Ant On A Rubber Rope Paradox

An ant is placed on one end of a rubber rope and he begins walking at about 5cm per second. As he’s walking, the rope gets stretched… and stretched… at a rate of 10cm per second. The rope is getting stretched faster and longer relative to the ant’s consistent walking pace.
Can the ant ever get to the end of the rope? Is he caught in an endless, impossible trek in which the end keeps getting further and further away?
This classic paradox has very real implications to how we understand our position in a rapidly-expanding universe.
********** LINKS ************
The Create Unknown Podcast: bit.ly/2TKVDdc
What Is A Paradox?: • What Is A Paradox?
Ant On A Rubber Rope Discussion:
bit.ly/2DYQ7it
Harmonic Series Proof on Khan Academy
www.khanacadem...
Harmonic Series Proofs
scipp.ucsc.edu/...
Harmonic Series Proof
web.williams.e...
***********
Written by Matthew Tabor, Michael Stevens and Kevin Lieber
Huge Thanks To Paula Lieber
www.etsy.com/s...
Get Vsauce's favorite science and math toys delivered to your door!
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Twitter: / vsaucetwo
Facebook: / vsaucetwo
Hosted, Produced, And Edited by Kevin Lieber
Instagram: / kevlieber
Twitter: / kevinlieber
Website: kevinlieber.com
Research And Writing by Matthew Tabor
/ matthewktabor
Special Thanks Michael Stevens
/ vsauce
VFX By Eric Langlay
/ ericlanglay
Select Music By Jake Chudnow: / jakechudnow
MY PODCAST -- THE CREATE UNKNOWN
/ thecreateunknown

[@michaelsgotguts]

i wish he hadn't even mentioned needing another arm and it just wiggled onto camera with no explanation or acknowledgement of it

[@noiber]

Video produ'tion HRs called for it my niño. (No idea what my niño meand btw)

[@shrug1250]

An arm unknowingly slumps into battle!

[@madampawsy1903]

He didn't explain it but he acknowledged it

[@zepfan9993]

He’s not Micheal

[@C00LI0DUDE]

vsauce is 50% off today

[@billyii2877]

I tried this experiment. In my version it ended with the rubber rope breaking and the ant being launched across the room, so yeah, no paradox there.

[@stonecoldpizza]

the ant reached the end

[@egg04]

@@stonecoldpizza 💀💀💀

[@jonnym4670]

well if you have a 3rd hand that doesn't happen

[@stonesphixsballer5786]

@@jonnym4670 🙂

[@iconofthicc6086]

Just like our universe wait *uh oh*

[@Brakathor]

Initially I seriously thought that the "paradox" would be that while the ant could THEORETICALLY reach the end, as you stretch the rope thin, its legs could no longer touch the rope, and therefore it would only be able to flail its legs aimlessly while flopping around on its belly... Yeah... A harsh reminder of the shoddy fundamental architecture in my brain that caused me to fail math.

[@tyleranderson3178]

Hey man, I like it. Outside of the box thinking. That’s the type of stuff they should encourage in school, creative thinking like that.

[@mirandapanda5439]

Same here man.. got that same imaginative mind that made me fail math time and time again lol

[@gem4036]

No, that's actually a really interesting take. If I was your teacher I'd give you extra points for creativity :)

[@gem4036]

@@mirandapanda5439 Thinking like you do gets you jobs other people can't do. Yeah they have the education but creativity is important in all we do. The great CEO's and inventors are creative

[@gem4036]

@@kittykat8485 Not necessarily for effort... their answer is right actually. Not the answer I would be looking for, but they’re right

[@berriee7848]

this guy's making me study when I'm supposed to be procrastinating

[@hf6553]

Underrated comment lmaoo

[@dannybee9068]

It's still procrastinating tho

[@Irish_Enderman]

Lol same

[@keekwai2]

procrasturbating

[@VivaMarianoRey]

I’m in vacations XD

[@balkar01]

So,Basically we can reach the end of the universe.

[@jacobschwartz8175]

Well, like he said in the video, due to the constraints of time and reality, no. And the fact that the expansion of the universe is actually accelerating and isn't constant.

[@nikhat6884]

OR CAN YOU? H 😂😂. IF YOU CAME TILL HERE WELL F Y'all made me do this I like how every once in a while someone reads this comment F @mindoftheswarm how much longer will you make me go F

[@gamekichiguy8821]

Damn, my teachers always said it would be impossible

[@miguelbaltazar7606]

Well yes, but actually no.

[@heetruparel3331]

@@nikhat6884 man😂

[@thelolminecrafter7830]

I have the solution for you: Just keep stretching the rope until the length suffers a buffer overflow and drops into negative values. Sure, the rope is now a nonexistent point in space, but so is the ant that was walking on it. Now the ant is standing on both ends of the rope simultaneously.

[@hawkbatcustoms3172]

Yes i understand

[@jellybeancupcake4020]

Yup, I totally understand this (I don't understand this)

[@ThatUnknownDude_]

?

[@thelolminecrafter7830]

​@@jellybeancupcake4020 Programming joke. en.wikipedia.org/wiki/Buffer_overflow

[@thelolminecrafter7830]

@@ThatUnknownDude_ Programming joke. en.wikipedia.org/wiki/Buffer_overflow

[@homebrewHousehold]

Kevin: first I wanna mention My headphones: *B A T T E R Y L O W*

[@josephaether377]

lol

[@eva-lota9701]

@SQ38 bluetooth headphones.

[@DogeisCut]

Can relate

[@ignzyriq]

@SQ38 cool bro

[@hometimemayhem928]

Wait you use the wireless Jlab rewind

[@kiriedawa]

LEVEL OF TRUST BETWEEN HIM AND HIS THIRD ARM IS UNREAL

[@migueldelmazo5244]

Don't ask where the 4th arm was...

