Why Rockets Don't Fly Straight Up

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  • čas přidán 17. 05. 2024
  • Why don’t rockets fly straight up? Neil deGrasse Tyson and comedian Chuck Nice explain the curious flight paths of rockets, how they get into orbit, and the Brachistochrone problem.
    What are the major differences between airplanes and rockets? We discuss rocket fuel, hydrogen fuel, and when rockets execute the roll program. Learn about how rockets get into orbit and how it looks like the Bernoulli brother’s Brachistochrone problem.
    Get the NEW StarTalk book, 'To Infinity and Beyond: A Journey of Cosmic Discovery' on Amazon: amzn.to/3PL0NFn
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    About StarTalk:
    Science meets pop culture on StarTalk! Astrophysicist & Hayden Planetarium director Neil deGrasse Tyson, his comic co-hosts, guest celebrities & scientists discuss astronomy, physics, and everything else about life in the universe. Keep Looking Up!
    #StarTalk #neildegrassetyson
    Timestamps:
    00:00 - Rockets v. Airplanes
    3:15 - Executing the Roll Program
    5:42 - How Rockets Get Into Orbit
    7:05 - The Brachistochrone Problem
  • Věda a technologie

Komentáře • 709

  • @StarTalk
    @StarTalk  Před 5 měsíci +185

    Did you watch the recent Starship test launch?

    • @RiseOfTheNew
      @RiseOfTheNew Před 5 měsíci +3

      No u missed it 😔😔

    • @Meatloaf_TV
      @Meatloaf_TV Před 5 měsíci +12

      I did and me and my friends collectively lost our minds it was one of the coolest rocket launches I have ever seen
      This and other rockets are what inspired me to do aerospace engineering in college right now

    • @Zbezt
      @Zbezt Před 5 měsíci

      In my observational understanding it was perfect aside from some electrical issues from possible internal condensation

    • @Meatloaf_TV
      @Meatloaf_TV Před 5 měsíci +4

      ​@@Zbeztfar from it something went seriously wrong with the engines on the booster after hostage and we still don't know why starship was terminated

    • @Zbezt
      @Zbezt Před 5 měsíci

      ​@@Meatloaf_TV stick to gaming and programming its evident you cant see through walls without an algorithim assist

  • @beau-urns
    @beau-urns Před 5 měsíci +439

    Sometimes I realize how much Chuck has learned… dude is VERY educated on these topics now. He’s always been bright, and now he’s bright AND educated. What’s a combo

    • @mctuble
      @mctuble Před 5 měsíci +25

      The importance of having smart friends

    • @usaintwinnin7312
      @usaintwinnin7312 Před 5 měsíci +8

      But just memorizing what someone told you doesn’t make you educated. Maybe a little more knowledgeable, but that’s it.

    • @mctuble
      @mctuble Před 5 měsíci +8

      @@usaintwinnin7312 I definitely doesn't hurt to have people talking to you about stuff like this and making you think rather than the latest internet challenge.

    • @beau-urns
      @beau-urns Před 5 měsíci +56

      @@usaintwinnin7312 he’s not memorizing. He responds to complex prompts and even when not prompted with analysis and input
      It’s clear he understands, doesn’t memorize. He’s also quick AF on his feet
      Not sure why you’re being a hater.

    • @jeffstorey9147
      @jeffstorey9147 Před 5 měsíci +12

      Remember that his job is the play the uneducated role, and set NGT up to go on a rant. He’s a real pro at it too.

  • @aaronscott1907
    @aaronscott1907 Před 5 měsíci +74

    I love how profound his off-the-cuff statements are: "When you're curious and you're mathematically literate and you know a little bit of physics, these questions come up in your life, and you're compelled to solve them." Perfect wording to convey why math and science are so important in a practical sense. Great episode!

    • @timtim4664
      @timtim4664 Před 3 měsíci +1

      @@HopDavid It is still a step in the right direction. People think Neil is a "God of Science".... which is a step above "Bill Nye the science guy". I'm just happy the common people still look for science as an explanation to the question of "why?". He may not always be correct, but he is in the right location...because a lot of people are lost.

