any other software can only plot the solution by solving the equations numerically.

[@vinimetal]

Holy shocks, you need an engineer degree to understand this video for real

[@andrewslater6846]

Well, from my perspective, I need this video to understand getting an engineering degree.

[@tahsin28]

'Well, from my perspective, I need this video to' pass 2nd year mech engg.

[@trueamerica911]

Thats why Tony Hawks first Skateboard Looping attempt was a total failure...he build a circular loop, which caused the high changes in g forces making it also extremly dofficult to ride.

[@Wintersghost135]

Dude, that is so observant. I saw that video too. I never realized. Also, I never though they’d have to consult with an engineer to design the shape of the loop. I could see him fighting the g forces at the bottom of the loop.

[@AmusementInsiders]

Always love how informative your videos are! Keep up the amazing work man!

[@ArtofEngineering]

Thanks so much!

[@brennaobrien9576]

I’m taking algebra 2 right now in high school and have wanted to be a roller coaster engineer since I could remember. I love this explanation. I can’t *comprehend* it, but I kind of understand it. This makes me want to engineer roller coasters even more. Mathematics is a fascinating subject.

[@AlexAn7]

No kidding, the most interesting video I've ever watched. I hope that you ll make more videos in the future that shows the practical use of mathematics. Keep doing!!!

[@matheusmatos8055]

Exactly what I had in mind! Keep up the good work, bro!

[@fintamin]

Finally an excellent video to showcase this matter! Not only does it explain the maths, but it also dives into other aspects of a loop as well. Very very good job. /engineer

[@nicholasr79]

We'll need to use a little calculus. * *busts out Diff Eq*

[@nicholasr79]

@Pizza Pie wrong. Differential equations don't solve for variables, they solve for a set of all possible variables... or equations. You're taking one possible example (f=ma) and using that to represent ALL diff eq. If diff eq was so easy, then why are some problems unsolvable? Don't bother answering, it was rhetorical.

[@DisHsDd]

😁🖒

[@spb1179]

Neat. I’m in calculus right now and watched this when it first came out but didn’t understand it then. I still wished you did more math to explain what was happening when you were doing partial derivatives

[@spaceboi2581]

Once he started talking about calculus my brain checked out

[@electric7487]

Hey at least the stuff you learned has a purpose

[@Krosstic]

El_Ectric I didn’t learn anything.

[@OkOk-tu3gc]

I learn more in these videos than I have ever learned in school science class. I sometimes wonder whether that is good or bad.

[@matthewclark4510]

Oompa Loompa this is beyond me but if I pay attention long enough instead of scrolling through the comments I’d could do it.

[@prestonhall5171]

It’s good because then you at least know the concepts and how to apply them and that’s the end goal of teaching. The how is less important than the what and why.

[@user-yr3uj6go8i]

Then the chances are too good that your school failed you.

[@cat1n282]

Interest = remembering

[@psu4127]

Love these videos! As an engineering student, it helps me put together all the components I learn!! Thanks!

[@kaydub0623]

Math: circular loops are dangerous for riders Schwartzkoph: *hold my beer*

[@aoeuable]

I always thought that Nessie at Hansa Park had a circular loop (and yes it's a Schwarzkopf), but looking at pictures that include the bottom, you can see that only the top section is circular: www.webstagram.one/media/BS-e6Blh352 Nearly 40 years and still running.

[@artya]

@@aoeuable its still not circular

[@PabloGonzalez-hv3td]

@@artya - The Mind Bender in West Edmonton Mall in 🇨🇦 is a Schwarzkopf and has perfectly circular loops I can confirm the Gs wiki says over 5 en.m.wikipedia.org/wiki/Mindbender_(Galaxyland)

[@CoasterCrafter662]

@@PabloGonzalez-hv3td no, Mindbender has a mixup of clothoïdal and circular loop, but it's not 100% circular.