[@alephzero1984]

@@migueldelmazo5244 LMAO NOOOOO

[@rileyday6025]

Use my third arm

[@LilBurntCrust99]

The amount of potential energy could be theoretically almost countably INFINITE when approaching

[@brainbrightenation449]

@@rileyday6025 well I guess nobody before becoming teenager will get it luckily

[@CptPatrik]

Its 1 am and i am watching video about ant travelling on a rubber rope

[@williamseurkamp2240]

Cpt Patrick me too fam, me too

[@ember4262]

2:10am and i am replying to a comment about an ant on a rubber rope.

[@terraplayer832]

Its 1:18 AM and I am doing the same thing.

[@CptPatrik]

@@terraplayer832 its 0:25 am and i am replying to comments about my comments about ant on a rubber rope

[@terraplayer832]

@@CptPatrik Its 1:44 am here and I need to sleep, you should go to sleep too.

[@thejesuschrist]

1:18 Don't worry, I'm still going through puberty for the last 2000 years.

[@kapjoteh]

Ded

[@coleslavpiesboi]

Jesus Christ

[@nottletottle]

😂😂😂

[@brtd8782]

Jesus Borne it's Jason Christ.

[@FSX3000]

Just saw you in phily D video, you're every where !!!

[@AjayyHatake]

Who else kept having anxiety that the rubber rope would snap lol

[@billstephens396]

It would snap after 3 stretches and thus the ant would only have to move twice... No paradox... No anxiety... The rope ALWAYS breaks after 3 stretches...

[@theolodger]

ϒϵα lϴl

[@Piraticgames]

oh gawd now i do

[@woomylover2006]

Aaryan xll me

[@DaRat100]

yes

[@do3807]

Seeing the proof, and demonstrations in an easy to understand manner, fills Billy with determination. Whether he gets there or not, he knows he's making progress and sometimes that makes all the difference.

[@roisingrant]

Undertale

[@playingwithdimethylcadmium2766]

​@@roisingrantIt's hard to tell if it's a reference or not.

[@theCodyReeder]

can I apply this to cosmology?

[@raynin96]

That's what I wondered about..if humans would speed up earth a little bit (like the ant is walking by herself), apart from it's normal speed in space (like the ant just sitting on the rubber band), would it somewhen reach the end of the universe?

[@turtlemaster680]

Hey Cody, love your vids man

[@ebreshea1337]

@Ricky Smith I think this principle still applies, but the problem emerges as expansion speeds approach infinity. That would mean the % covered by light's own speed approaches 0%. We may yet be able to see more galaxies than we can right now, but over time, that would stop happening.

[@SLA-yo4is]

Apply this to quantum mechanics

[@pranaymondal1950]

@@ebreshea1337 yes it might get close enough to infinity, but not gonna be absolute infinite ever, so the lights which have already been covered almost the complete path between their source and us will still overcome the expansion rate of the universe and come to your eyes. You can simply think, lights are not discretely coming to us, it comes continuously, the rate of their approaching to us will just slow down. The light will take more time than before to come to us, and as a result, the time will apparently slow down for any distant star.

[@frankking6971]

When the rubber rope snaps, rubber bands back and hits Billy in the face at the speed of sound... Yes he will reach the end of the rope, as it knocks Billy back to yesterday.

[@kjkh3104]

The end of the rope will reach billy

[@NStripleseven]

Well, Billy won't need to do that, because the rope will come to him.

[@pebble1481]

Why did I read this in a pryocinical voice

[@Okaiako]

if it hits him at the speed of *light* (or faster) it very well could send him back to yesterday quite literally lol

[@kaninneko4895]

Technically wormholing the rope, since he skipped the rest of it to get to the end.

[@Keine]

For the longest time, I've wondered about light traveling in an expanding universe, but could never really wrap my brain around it. This video finally helped me understand it. This is a terrific explanation of the proof and you surprised me with real world application. Great job!

[@shanek6582]

I can't grasp how if the farther things are away, the faster they go, at some point matter would have to reach light speed wouldnt it? But matter can't go that fast right? So I'm missing something or the rules of light speed or the expanding universe is wrong, (I'm definitely assuming I'm missing something)

[@EebstertheGreat]

Dave, I thought this was a good video, too. It also gives insight into why the observable universe is nearly 47 billion light years in radius even though the universe is less than 14 billion years old. Light has been able to cover a much greater distance than you might expect because space has been expanding behind it as it traveled. Of course, light in an exponentially expanding universe cannot get infinitely far, but it can still get quite far.

[@antonispipo1]

@@shanek6582 you are indeed missing something. If i got this right you're wondering about how stuff in out expanding universe can travel faster than light. Well you are right, nothing with mass can reach light speeds and yes, the universe is expanding faster than the speed of light and even is getting faster by the second. How can this be? Well stuff isn't actually moving. Don't think of this as stuff moving apart but as more space being 'created' in between them. Its not rubber stretching. There's not really an analogue to this in our every day life so its very difficult to wrap our head around. I hope i could help you understand this a little better and obviously this is an over simplification of it. I would suggest looking it up yourself as it is a very interesting part of cosmology and very mind bending.

[@EebstertheGreat]

@@shanek6582 This has to do with the way we define speed. To define speed, we need a reference frame in which to measure it. In special relativity, we can pick any inertial reference frame and define it globally, so we can measure the speed of anything anywhere in the universe relative to that reference frame. And indeed, this speed is never greater than c. But in general relativity, these inertial reference frames can only be defined locally in general. Metric expansion is one example of why they cannot be defined globally, and over scales at which this is significant, it is no longer the case that objects can only be receding from us at a speed less than or equal to c. Another example is a black hole, as speeds for objects inside a black hole cannot be defined for observers outside it. The important fact is that if you get close enough to the moving object, you can define a reference frame locally there, and in that reference frame, no matter which one you pick, it will not be moving faster than the speed of light. Locally, spacetime in general relativity must resemble spacetime in special relativity. Another way to describe this is that space itself is expanding between the observer and the distant moving object, and this explains the apparent recession; the object is not actually moving "through space" at that speed. Also see my reply to Dave.

[@lucywucyyy]

doesnt light have a constant speed in a vacuum tho so surely that doesnt work the same way as this

[@michaelgordon8142]

A paradox is just when you try to squeeze a logical answer from an impossible question.