  • @silentblackhole
    @silentblackhole Před 5 měsíci +106

    This is the first time I've ever heard of the Brachistochrone problem, and then when you flipped it up side down for the orbit solution it blew my mind! Loved it! Thanks guys

    • @RollMeOne_Kenobi
      @RollMeOne_Kenobi Před 5 měsíci +7

      Vsauce has an excellent video about it where he goes way in depth and it honestly only gets more mind blowing lol

    • @Svensk7119
      @Svensk7119 Před 5 měsíci

      Me too! Well done, doc!

    • @stephenselby4252
      @stephenselby4252 Před 5 měsíci

      Hang on. For going up, wouldn’t the rocket gain more velocity going sideways first and then flipping upward? But then it would have to overcome the friction of the denser air on the surface. Is the Brachywhatever really relevant?

    • @michaelslee4336
      @michaelslee4336 Před 2 měsíci

      The Curiosity Show with Rob and Deane (in Australia) from when I was a kid did this, mind blown even back then.

    • @falklumo
      @falklumo Před 28 dny

      @@stephenselby4252The goal is not for going up but going fast. Velocity is where the energy is needed for, height is nothing (less than 1%). But you also need the right direction if not going to the moon. And you’re right, you need to be out of atmosphere as quick as possible too or you just melt …

  • @hidasineto
    @hidasineto Před 5 měsíci +75

    The new "edited" videos are much better than the past non-edited videos. Please, keep investing on that. Love you guys! Been watching you since the big bang

    • @BJSMOOVEyt
      @BJSMOOVEyt Před 4 měsíci +1

      13.8 billion years plus?!

    • @gm2958
      @gm2958 Před měsícem

      Except they would be much better without the added “music”.

  • @RandomInTheChat
    @RandomInTheChat Před 5 měsíci +41

    Hi guys! It's RJ,while I was watching this video I noticed something I found very relatable so I figured I'd share my thoughts and findings with you and absolutely excited to do so.I've been skateboarding for going on 30 years and I can confirm that dropping in ( rolling down) a quarter pipe ( quarter circle ramp) one definitely obtains more speed than if one was dropping in on a bank ramp (triangle ramp) that is the same height so long as all other variables are the same ,no matter what human powered vehicles happens to be their psychotic hearts desire from bikes to scooters,roller blades and my personal favorite skateboards the results are the same. As a long time and multiple account subscriber ( not great with passwords lol) and lifetime learner I want to thank you both for making learning fun and things like science and physics approachable not just for myself and my 4 boys but for so many others Thank you Neal, Thank you Chuck .I hope that you two continue to do what you do for many years to come!

  • @RiseOfTheNew
    @RiseOfTheNew Před 5 měsíci +89

    Love watching star talk. Learn, laugh and get my mind blown everytime ❤❤❤

  • @nagasreekanth7178
    @nagasreekanth7178 Před 5 měsíci +13

    The Brachistochrone Problem is the coolest thing ever I heard in this channel

  • @jesperlett
    @jesperlett Před 5 měsíci +23

    The reason for the brachistochrone curve in space flight is not immediately obvious, however. The reason to go vertical in the beginning is to get clear of the dense atmosphere as quickly as possible to reduce drag.

    • @VonJay
      @VonJay Před 4 měsíci +2

      I’d argue that the brachistochrone curve in space IS immediately obvious, since the craft needs to fall towards earth and the fastest way to get that started (and for efficiency reasons) is with a curve of shortest time. The curve is not practical in heavy atmosphere, and that’s exactly what they covered.

    • @shrodingerschat2258
      @shrodingerschat2258 Před měsícem +1

      KSP taught me that!

  • @SunilNayak714
    @SunilNayak714 Před 5 měsíci +11

    Lets take a moment to appreciate the effort of Bernoulli brothers 👏

  • @DonCarlione973
    @DonCarlione973 Před 5 měsíci +11

    Out of the scientists of our era I feel that Neal D. Tyson is one of the most open-minded and honest of them. He's very unbiased which is a trait of a great scientist. Nothing is impossible to him and I love how he breaks everything down for us. I have quite a bit of respect for this man I appreciate what he does and the true person that he is.

  • @josephmarino2654
    @josephmarino2654 Před 5 měsíci +27

    Thanks Neil and Chuck. I always enjoy StarTalk.