[@PabloGonzalez-hv3td]

@@CoasterCrafter662 - Much more circular than clothoidal they're more intense than anything at Cedar Point or Canada's Wonderland I know from experience

[@bleeve9425]

This is why us unprofessional coaster designers use FVD and NoLimits2. It spares us from this headache of math.

[@germanlover4561]

Bleeve 942 are you a roller coaster designer?

[@bleeve9425]

German Lover Unprofessionally, yes. It’s a whole community.

[@AgateBrick]

You forgot Planet Coaster.

[@bleeve9425]

AgateBrick97792 While in PlanCo you can build a roller coaster, it’s a lot harder than in FVD and NL2. To build a realistic coaster you have to spend 10 hours building it.

[@AgateBrick]

@@bleeve9425 I agree 100% it is harder to get a realistic coaster. Smoothing in PlanCo is the worst culprit in this task. I think part of it is just how the PlanCo editor works in general. NL2 has a bigger learning curve in using the editor at least Imo.

[@clcoolgaming]

This guy really deserves to be appreciated the level of depth in his technical analysis is just amazing! Keep the up the good work man!

[@spencebot]

I don't know how I hadn't seen your videos before, they fit perfectly with my viewing history. This is my favorite kind of video, basic math and engineering to explain things I hadn't really thought about. Please keep making awesome and informative videos.

[@rcmaniac25]

Holy Math, Batman! That was an amazing video. As a software developer and coaster/theme park enthusiast, I've often gone "I want to play Roller Coaster Tycoon, Parkitect, Planet Coaster, No Limits, etc." but once I hit the limitation of them, gone "what is needed to make my own version of these games?" but get stuck at the math. I don't have the time to make such games, and don't know how they came up with their calculations or similar, but to see the math that is needed just to do a loop is awesome. It makes me want to try and make my own game (again... no time...), but it's so interesting to see. Also interesting to think about how early designs such as Arrow and Schwarzkopf were done before computers really had the computational capabilities to make such designs, and to think how many limitations of the formula come from "the train doesn't have enough flex to be able to do that tight of a loop" and still needing it to fit within safety calculations.

[@NguyenHaNhutLong]

Thank you very much for giving us an insight into these mathematical derivations. It gives me a better insight into how engineers work out a design of a specific task. I hope you will keep up doing more good contents like this!

[@TimMc333]

Amazing video! Maybe do one about the physics behind zero-G/heartline rolls?

[@user-yg97f5hfvh]

Looks cool :)

[@NipkowDisk]

I sometimes deal with the clothoid or Euler spiral in highway surveying and heard a while back that they were also used in roller coaster loop design. Very good explanation, and major kudos for pronouncing Leonhard Euler's last name correctly.

[@zagaberoo]

Great video. The insight into design aspects of a real coaster is super cool.

[@hedgehog3180]

This was an amazing indepth explanation of the physics at work here, love it!

[@Zigaroula]

Your videos are insanely good. Keep going !

[@Crossedkiller]

I don’t know shit about physics but I’m stoll fascinated by this video lol

[@joepeters4032]

That's very interesting about all the different formulas for the different shapes of loops, as well as friction energy losses and positions of the cars. Those are much more precise than what we learned in school.

[@tristanmoller9498]

THIS is great content

[@rosem4897]

I’m starting my studies in civil engineering in Quebec in september and your videos inspired me to choose that field of study! So thank you for that!!

[@marcovillalobos5177]

I really liked that you got into the meat and the actual equations! :)

[@brandonelliott7297]

Hey man, really really insightful and informative. Although I may not have made it past algebra 2 in high school and thus don’t really understand most of these equations, I was still able to follow along all the way through and understand the message you were trying to convey pretty easily. Well done!

[@rcb3000]

Please make more of these. I love them!!!

[@2222paulr]

Love it!! Wish I found your videos when I was a much younger engineering student!

[@utuberaj60]

Hats off for sucha good and crisp presentation of this fun subject with real physics and math. Got hooked!