[@michaelsurratt1864]

I didn’t get it at first until I understood that when the rope is stretched he’s still connected to the rope so he’s getting pulled forward. We were just adding a kilometer onto the end. You would never reach him.

[@theobromyn]

Its been a year since i watched this, now that i rewatched it.. but seeing the clip at 2:46 i feel bad for the magical hand for getting hurt because of the rubber band lol.

[@itismethatguy]

Lol

[@nozerav]

Poor magical hand

[@escanorhellfire8088]

I bet that hand has a cute body attached to it

[@colterbayle3279]

Anxiety for rope snapping

[@TheInnerSpark]

😂🤣🤣😂🤣🤣😂🤣

[@literaltrash9144]

There are ants alive that are older than me :(

[@trickytreyperfected1482]

Let's torch 'em!

[@danandchristineharbour2538]

@@trickytreyperfected1482 noof [no+oof]

[@trickytreyperfected1482]

@@danandchristineharbour2538 I have no clue why I said that. Was I referencing the video because that seems wildly out of character for me to say (granted, I did say it as a joke).

[@blubasnurk4241]

Poopy

[@trickytreyperfected1482]

@@blubasnurk4241 stop. Get some help.

[@pluffaduffYT]

To think we're finally at the point where Vsauce2 uploads more frequently than Vsauce

[@lucasbueno7534]

we are at that point for longer than one year. Vsauce 1 is disappointing

[@adonismoy721]

Vsauce 1 posts mostly on the channel DONG

[@HideorEscape]

Vsauce 1 died when it made those paid episodes.

[@mikes333]

Yah, but if Vsauce2 uploads 2 videos every 1 month and Vsauce uploads only 1 video every 2 months, will Kevin ever equal or even surpass Michael's popularity. I think we'll need to break out the calculus to prove it....

[@WhimsicottFanatic]

This comment is funnier with the fact you've got JonTron as your icon

[@meltymooncakes]

Crazy? I was crazy once, they put me on a rope, a rubber rope, rubber rope with ants, and ants make me crazy.

[@geraldgodoy7600]

Crazy? I was crazy once, they put me in a strange thing, a rubber strange thing, a stretching strange thing, and stretching makes me crazy!

[@jiminboo]

Kevin, you drew me a potato one day, years ago. I cherish that drawing.

[@omegalord]

It was your portrait.

[@minotaur470]

*dabs*

[@egormatuk3786]

Draw me like one of your french fries

[@BigBahss]

@@egormatuk3786 Your comment wins 2018

[@GAZAMAN93X]

@@egormatuk3786 what about sandwiches?

[@kristijantodorovski5790]

1:01 "This ant's name..." Me in my head: Billy "BILLY" ME: DAFUQ?

[@ytterbius2900]

Exactly!

[@shadowjedi4602]

Same I’m still looking for paradoxes in comments like this

[@kiiyoko]

r/thathappened

[@youraveragedegenerate1]

69 likes

[@sophiafritz1760]

I did that too

[@RandoRy]

5:20 you have the divergent series containing 1 over 1 which is 1 and then proceed to say that it will eventually surpass 1 but the first fraction is already 1

[@simongolddrone]

I was wondering that too 🤔

[@DarthJJ777GMAIL]

Saw that too, I assume it just wasnt supposed to have the 1/1

[@JivanPal]

The actual useful fact is that, since it is a positive divergent series, the partial sums become arbitrarily large; that is, the series will eventually surpass any positive number you can think of. When it comes to the final proof, this means that a/(kc+kv) [ 1 + 1/2 + 1/3 + 1/4 + ... ] eventually surpasses 1, because 1 + 1/2 + 1/3 + 1/4 + ... eventually surpasses (kc+kv)/a, which is a positive number.

[@ge2719]

Nevermind that part, because a half and a third and a quarter is already larger than 1 also. He meant will it reach one once its been multiplied by the scalar for the specific length of rope and stretch length. The actual sum of the series shown approaches a number much bigger than 1.

[@JivanPal]

@@ge2719 - *_"The actual sum of the series shown approaches a number much bigger than 1."_* Indeed, it approaches infinity!

[@TwiDashFTW]

The scary part about this all is I actually remember learning that math.

[@Vsauce2]

I GREW AN ARM FOR THIS VIDEO. Here's the link to my podcast please subscribe thanks: bit.ly/2BCLhoK

[@oofire2480]

Woah impressive

[@proffesseurevil]

Btw. An ant going to the "ant" of the rope. "Antbitions" till the"ant" of their life? Really _._

[@tman11122]

Will we ever see another Mind Blown?

[@adonismoy721]

5th

[@VariantAEC]

So doesn't that last statement prove that light will have a definitive cutoff point in our universe? If light travels a constant rate and the universe expands at a accelerating rate light will have a definitive cutoff distance from the point of observation and as that time increases light from distant sources will continue to get harder and harder to see. That means that objects in the night sky regardless of their age and actual activity at a point of origin in time the light from any given source moving away from our galaxy will not only be blue shifted, but continuously dimming. Continuously diminishing light doesn't seem to be a thing in space however. Obviously the only other explanations that support these hypotheses is that we simply haven't existed and recorded data long enough to support the possibility of a universe expanding at an accelerating rate.

[@petermarsella6537]

Alternate title: Man keeps ant from crossing rope for 12 minutes and 9 seconds

[@svetafeo]

Lol😂😂😂😂😂😂😂😂😂😂😂😂

[@ignzyriq]

@@svetafeo well... you like emojis, don't you?

[@svetafeo]

@@ignzyriq yes.......but it usually is a rule that I follow when just reading comments when I make a reaction I have to reply with that reaction

[@ricobrawlstars4880]

And gives a name to it

[@vinaythakur4742]

13:57 ? It's now only 12:09 What?

[@zockerhdgarantiert]

So you want to tell me that the ant is faster than my soul speed 3 shoes on soul sand in water?

[@hehdivorce2878]

With depth strider and dolphins grace

[@TheDeadOfNight37]

@@hehdivorce2878 and speed II

[@brandonnadel4298]

And riptide 3 trident

[@Republica_de_Panama]

@@TheDeadOfNight37 and if you use the effect command to have speed 255

[@BaalTomekk]

It depends on whether the soulsand you are walking on is on the rubber band or not.