  • @sapelesteve
    @sapelesteve Před 5 měsíci +51

    Terrific explainer Neal! Like many others, I never knew the exact way a rocket was launched into space. Thanks & I hope that you, Chuck & your families have a terrific Turkey Day & Happy Holidays. 👍👍🦃🦃🌲🌲

  • @Famezonen
    @Famezonen Před 5 měsíci +8

    7:17 never ent to college. But Neil will always be my fav teacher. Mind boggles me in every video.

  • @nickdsp8089
    @nickdsp8089 Před 5 měsíci +58

    For those wondering how they came up with the word 'brachistochrone'. It is from 'brachis' = 'βραχύς' in greek and its meaning is 'sort' combined with 'chrone', 'chronos' from the greek word 'χρονος' which translates to 'time'. Βραχυχρόνιο = sort time. We use this word even in the modern greek.

    • @jpdemer5
      @jpdemer5 Před 5 měsíci

      Not βράχιστος χρόνος?

    • @nickdsp8089
      @nickdsp8089 Před 5 měsíci

      @@jpdemer5 maybe superlative but not used

    • @Paolo_De_Leva
      @Paolo_De_Leva Před 5 měsíci +2

      _"A _*_brachistochrone curve_*_ (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time')..."_
      From Wikipedia (en.wikipedia.org/wiki/Brachistochrone_curve)

    • @nickdsp8089
      @nickdsp8089 Před 5 měsíci

      @@Paolo_De_Leva
      it is like : good, better, best
      as i said βράχιστος is superlative of ΄βραχύς not a completely new meaning.
      Ιnstead of βράχιστος we prefer βραχύτατος nowadays which has the same meaning.
      BTW the comperative is βραχύτερος

    • @Paolo_De_Leva
      @Paolo_De_Leva Před 5 měsíci

      ​@@nickdsp8089 You wrote _"maybe superlative."_ Wikipedia gives a better answer.

  • @6thdayblue59
    @6thdayblue59 Před 5 měsíci +8

    “Mike Drop”
    I’m in tears laughing………
    Chuck is a legend.
    Thank you guys x

  • @TonyVSTheWorld
    @TonyVSTheWorld Před 5 měsíci +8

    The illustration of the Brachistocheone curve blew my mind when you flipped over!!!! Made all the sense in the world!! Crazy!

    • @bodan1196
      @bodan1196 Před měsícem

      We Kerbals calls it a gravity turn.

    • @fiercefix
      @fiercefix Před měsícem

      ​@@bodan1196Anti gravity turn !?

  • @newbornmaple87
    @newbornmaple87 Před 5 měsíci +8

    ""Bernoulli Brothers were the life of the party"! 😄

  • @SoupedUpCustoms
    @SoupedUpCustoms Před 5 měsíci +3

    Chuck Nice really adds up a whole lot more to these kinds of explainer vids. More of this and chuck nice as well!

  • @henocktaddese6892
    @henocktaddese6892 Před 5 měsíci +2

    Thank you Neil, you remind of questions we never bothered to ask, and love the mic drop! Thank you Chuck for all the fun, you never miss a beat!

  • @Carverman08
    @Carverman08 Před 5 měsíci +13

    Just started the episode but I know it’s going to be another great video!

  • @yournway
    @yournway Před 5 měsíci +5

    That was certainly one of the geekiest but most elegantly explained maths translated to actual rocket science explanation I have ever heard, even on this channel.
    Thank you!

    • @Shaw_Cardboard_Treasures
      @Shaw_Cardboard_Treasures Před 4 měsíci +1

      You know the phrase it's not rocket science. Well this makes it a lot simpler to understand and explain to even a 6yr old.

  • @Cristianotorres1998
    @Cristianotorres1998 Před 5 měsíci +8

    Wow, that’s great to learn, Thank You Neal ! 👏🏼🙏🏼

  • @toon3ry
    @toon3ry Před 5 měsíci +13

    Love and light to you and the family Neil. Condolences to you on the death of your mum❤🕊️

  • @flyingo
    @flyingo Před 5 měsíci +4

    That announcer from the Hindenburg disaster was from my hometown Wichita, KS. Just a fun fact.. well, to me anyway.

  • @Darren51283
    @Darren51283 Před 5 měsíci +4

    So, what do a solid fuel rocket booster and an EV battery pack have in common? Once they ignite, you can't turn them off.