[@satyris410]

Incredible video, very well described mathematics and engineering principles. Thank you

[@sarahfleischer3571]

Thank you so much for this video. Can't find many others explaining Rollercoaster G forces

[@sigmunddogbert7418]

Thank you! It was too good, really. Great explanation. Finally an engineer who is not afraid to show his calculations!

[@sebastianboss6865]

I'd love to watch a similar video about camelbacks, zero g rolls and heartlining in general

[@williamlee7119]

I was working on a project where you could input g-force and the output was any rollercoaster design element you could think of

[@3godzilla3]

Whoa. This actually really explains why I love pretty much every coaster I've ever been on with the exception of Scorpion at Busch Gardens. The one loop on that ride is a near perfect circle, which explains the massive headache I get after riding. 😩

[@thomas.02]

TLDR: sit in the middle for the most consistent experience; sit on the ends of the cart for more extreme Gs amazing derivations and explanation of the topic, would love to see more :D

[@xMarto96]

Amazing video man!! As soon as i saw it i open matlab to create my little coaster editor and it is coming along really nice :)

[@bpark10001]

This is a most excellent analysis! But there are other loop variations you should cover in another video. The first (and most important) is the helical loop, stretched out along the axis of the "circle". This permits reducing G-loads while maintaining the circular path of the loop (viewed from the axial direction). An example of this is the coaster that was at Knott's Berry Farm "Corkscrew" (which was the first coaster to take riders upside down). The second (less elegant) way to do this is to tilt the plane of the "vertical" loop at an angle. (In the extreme, the loop is horizontal!) An example of this is Knoebel's amusement resort's "Twister".

[@josephtraverso2700]

Thank you so much for this video!

[@jeroenkodde2438]

Wow, designing roller coasters with differential equation. Just stunning!

[@Adamhc75]

Awesome video!!! Makes me wonder about the crazy math that went into the Premier Rides Sky Rocket II design like The new coaster Tigris at Bush Gardens Tampa.

[@tweedyburd007]

I could use this to study for my engineering exams

[@Jet-Pack]

Hey, this was a really good explanation! Thanks :D

[@Coyotek4]

Whoa, didn't expect to stumble into a 3blue1brown video! Awesome work!

[@mtado4933]

physics exam tomorrow, mad respect for the vid, really helpful

[@raygreen4675]

I got my degree in meteorology ( atmospheric/engineering science) way back in 82. It took forever, if ever, to figure out why I had to have 4 semesters of calculus including Dif EQ. Especially since I went into television and not research. The $$$$ you know. Lol. Thanks. Much enjoyed.

[@azaelue5]

AOE : Explaining complex maths My brain : NEEEED COFFEEEEE

[@ikewillis2687]

This is a fantastic video. Way to whip out the diffey eques

[@lubieplacki7409]

Omg man this channel is a banger. I love it!!!

[@Andrew-ep4kw]

Excellent explanation of the shapes of roller coaster loops. Also, this video demonstrates why advanced math skills are essential to an engineering career.

[@vg5028]

Really good and in-depth video!

[@gigi199596]

Excellent topic! I would really appreciate other computations on other type of coaster elements :)

[@seriomarkj]

I so wanna try a circular loop just once to feel that sharp change in g's

[@scs998]

I would love to see this kind of mathematical break down of sidways loops, like cork screws, barrel rolls, heart line rolls, and how changing the point of rotation changes the way you experience a curve, and also how sitting on different sides of the train would feel. I guess that's a lot 😅 but do whatever you want.

[@zeeshanali579]

Well explained 👍

[@floydpinkerton7954]

Great video. Very informative and well-paced. One point of clarification - for the Constant G loop equation derivation around 13:00, you input one of the linear kinematics equations to substitute for velocity. Can you explain why that is valid even though we are not exclusively working with linear motion? Thanks.

[@Saki630]

I wished this was a problem in my engineering courses. Instead I had to solve double pendulums and a 2D cart with springs, dampers, and rotation about its axis(prob 6 degrees of freedom) as it goes over obstacles. That was the only time I had to solve nonlinear DE's or PDE's in the case of vibration theory. This video would serve as a great problem to solve after learning some of this stuff and allow for good writing and analysis. Good job.