[@wikitt5801]

No matter what, even though it will take a long time for billy to reach the end of the rope, at least he's getting some great cardio into his life

[@gekkotadpole2478]

5:18 Kevin: "The sum of these fractions eventually surpasses 1." Me: Wouldn't... 1/1 + 1/2 surpass 1 immediately?

[@rie66s]

good point

[@SoməøneXD]

thats what i was thinking the entire video

[@TheFinalChapters]

The "fractions" was a/(v+c) *times* (1/1 + 1/2 + 1/3 + ...).

[@hubertszy]

THANK YOU!!! Yeah it's a mistake :P He did it twice, the second time it would make sense if he took a/(v+c) into account :)

[@mekb-the]

even 1/2+1/3+1/4 > 1

[@callumz1352]

Can we talk about how that “pizza” looks

[@hclyrics]

I'm from NY and my first thought was wtf is that??

[@tarnishedecho]

no

[@jeralgrant]

That's gotta be a microwavable Jeno's.

[@PikKraken8]

It looked like a cheesy blob.

[@gameplays2676]

🤣🤣

[@ididafewthings]

5:23 I‘m no scientist, but I‘m pretty sure that the sum surpasses 1 after the first element.

[@ididafewthings]

Oh god this is a reoccurring theme in this video...

[@zesstrin00]

thought the same in the instant he wrote it

[@icicleditor]

2

[@yashuppot3214]

He meant 2

[@Doom8810]

I think he means 2

[@LoriPeace]

I struggled with math throughout high school; took remedial math in college as the easiest possible course to get the credit that I needed to graduate. I absolutely LOVE that your videos make math not just doable but fascinating to me! I wish I could show them to my 11th grade self as I struggled with algebra 2 -- although that was 1977-78 and it would have blown my mind to watch a VIDEO on a COMPUTER that could sit on my desk... I hadn't even heard of videotapes at that point! LOL

[@MagikarpMan]

The harmonic series:Exists Me: 1/1 is 1

[@mydicchurt7136]

Yeaaaaaaa

[@alexiswong7335]

I know and even if you remove 1/1, 1/2+1/3+1/4 is more than one

[@alexiswong7335]

Btw I think he means 1/2+1/4+1/8+1/16...

[@JackAttack-jn4op]

Alexis Wong he doesn’t

[@hassanakhtar7874]

@@alexiswong7335 the point was it goes to infinity so definitely not the second one.

[@christopherlyerly4631]

5:05: Of course your harmonic series "eventually " exceeds 1 - you STARTED with 1/1!

[@jugodats9990]

Thank you

[@dustrider9306]

You passed the test :)

[@thatnike2604]

Thank you that's what I was thinking

[@ejgoldlust]

I caught that too. It's actually really crazy sounding: that sum will actually become infinitely large.

[@thatnike2604]

@@ejgoldlust -it will barely reach 2-

[@zeldafreak2232]

Did you know you can tell an ant's gender by putting it in water? If it sinks, then it's a girl ant, but if it floats...it's *buoyant*

[@SoundlessScream]

If it sinks it's not a witch

[@diy-projects]

Lmao

[@simonshugar1651]

@@pixiepandaplush I think his formatting is fine. I understood it with no problems

[@lkajsdflkasjdf1597]

@@simonshugar1651 Same. It make me laugh.

[@benchapple1583]

@@lkajsdflkasjdf1597 What if it's transient.

[@carriemcclung4615]

So, I’ve always wondered how for example, an ant can ever reach the end of a rope if he must first traverse half of the remaining distance? Isn’t there always half of the distance left to cross, and then half of the new remaining distance left to cross after that in perpetuity? You’ve come the closest to making that make sense to me in 30 years, but I’d love full clarity?

[@deltalima6703]

Too hand-wavey, I agree. Not all functions make it to 1, just because his first example did proves nothing. Sum (1/(2^n)) for n approaches infinity would get really close but Sum(1/(3^n)) for n approaches infinity would not. Unless I am wrong, but I would like to be convinced, and hand waving wont do it.

[@TheFinalChapters]

Instead of looking at the rope in meters, look at it in % traveled. The % traveled does not change when the rope stretches, which allows us to use the harmonic series he explains in the video to prove that eventually the ant will, in fact, cross half the distance remaining and soon after reach the end. I should also note that the summation of 1/(2^n) approaches 1, not infinity. However, the summation of 1/n, i.e. 1/1 + 1/2 + 1/3 +... does approach infinity.

[@leightonpetty4817]

What you just described is Zeno’s Paradox, also known as Achilles’s Race. And it was originally made to show the fallibility of theoretical calculus when applied to the real world-obviously, in reality, Achilles will still overcome the halfway point and beat his opponent. While math dictates that there will always be a halfway point, on a physical level there _is_ in fact a “Smallest unit of measurement that cannot be cut in half”-the Planck Length. Reality is not capable of moving half a Planck Length, and from that the paradox crumbles in a real world setting to the obvious conclusion (overcoming the halfway point).

[@thine.]

i think the "supertasks" video from Vsauce 1 could make sense here, essentially its a task that cannot be ended because you can always divide it in half

[@lyricalap]

@@leightonpetty4817 I'd like to clarify this: You can go smaller than Planck length, infinitely smaller ( to our knowledge ). The Planck length is just the smallest distance in which measurements make sense ( also meaning that its the smallest distance in which our natural laws apply and classical mechanics can be used ). In short it is theoretically possible to move smaller than a planck length.