  • @BogdanSilviuAron
    @BogdanSilviuAron Před 5 měsíci +2

    I follow a lot of scientific channels... A lot... And always learn new things or understand the new things I've learned... But in this channel I understand everyday things, that I thought I already understood, but I find out that I was wrong, or not completely correct, and after seeing your videos I completely understand them correctly. This channel it's Unique! Thank Neil and the other guy 😜

  • @andrewm8429
    @andrewm8429 Před 5 měsíci +2

    i was just thinking about why they roll the other day! Perfect timing!

  • @danielludlow8960
    @danielludlow8960 Před 5 měsíci

    I love these explainers so much...I literally talk out loud like you can hear saying how awesome it is!

  • @diegofernandez4789
    @diegofernandez4789 Před 5 měsíci +2

    Great explainer guys. Loved it!

  • @whistletutor
    @whistletutor Před 3 měsíci +1

    I've been a space nerd my whole life and I've never heard of the Brachistochrone problem and how it relates to trajectory - that's why I love Star Talk, always something to learn!

  • @michaelwoo4279
    @michaelwoo4279 Před 5 měsíci +1

    Thank you for starsplaining! Love it every time!

  • @ra2186
    @ra2186 Před 5 měsíci +2

    08:05 There's usually not a concept completely new to me when I watch these. This fact about rocket launches blew my mind!

  • @michaelccopelandsr7120
    @michaelccopelandsr7120 Před 5 měsíci +5

    Neil and Chuck for 2024!

  • @markousey5564
    @markousey5564 Před 5 měsíci +1

    Wow you guys explain and simplify the science enough to make it digestible. I love the rapport and humour making it entertaining too. Simply my favourite CZcamsrs. Thank you.

  • @likemeordont5951
    @likemeordont5951 Před 5 měsíci +2

    Thanks for letting me hang out with you guys, and you know, teaching me stuff.

  • @ghahrai
    @ghahrai Před 5 měsíci

    Thanks Neil & Chuck . I have learned more from your explainers than i could have from a hundred books. ❤

  • @sosheeanand3537
    @sosheeanand3537 Před 5 měsíci +3

    This is mind blowing explanation honestly.

    • @VonJay
      @VonJay Před 4 měsíci +1

      It’s probably one of the smoothest explanations I’ve ever heard. I didn’t have to struggle to understand anything.

  • @Madhamz
    @Madhamz Před 5 měsíci +1

    From one Chuck to another Chuck. Watching my two favourite shows on youtube,Inside the NBA and Star Talk.

  • @PedroFernandes1984
    @PedroFernandes1984 Před 5 měsíci

    I really love this channel. always learn something new, and the comunication is awesome! Congratulations!

  • @kirandeepchakraborty7921
    @kirandeepchakraborty7921 Před 5 měsíci +1

    One of the best episode ever. ⭐⭐⭐⭐

  • @eldwdubu6968
    @eldwdubu6968 Před 5 měsíci +11

    Learn and laugh with these two. Always fun, Thanks!

  • @thomchristensen990
    @thomchristensen990 Před 5 měsíci +2

    Great episode. The editing on this is superb. Thank you.

    • @thomchristensen990
      @thomchristensen990 Před 5 měsíci

      Also looking forward to your year in review talk on Jan 4th at the Museum. Last year was eye opening.

  • @mglasson
    @mglasson Před 5 měsíci +7

    "When the pitchforks come for Elon..." 😝

  • @brawlwith_me
    @brawlwith_me Před 5 měsíci

    Nice new intro music my favourite was the first one you guys had used in older videos ❤

  • @helpus1456
    @helpus1456 Před 5 měsíci

    First time watching. I really enjoyed this segment. Didn't understand more than have of it. But, you two were fun to watch.

  • @BobbyJOnline
    @BobbyJOnline Před 5 měsíci +1

    If we had these videos when I was a Kid in the 80’s.. I’d be GLUED to the TV 24/7

  • @EdwardKilner
    @EdwardKilner Před 2 měsíci

    Excellent. Learned stuff. BASc EE ‘71 and still love learning.
    Granddaughter is new Systems Design Engineering grad, now in Masters program. Third generation Engineer. My parents never met her, but would have been delighted with her and astounded by us all.

  • @Stisse12
    @Stisse12 Před 5 měsíci +3

    Best Science YT channel !

  • @peterjackson2722
    @peterjackson2722 Před 5 měsíci

    That was fascinating, thankyou!