[@thisnicklldo]

The maths doesn't help if the welder has a hangover

[@Krebzonide]

We were just talking about this recently in my physics class, and we will be going to six flags on Friday for an end of year trip.

[@douadouard1009]

Teachers finding excuses to go to amusement parks with their class, I like that

[@twinturbostang]

Fascinating!

[@AceNinja2112]

Awesome video! I feel smarter by watching it!

[@dosmastrify]

I remember the trig names and limits but no Mas, still near vid though

[@seriouscomment8311]

Thanks, now I know how to make ideal loops on planet coaster using this concept!

[@seandemers4207]

Awesome video!!

[@Wayne_Robinson]

Roller Coaster Tycoon Death Coaster design tips! I appreciated this reminder of physics calculations that I haven't done in, umm, a few years. edit: Thanks for the analytical solution to the "front or back row?" question.

[@christophervolosy7100]

FANTASTIC VIDEO!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@nicolebrown1927]

Alrighty young man. I've never been interested in math but I honestly enjoyed this video. Thanks for explaining how roller coasters work and keep it up. Your a very talented and smart young man.☺

[@Weminan]

Makes me love maths and physics all the more

[@TripleTSingt]

There are still modern roller coasters with a partially circular loop, as this shape gives great hangtime (the feeling of falling "up" while the train is upside down). On the other side, constant G shapes are used, if the desired effect is an intense, but smooth loop. It depends on the manufacturer, what effect they want (eg. Gerstlauer and Mack Rides seem to favor Hangtime, while old-school B&Ms are very intense).

[@Silacide]

Faxmachine.mp3 Awesome vid! Really didn’t know there was... THIS much math behind a ‘simple’ loop! *Good to remember for my backyard rollercoaster that I can’t build yet*

[@Silacide]

@Eric Thomason No young, I'm 16 and I live in Belgium so I don't have a gigantic lawn like americans have 😂 mine is +- 20x15m. But my poney and my moms horse are standing there in winter.

[@Silacide]

Eric Thomason Yes. Yes it does.

[@m.pavankalyan2650]

Seriously amazing 😍😍😍😍😍🙌

[@wobius]

great video!

[@bwabbel]

i'm not gonna pretend to understand any of this, cause i don't. but it makes me appreciate the guys who design roller coasters in a way that makes me enjoy them and not get whiplash or die

[@gaeb-hd4lf]

Awesome video! You earned my suscription lol

[@user-yv5mt9rm3d]

What a video, thanks!

[@BryanzHAn03]

17:23 "Medusa" in Six Flags Discovery Kingdom

[@CHEESYhairyGASH]

This is great.

[@justdoingwork4072]

Do you think you could do the science behind flying coasters? Like tatsu from magic mountain and some other ones from B & M.

[@intotheunknown804]

Loved it

[@iamsecrets]

Great video

[@just-Lauren]

Wow this was cool!

[@nicolasberginiat2959]

Wow! Impressive video ! But how do you know all of that ? Have you just made the operations because you had the time or it's part of your daily job so it's everyday knowledge for your?

[@loganolson3968]

I knew everything you where calculating until you brought up Eulers method. Hahaha....I need to learn calculus. Great video. Also, what's the difference between centripetal and centrifugal acc?

[@HPD1171]

great now to apply this to my rides in planet coaster

[@ignasquattro]

THANK YOU.

[@makemeink7373]

What about the first loop on the Texas Tornado at Wonderland in Amarillo, Texas?? I would love to see the physics and maths behind that unique loop😂😂

[@kcwz]

well done.