[@thomaschambers3588]

Watching a fellow left handed person awkwardly struggle to write on a white board gave me flashbacks of school

[@harry_page]

If you like differential equations, here's how to find out how long the ant will take: Using Kevin's variables, with a little tweak: let the distance travelled by the ant be s, and the rope length be C, both functions of time, with initial length L , so C = vt + L for constant stretch rate v ms^-1. If the ant's velocity relative to the rope is a, then its velocity relative to the start point has another component; the stretching of the rope. Since it is stretching uniformly, this stretch velocity is proportional to s, and its easy to show that this velocity is vs/C = vs/(vt + L) Putting that together, we get the differential equation ds/dt = vs/(vt + L) + a This can be solved with the integrating factor method; the factor is 1/(vt + L): 1/(vt + L) * ds/dt - vs/(vt + L)^2 = a/(vt + L) d/dt ( s/(vt + L) ) = a/(vt + L) s/(vt + L) = (a/v)*ln(vt + L) + d When t = 0, s = 0 so d = -(a/v)*ln(L) s/(vt + L) = (a/v)*ln(vt + L) - (a/v)*ln(L) = (a/v)*ln((vt + L)/L) s = (a/v)*(vt + L)*ln((vt + L)/L) The ant has reached the end of the rope when s = C = vt + L so we get: vt + L = (a/v)*(vt + L)*ln((vt + L)/L) 1 = (a/v)*ln((vt + L)/L) (vt + L)/L = e^(v/a) vt = L(e^(v/a) - 1) t = (L/v)*(e^(v/a) - 1) So for the first situation, where a = 0.05 ms^-1 , v = 0.1 ms^-1 and L = 0.2m you get t = (0.2/0.1)*(e^(0.1/0.05)-1) = 2*(e^2 - 1) = 12.7 seconds Now the second situation with a = 0.01, v = 1000 and L = 0.2: t = (0.2/1000)*(e^(1000/0.01)-1) =1/5000*(e^100,000 - 1) =5.61*10^(43,425) seconds =1.78*10^(43,418) years Odd, my answer's a few orders of magnitude away from Kevin's. Maybe he worked it out from a more discrete method than my continuous one

[@harry_page]

Also, here's a graph of time against rope stretch speed, with ant speed at a constant 0.05 ms^-1 and initial length 0.2 m: imgur.com/7ylrSSt Notice that as rope stretch speed tends to zero, time taken tends to 4 s (as in the start of the video) and the solution for when it's stretching at 0.1 ms^-1 is about 12.8 s, growing pretty much exponentially. Also, an interesting middle ground solution I notice is that for ant speed 0.01 ms^-1 and rope stretch speed 0.072 ms^-1, the time taken is about an hour and if the stretch speed is 0.2073, then the time is about 30 years, the maximum lifespan of the ant!

[@ryan-ci3fz]

What I see: Hubbysjsjwn+jdnyxh=lmnop

[@nekogod]

Yeah I believe the rope stretches in steps rather than continuously in his example. So at the end of every second it instantly stretches 1km.

[@hacker1oo173]

@@harry_page Im just guessing here but you have at least 2 brain-cells.

[@harry_page]

@@hacker1oo173 2 brain cells and no life by the looks of it. Good god, why did I type all of that? xD

[@hexerin]

Should’ve been A, N, T for the variables. Missed opportunity!

[@vikranthguda8473]

although he had k for seconds. k for Kevin and second referring to vsauce2.

[@lumonox]

Pooping💩

[@logangaastra4679]

k is actually just a variable commonly used for indexing, i.e. representing 1,2,3,4,...

[@jacobbenz5833]

he also shoulda named the ant ant(h)ony

[@potatokingtheexistent8994]

If you didn’t understand here’s a quick explanation: basically when the ant moves it moves a fraction of the rope and when the rope stretches it takes the ant with it. That means the ant has still covered the same fraction but the amount it covers is becoming smaller and smaller of a fraction but it does eventually reach the end.

[@robertoespi3500]

not really, the rope stretches, so a distance represented by 1cm now will not mean same distance traveled later, there will be new gaps in the rubber band from the stretching so there will always be more new lenght to be travelled.

[@bikeman123]

​@@robertoespi3500did you watch as far as 3:37

[@Quadrocephis]

1:18 *VOICE CRACK*

[@thatcrystalpie]

harry pOTter

[@szymsonthatmemeguy3181]

Harry pAHter

[@jhNic]

Harry POoreeeTtEr

[@egmusic1925]

harry p *AH* ter

[@stgeorge5862]

Hairy Pothead

[@Slideyslide]

I just wanted to see a real ant on a rubber band... 🐜

[@fivedfiva]

same dude

[@nathanlye9279]

Ants don’t like rubber ropes... or the smell of it.

[@xarran]

Imagine, if after reaching the end of the rope he has to come back.

[@n01rsc3n3]

It just shrinks and it’s a speedrun

[@owenstauble6370]

That’s when someone releases one side of the rope and it snaps back like a rubber band, shaking the entire universe and killing the ant instantly

[@Lady-Anubis]

Or the rubber rope is actually a rubber band

[@MadScientist267]

Gotta be easier than sitting thru another video with this drama queen

[@bramvanduijn8086]

Ants can bite, he'll just bite the finger holding the rope and be flicked home nigh-instantly.

[@CamoGirlPlayz]

As a college student currently in calculus 2, this was the best and only real world application I've ever seen of this stuff.

[@jasonlarsen4945]

Yup. Major college calculus flashbacks, and I only took Calculus I stretched over two semesters.

[@vanskis7618]

Who else thought that the ant's name will be Anthony.

[@alialomar39]

I'm more disappointed than I should be that the ant wasn't called anthony

[@Anthony-tu2mm]

That's ok

[@alialomar39]

@@Anthony-tu2mm I'm glad you are called Anthony

[@alialomar39]

@@Anthony-tu2mm IT CALMS MEEEE TO SEEEEE ANTHONYYYYY

[@patsmith6867]

I would have said Alvin . LOL . I think his Joke was Funny , Extremely Lame , but Funny . He chose a name that started with "B" an intetional Joke .

[@DarklingReborn]

there are ants older than me.....

[@DPLSHR]

Respect ants !

[@banksy7376]

Aunts*

[@smackclack192]

Same

[@piano_covers4487]

Same... wow! I will never disrespect ants again 🐜

[@lightestseed]

I don’t know why I thought billy was a real ant for the first minute and a half

[@BlackLegVinesmokeSanji]

Billy is a real ant just belive

[@dacat2880]

@@BlackLegVinesmokeSanji did you mean... BILLYve??