  • @RodrigoM3llo
    @RodrigoM3llo Před 5 měsíci +1

    This video gave me a completely new perspective on explaining rocket's optimal trajectory

  • @GAMEOVERTBH
    @GAMEOVERTBH Před 5 měsíci +1

    Short video but gold information 😜👍🏿💯

  • @mattgraham5992
    @mattgraham5992 Před 5 měsíci

    Thanks again. My favorite way to start my late night.

  • @Shaw_Cardboard_Treasures
    @Shaw_Cardboard_Treasures Před 4 měsíci

    That was very informative. And not over my head.

  • @mickeybrumfield764
    @mickeybrumfield764 Před 5 měsíci +2

    Long live Neil, Chuck, and Star Talk bringing us such good information about the party oriented Bernoulli brothers and rocket science. ❤

  • @TheKobiDror
    @TheKobiDror Před 5 měsíci

    Fantastic vibes between the two of you AND getting to know a little bit more each time. I didn't knew that fact and it goes straight into my useless knowledge drawer within my memory 🤓

  • @charlessukati4866
    @charlessukati4866 Před 5 měsíci

    Brilliant lesson from these two ❤❤❤🎉🎉🎉

  • @zelmoziggy
    @zelmoziggy Před 5 měsíci +1

    Apparently Tyson saw the TikTok video that incorrectly asks which path is the “shortest” among three choices for a ball to roll.

  • @tamirleibovich
    @tamirleibovich Před 5 měsíci

    8:33 classic Neil/chuck good vibes 😍⭐️ love you guys !

  • @GAMEOVERTBH
    @GAMEOVERTBH Před 5 měsíci +1

    Great information 👌🏿100%

  • @alafayette1100
    @alafayette1100 Před 5 měsíci

    Love these guys so much ❤

  • @DJ-xu5fv
    @DJ-xu5fv Před měsícem

    Best 10 minutes of my day! Thanks guys.

  • @SureshKesraniOfficial
    @SureshKesraniOfficial Před 5 měsíci

    Amazing truly amazing discussion

  • @yaelbj
    @yaelbj Před 2 měsíci

    That mic drop was really good! I am very glad I watched this! 😌

  • @cliftonjohnson8069
    @cliftonjohnson8069 Před 5 měsíci

    Great Star Talk!!!

  • @tokonjudo
    @tokonjudo Před 5 měsíci +1

    it is heart warming when you laugh at chuck for real :)

  • @akubuosamuel4194
    @akubuosamuel4194 Před měsícem

    Today I know about brachistochrone problem and rocket trajectory. ❤❤
    Thanks Sir!.

  • @OnceSomeFunATime
    @OnceSomeFunATime Před 5 měsíci

    That's the coolest thing I learned this year

  • @abelmedina-aispuro3716
    @abelmedina-aispuro3716 Před 5 měsíci

    I can’t stop watching this episode. I love it

  • @davidconlee2196
    @davidconlee2196 Před 5 měsíci

    That bit about the Brachristochrone problem was pretty cool

  • @Goodjobeveryone
    @Goodjobeveryone Před 5 měsíci

    Thank you guys!!!

  • @BakedPhoria
    @BakedPhoria Před 5 měsíci +2

    Background music is nice for recaps but the silence is better

  • @JamesWilliams-gv7zd
    @JamesWilliams-gv7zd Před 27 dny

    I used to live in Titusville Florida. It was awesome to see and hear the the shuttle coming and going

  • @fiddlefiver
    @fiddlefiver Před 5 měsíci

    I'm learning so much, Thank you 🎉

  • @petergreen5337
    @petergreen5337 Před 5 měsíci

    ❤thank you very much for this beautiful lesson

  • @StofftiereLindner
    @StofftiereLindner Před 5 měsíci

    Love your talks

  • @Bojeezy
    @Bojeezy Před měsícem

    Neil and Chuck need to do a standup special together. I would watch that also.
    Their banter is just as entertaining as the information being provided. Like they are just 2 friends talking over a cup of coffee.

  • @gregkent8955
    @gregkent8955 Před 5 měsíci

    Subject: roll at Shuttle launch: I was part of "lunch with an Astronaut" at Kennedy Space center a few years ago and it was with Chris Hatfield. "any questions?" he asked. I asked "Why does the Shuttle roll after launch?" Answer: The launch pad is facing the wrong way for Shuttle launches because it was orientated for rocket launches. The Tank needs to be on the Nader side for safe release. This concludes my 'explainer'. You and Chuck are a perfect match. Chuck keeps you from going too far by using humor. Keep 'em coming.