[@Wintersghost135]

I think I got it. Centrip force = (mv^2) / r. The car has the highest velocity at the bottom of the loop (both on entering and exiting the loop). With a very high velocity at the bottom of the track to cause a very high g force, you need to negate that by raising “r” in the denominator. Then, the car has lost most of it’s velocity at the top of the loop, (having converted its KE into potential energy) so you need to lower the radius to maintain the same g force. Gravity will also counter g force at the top. So, rearranging the equation and ignoring m, as you did, you get r = v^2 / Force. V is known (from KE minus potential energy) so you could possibly plug in whatever g force you want (say 3) then that will give you the desired radius for each part of the track (whilst adding or subtracting the gravity vector - add the gravity vector for the bottom half and subtract gravity from the top half of the loop). Granted, it’s more primitive and clunky, but may work for those of us that don’t know how to apply calc yet (most of us). Chain bear talks about F1 turns with increasing and decreasing radius. Except those loops are flat and not vertical. Your channel is like his - only cooler with more math. With respect to him, he had my favourite channel up until now.

[@YorrickLiu]

0:19 this one in Paris' Asterix park is one of the best in France

[@theflyer0504]

Enjoyed the video and it was very insightful, however at 12:50 I think it was G=(v^2/(r*g))+cos(θ) and not G=(v^2/r)+g*cos(θ), then multiplying both sides by g gives g*G=((v^2)/r)+g*cos(θ), then the expression for r becomes r=((v_0^2)-2*g*y)/g(G-cos(θ))

[@bpark10001]

Comment about your analysis and how longer trains affect the results: you address longer trains, but only from the standpoint of the rider not being in the center of the train (and how this tends to skew and increase the peak G-force experienced), but still treat the train's mass "being at a point" as far as speed is concerned. I think you have not accounted for the length of the train's affecting the speed because the mass is distributed in a way that lowers the train's CG. To explain, I will assume perfect circular track and no friction losses, as you did initially. Case 1: train "very long compared to the circumference of the loop". In this case (taken to extreme), the train's velocity does not change at all from loop entry speed because most of the train remains at loop entry height. You have a constant centripetal acceleration toward the center, and the rider experiences 2 more G's at the bottom than at the top, like your case of the constant centripetal acceleration loop with a short train. (This is the same as many vertically spinning "flat rides".) Case 2: train length equal to the loop's circumference: centripetal acceleration varies by 2 G's toward the center (the train's CG only rises to the center of the circle as train crests the top), and rider's G's vary by 4 G's bottom-to-top. Case 3: train length is 1/2 the circle's circumference: centripetal acceleration varies by ~3 G's toward the center (the train's CG only rises ~ midway between top and center of circle), and rider's G's vary by ~5 G's bottom-to-top. By combining a longer train with the loop profiles you describe (along with helical loops), it is possible to make "more circular" loops with tolerable G-forces.

[@smakkacowtherealone]

IV'E SEEN THAT JEEP SUV AD SO MUCH

[@hazonku]

I actually did my 8th grade science project on this back in the late 90s. Partly because science and building a rad display with foam insulation tubing & marbles, but mostly just because I wanted an excuse to go gather "real world data" at Six Flags. LOL. Either way, I got first prize at my school science fair, my district, and my county. Had no idea I'd have to figure out trigonometry and calculus to pull it off but it was worth it. Oddly enough my main go to for real data was Revolution, the coaster from 1:04 for its history & the fact it was a bit of a yawner compared to everything else at the park by then. It wasn't uncommon that you could ride it a few times in a row without having to get back in line before they finally kicked you off.

Hi, love this video its so interesting! I just have one question I cant seem to wrap my head around, how could you have a rollercoaster loop with constant G force experienced the whole way around, since wouldn't you always experience 0 G's at the top of a loop ( since the normal force = 0 at top of loops, where you get that weightless feeling, and just start to come out of your seat)?

[@BigDBrian]

Some more things to take into consideration: There is essentially a minimum radius before your trains cannot handle it any smaller. For instance, the top of the carts would be pushed into each other due to the curvature. Why you would choose one shape over the other. Blue Fire in Europa Park is an excellent example, using its shape to create hangtime at the top to create thrill. Obviously this is much more subjective, but still interesting.