[@knoniezzimlou2193]

He…he’s real to me 😫

[@knoniezzimlou2193]

@@dacat2880 oh my lord get off the cite you dork 💀

[@pioneer_ten]

@@dacat2880 no stay on the site you very funny person

[@mansumkashyap4090]

when he said " oh and this ant's name is..." i literally was thinking about the name billy and then he named it billy ._.

[@Skelly57]

this channel is the only main VSauce channel that uploads consistently the others aren’t dead (their twitter accounts are still active), they’re just working on big projects right now

[@milkywegian]

:thonk: big projects such as?

[@jerry3790]

Only Vsauce 3

[@jonathanodude6660]

@@milkywegian CYSTM: mad max

[@BobSamson147]

It's ok, Kevin is my favorite anyway

[@milkywegian]

Jonathan Odude mad max is already done

[@GK-Fred]

5:17 "Where the sums of these fractions surpases 1" Hmmmm... the first fraction is 1... Upsss...

[@DorthLous]

I was about to say...

[@ayueshi_]

I mean i did 1/2 + 1/3 + 1/4 and it's already 1.08333...

[@itismethatguy]

Lol ya he must have accidentally did that cuz if we remove it it is still more than 1

[@Chino_The_Muted]

s u s

[@jochemverzijl2601]

@@ayueshi_ you have to take steps of two. Like 1/2 + 1/4 + 1/6 + 1/8. Maybe that is The solution

[@Astro_Guy_1]

Vsauce 2 is here to fill the gap in my heart that Vsauce (micheal here) left.

[@ynico1858]

Agent 47 :,(

[@redundantjovianthor2225]

1:03 in and im thinking: "if the ant is ON the "rope" and you're stretching the physical body of the rope, then there's 0 chance that you are not also simultaneously dragging the ant forward and actually AIDING his progress more than inhibiting it BY stretching the rubber "rope"." So I'm already having a hard time fathoming how this is paradoxical... *save to watch later*

[@michaelsurratt1864]

Yeah, it didn’t make any sense until that I figured that out. if you were just adding distance to the finish line, then he would never make it. not a paradox at all. It’s just a trick phrase.

[@ham_fast]

Thank you very much, Kevin. You just helped me write a college term paper. I appreciate all the work you put into this.

[@bellhop_phantom]

Well, what grade did you get?

[@78tag]

@@bellhop_phantom Does it matter? The process motivated him to think! Even if his work was judged (by some arbitrary acceptance that the professor knows something) to be a failure, he still learned something by the effort. Good on you Ham.

[@kennycano1269]

@@78tag it matters

[@StarlightNkyra]

​@Kenny Cano Honestly, in my opinion grades are just letters used to get you diplomas.

[@cooldes4593]

5:21 1/1 + 1/2 is already > 1

[@cassiopeia9701]

true, but he meant with a small factor in front like 5[cm]/(40[cm]+10[cm]) or whatever you plug in

[@cooldes4593]

@@cassiopeia9701 what do you mean? I see no indication of this

[@cassiopeia9701]

@@cooldes4593 later in the video, when he compares the realitve distance the ant has gone. Around 7:38. He "normalizes" the series through the fraction he puts in front of it. But technically you are right, he even says it at the part: it diverges so it must go to invinity not 1.

[@henrydaniel6420]

He meant to say surpass 2 , after an infinite number you can reach 2

[@Biggie_Butters]

kzbin.info/www/bejne/aqrcfZmoqLiLh7M

[@loganpowell6510]

should have named him "antony" edit: i did not think this was gonna get as many likes as it did lol :D

[@laysone346]

That's from the movie "Antman" so it's an unoriginal joke

[@dara-bk5rh]

@@laysone346 shut up

[@kougaon8513]

@@dara-bk5rh Glad you contributed to this conversation, have any other sagely advice to give?

[@cursedsound]

@@kougaon8513 Do drugs they are fun

[@Face08]

That is a bad joke there.

[@spitalhelles3380]

For the 10cm/s example: The end of the rope moves with linear speed 10, so š(t)=10t+20 the ant: v(t)=s'(t)=10s(t)/(10t+20)+5, s(0)=0 differential equation solution: s(t)=5(t+2)log((t+2)/2) to solve, we equate: s(t)=š(t) 5(t+2)log((t+2)/2)=10t+20 solution: t=2(e^2-1)≈12.778 for the 1km/s: s(t)=((5000t+1)log(5000t+1))/1000 š(t)=100000t+20 equation: s(t)=š(t) solution: t=(e^20000-1)/5000≈1.55*10^8682

[@Релёкс84]

I have seen many symbols for functions, but it is my first time today seeing š(t)

[@spitalhelles3380]

@@Релёкс84 I would have put s̃, but that's not on my keyboard

[@samponor4341]

2:46 that poor mystery hand😪

[@albertskunik]

it snapped the hand got hit by the rubber poor hand

[@voilet-the-non-violet-vulpix]

You can prove the first half of this with a lot of video game leveling, sort of, if you’re in the right mindframe. The progress bars keep getting longer and longer, and eventually, in a game where you got your first fifteen levels on the first day, it’s taking a week to gain a single level. But you still made progress. You’re still never going to have to repeat lvl 23. You’re still closer to the level cap, even though the same amount of time and effort is no longer yielding levels as often.

[@djliam_9042]

Warframe moment

[@Duplicitousthoughtformentity]

Diminishing returns. The bane of all gamers.

[@awesomedudeplayz3950]

Are you a furry

[@3217491]

so you didn't understand anything explained, gotcha. (level cap doesn't keep moving away from you constantly)

[@thomasp506]

@@3217491 But the amount of XP needed to level up increases for each subsequent level. I think OP understood it better than you did.

[@mykeva1]

28 “or” 30 years, so not 29 years?

[@brandonkey181]

ONLY 29 or 30.