  • @JKa244
    @JKa244 Před 5 měsíci

    Launch trajectories are beautiful. Rocket, ballistic, air breathing - such a beautiful interplay between simple physics and more complex fluid mechanics

  • @jamesonrichards5105
    @jamesonrichards5105 Před 5 měsíci +1

    Neil should do a video with chuck to see how much he’s actually learned. I’ve seen some other medical channels do the same with their video editors and it’s always great to watch. Either a physics, astronomy, or general knowledge questions about space.

  • @kevinfletcher1999
    @kevinfletcher1999 Před 23 dny

    Even on topics I know something about, Neil always has some great facts I knew nothing about. He’s my favourite educator.

  • @KokoroInt
    @KokoroInt Před 5 měsíci

    This was awesome!

  • @joeyannece6197
    @joeyannece6197 Před 5 měsíci +1

    The kids in the class were watching Challenger blow up.

  • @VikingTeddy
    @VikingTeddy Před 5 měsíci

    Oh yess! Thank you Neil. This will help my Kerbaling so much!

  • @christianmarquez3143
    @christianmarquez3143 Před 5 měsíci

    Thank you so much. Love you guys.

  • @shantanuaphale3966
    @shantanuaphale3966 Před 2 měsíci

    Gr8 explanation by world's greatest of educators ❤

  • @sherwincantago6384
    @sherwincantago6384 Před 20 dny

    Thank you!..
    I learned something today.
    😊

  • @phillippolack145
    @phillippolack145 Před 5 měsíci

    AMAZING WORK!

  • @MedicalMyke
    @MedicalMyke Před 5 měsíci

    Man, I love how much love you both have!

  • @BurqueMD
    @BurqueMD Před měsícem

    These are so wonderful, thank you!

  • @msbudmsbud7593
    @msbudmsbud7593 Před 5 měsíci

    Very cool guys, thank you !!

  • @Grubik
    @Grubik Před 5 měsíci

    8:25 was the ooooooh moment for me how did i never think of that. I have had same feeling when i saw at some other video explanation of propeller as two sails on cylinder.

  • @OptemusPr1meXBox
    @OptemusPr1meXBox Před měsícem

    Sooooooo dope!!! Thx ND-T

  • @pratikmali9277
    @pratikmali9277 Před 5 měsíci +2

    I came here thinking I’d know everything he has to say about brachistochrone (because I watched about it in Vsauce) but that thing about rocket launch trajectory blew my mind 🤯

    • @hassaniq0777
      @hassaniq0777 Před 5 měsíci

      ​@@HopDavidas a physicist, I can say we all sometimes like to make stuff up.
      I made this up as well

    • @VonJay
      @VonJay Před 4 měsíci

      @@HopDavidI was trying to figure out how the b problem relates and I finally figured it out though it was plainly stated in the video. I keep forgetting, even though it was explained, that to stay in orbit you have to fall to earth fast enough to keep missing it. I think the point that confused me was the frame of reference since the shuttle is no longer trying to escape earth and is just simply trying to fall toward it. So the b. problem is really about efficiency (as stated in the video) and continuously missing earth as you’re falling toward it (as fast as possible initially).

  • @6dmiller
    @6dmiller Před 2 měsíci

    Hmmm, for a very long time I thought a cycloid was the fastest path down a hill, but I don't remember where I got that. Glad I watched!

  • @vidtrax662
    @vidtrax662 Před 5 měsíci +1

    I can see how the basketball evolved to Saturn 😂😅
    Always enjoyed and learnt something from your channel 👍

  • @jeffreysokal7264
    @jeffreysokal7264 Před 2 měsíci

    Excellent....as always!

  • @ct1igu401
    @ct1igu401 Před 5 měsíci

    Brilliant & funny, thanks.

  • @chernohmansaray402
    @chernohmansaray402 Před 5 měsíci

    You are amazing. Modern genius.. classic demonstration

  • @miguelraulgonzalez7744
    @miguelraulgonzalez7744 Před 4 měsíci

    Beautiful explanation. Thank you Neil.