[@maxie1199]

He meant 28 to 30 years I’m assuming, same as how some dogs typically live 10 to 15 years of age

[@jahraccoon1396]

@@maxie1199 r/wooooooooooosh

[@snake4322]

@@maxie1199 r/whoooooosh

[@WhoTookThomas112]

Yep

[@findlaycoates3442]

In the harmonic series my understanding is that it's adding fractions to make 1 eventually but it starts off with 1/1 which means it's already reached 1

[@theerussianraccoon160]

*Vsause 2 uploads a video* Sleep: am I joke to you

[@ruileite4579]

am I a* I'm just being anoying

[@HildrTV]

@@ruileite4579 just stop...we all knew what he ment...

[@xxdudexx20]

Am gey lol

[@R.G.962]

Literally me now at 4am

[@ruileite4579]

Sinister Steel no u

[@syruptalk]

6:26 I'm dying the way he's says after "eafter"

[@BigBrotherMateyka]

It's like the Ant version of Odysseus, only Penelope is dead, life on earth has become extinct, the earth has been devoured by the sun, the light from all the stars and galaxies have gone out, and the only remaining things in the universe are a few scant positrons and antimatter particles hovering at infinitesimal fractions of a degree above absolute zero. But, by God, Billy will reach his destination. As will we all.

[@Nillowo]

BigBrotherMateyka this comment needs more attention and love.

[@jakobyarns4369]

I dunno why but learning that some ants are older than me is really mind boggling

[@christianphillips7695]

Out of all the things they could teach us about life in school, this is basically the stuff they decide to teach us

[@erent.2020]

They taught us this in university.

[@DrtyTreeHuggr]

A bunch of nonsense

[@JojoJere]

@@DrtyTreeHuggr You sound like a african aunte EDIT: No Offense

[@StarlightNkyra]

Yep, almost completely useless that only makes you feel like you "learned" something.

[@robinsinhaxii-a3848]

_No ants were harmed in the making of this video_

[@lkajsdflkasjdf1597]

Plastic ants lives matter.

[@Lena_M]

_throws ant at camera in the end of the video_

[@reedplaysgames]

Except for Billy.

[@alexbroxman]

I think this is one of your best videos yet. Even though I was extremely familiar with the subject as a math student and pretty much knew what you were going to do since I saw the original problem your way of presenting it made it incredibly entertaining to watch. I really loved the connection to starlight not reaching us due to the accelerated expansion of the universe at the end of the video. It was a very satisfying way of relating seemingly abstract mathematical problems with understanding the universe around us and I certainly hadn't thought of that one before. By the way this is the first time that I've noticed that you're lefthanded. Lefties unite!

[@guillermomarturetfendt9037]

Yeah me too, although as a student, I´ve suffered a lot by not writing the math in a formal way, and seeing this very informal math makes me cry in pain...

[@someaccount3438]

I see you are a comrade as well

[@andrewdrake7722]

lefties unite

[@Skelath]

The exact same principle can be applied to downloading something from the internet, as the speed of the download keeps getting slower and slower, the percentage of the downloads completion will continue to climb no matter how long it takes to download.

[@gamer_kid_naz4942]

LEFTIES UNITE

[@zavalanovakidue]

The thing about this little problem is that you're not extending the end of the rope, you are stretching the rope itself, and so every single millimeter of it moves and not just the end

[@Waffles1365]

Isn't that what he said in the video? I don't think you understood what he was saying.

[@bramvanduijn8086]

@@Waffles1365 People are allowed to rephrase a concept.

[@JamaalDaGreatest]

Hence the point of why the ants relative positioning on the rope stays the same.

[@ShaneGabaree]

He should have called the ant ANT-hony not billy

[@bjp4869]

No

[@testicleetus5298]

No

[@quadeemon3986]

No

[@adog5361]

No

[@jellybeancupcake4020]

That is brilli-ANT

[@IsomerMashups]

Maybe I just don't know the definition of a paradox, but this all seems perfectly clear.

[@Silver4Hire]

Same

[@nobaraa]

this isnt a paradox

[@adoniskostopoulos848]

*No ants were harm during the making of this video*

[@cdgonepotatoes4219]

Just mildly annoyed

[@yrok244]

Who would name their ant harm

[@anythingandnothing1996]

he squished him

[@brandonkey181]

*only before the making of the video*

[@buffalocrackerdong6978]

except red ants, they were always harm.

[@bestopesto]

Kevin talking about stuff, and then randomly: Oh this ants name is Billy.

[@TechSupportDave]

The presentation, the simplicity, vsauces always make the best videos.

[@yoseflowy958]

Simplicity?

[@TechSupportDave]

@@yoseflowy958 well yeah, he makes complicated problems very often very understandable. He doesnt just assume his audience understands everything, he explains every element in detail.

[@lordlajos3]

but who cares about an ant moving on a rubber rope

[@monrogthegamer8417]

EVERYONE!

[@manacress85]

*Vsauci

[@MattJasa]

That seems a bit of a Stretch, let me ask Googol

[@vyliad]

lol googol

[@bishissme6659]

EnderWizard413 lol

[@morgandebruler9281]

Yea I thought it seemed a bit ropey

[@canuckeraust]

2 puns in a sentence... how? Sans? G A S T E R ?

[@bobbyray911]

Knew I could count on someone to make math puns. He knew making a video on math would be a calculated risk.

[@kjoege1000]

and as always, ants for watching.

[@buckyspesh6620]

WeirdWolf ant as always

[@Invalid-user13k]

When Paradoxes are created from impatience to finish something

[@bluemooneatlions]

Just realized he is left handed

[@arcticbeak]

And I thought middle handled.

[@originalname8541]

Left hANTed

[@ignzyriq]

@@originalname8541 no... just no

[@originalname8541]

@@ignzyriq yes

[@meat_doughnuts3457]

Original Name left hanted?

[@barrettcarpenter1745]

5:25 by adding 1/1 to 1/2 and so on, you automatically have achieved a sum of 1. The series starts at 1/2 and goes to 1/3 and so on

[@andradianamaglas983]

1/2 + 1/3 + 1/4 = 13/12 which is already >1. I don’t get what he’s trying to do...

[@BrienCoffield]

Yes, he's not speaking correctly. What he should have said (or means to say?) is that the series does not converge to a particular number. Some series converge to, say, 1, as K approaches infinity. A divergent series does not converge to any particular number as K approaches infinity. i.e. this series does not converge. His explanation does a rather poor job of explaining the most basic concept of calculus, but I understand his intent I guess. Math is hard and stuff.

[@Nofro02]

@@BrienCoffield the series i a/c+v multiplied by (1/1 + 1/2 +...), he wrote it in the wrong way, if a < c+v and you stop a the first number in the series the result is a/c+v that is less than one, there isn't any mistake here, the only thing i didn't understand is why a/c+nv can be considered the same of a/nc+cv

[@DrtyTreeHuggr]

@@Nofro02 cus u can't question math. Jus believe what books tell u

[@Nofro02]

@@DrtyTreeHuggr in math if something isn't an axiom it has to be proved (unless it is obvious in the context where it's told) i saw the video 9 months ago so i don't remember how he explained that equivalence.

[@tomysamoa]

to think what I learned in Integral calculus would work for something 🤔

[@nishantdewaney5959]

tomysamoa I know that feel

[@WhimsicalLittledawg]

This is one of the few times where seeing the first person perspective of someone writing is normal to me, because I am also left handed.

[@timsullivan4566]

Ant's name should be spelled "Billie" not "Billy" 'cause all ants are females EXCEPT for a very few winged males who (apart from nuptial flights ) NEVER venture from the nest. But EVEN IF one did, given that a male's ONLY value to the colony is his fertility, he'd surely avoid ALL things rubber.

[@etherealstars5766]

Nah, Trojan is attractive to ants as well.

[@bernd8608]

im triggered now

[@DaP84]

We humans think we know everything. Science is just "the truth of the day".

[@GonzoTehGreat]

Tony (or Toni if you want a unisex version) is a better name than Billy. Can you guess why?

[@itz_siena2822]

Tim Sullivan billy can be a girl name and be spelt that way

[@captainwaa7274]

“Calculicious”

[@littolicce]

Why did I know the ant’s name before you said it? WHY???

[@thievingpeppers1914]

littolicce Billy is a common name to think of for fake people. Just like if you ask people to name a colored tool most of them will say red hammer

[@trumpetperson11]

lol same

[@lawrenceredmacher4382]

you're a wizard harry

[@VictorMartinez-zf6dt]

I kNOW RIGHT?!

[@vivimannequin]

I was thinking it would be jimmy

[@lucaspawprint1888]

Small numbers: okay Bigger numbers: naw theres something wrong here

[@jay-tbl]

For the 1cm/sec ant and the 1km/sec rope, I think I found a solution before watching the video: Say the ant was not ON the rope but nearly next to it. So the first second, the ant walked 1 cm and the rope is 1 km The second second, the ant is at 2cm and the rope is at 2km The third second, the ant is at 3cm, the rope 3km No matter what, the ant is .1% the distance of the rope. Therefore, we can confirm that he is not getting farther and farther away from the end of the rope. But, the ant is ON the rope, so when the rope stretches, he moves forward a little bit. That means that since he can't get any less than .1% of the rope, and he moving forward at technically faster than before, the percent of the rope he has travelled will slowly go up, and eventually hit 100% edit: damn that was surprisingly similar to the actual solution

[@fusrodahdaily9664]

*Plot twist:* Kevin was born with three arms.

[@Azimii]

FusRoDah Daily jokes on you I have three legs

[@fusrodahdaily9664]

@@Azimii And you don't use it to stand *Ayyyyyy*

[@sirkiz1181]

Top 10 Anime Twists

[@MilkySamm]

FusRoDah Daily *crab song intensifies*

[@hunterbrandal807]

FusRoDah Daily Third arm instead of third leg

[@SanjayPatel-gf1qh]

And I thought he was gonna make the rope into a circle so the ant could never actually reach an end

[@nowonmetube]

I thought this was what the video was about!! lol

[@neutronenstern.]

2:54 it will probably snap, and billy will get killed by it snapping. Poor ant

[@buurbloxproduction5434]

And then the rope breaks and it lashes onto billy and he dies The end Edit : YEET (Thx for likes :D)

[@DrAdityaReddy]

LOL

[@ThatOneRandomDude69]

You broke the paradox!

[@m2xsz441]

*_SPLASH_*

[@DanJFilms]

Too far man

[@chillbringer169]

Buurblox Production idk why I laughed so hard

[@zthirtyblue]

He makes math feel strange, dark and mysterious told in story format

[@Erilan5]

The key to understanding this, which should really be mentioned, is that the 1 cm the ant moves doesn't stay 1 cm but instead gets stretched along with the rope. So it ends up moving more than 1 cm with each time interval.

[@DrtyTreeHuggr]

That would defeat the argument of the paradox.. If the length/speed the any moves increases with the stretch, the ant would reach the end in the same time. No matter how much it was stretched

[@Erilan5]

@@DrtyTreeHuggr The length/speed of the ant doesn't change. But the total distance covered already does change. If up until now it had traveled 10cm while moving at 1cm speed and we stretch the rope by doubling it's length then after the next time interval the ant will have traveled a total of 21cm already.

[@semiautoduck7430]

Ah yes, I have now learned how to travel space and time. Thank you, ant on a rubber rope.

[@tmfd9476]

So eventually, there’s light from a star that we see today but there will be a time when it’s light fades due to expansion?

[@Ben-pp5tt]

Only if it’s light from a star that’s in another galaxy. We move along with the stars in our own galaxy.

[@leoneldavidsantiago1815]

@@Ben-pp5tt even then, eventually the super-massive blackhole that keep the galaxy as one will fade out, and then everything on the space will get distanced form every other thing

[@tmfd9476]

@Melchior Magni “we” my friend, we will be old as hell if we see that night.

[@Swordmaster7th]

Andromeda galaxy however is going to collide with ours however at some point. So our galaxy will have far more to it at least for a time.

[@SATO_FD2R]

*_You were the real reason I watched KZbin back in the day!_*

[@nuka5619]

アレキサンダー 佐藤 EK9!!

[@jjmetrejhon1743]

My mouth literally dropped open at "the real world analog to the ant on a rubber rope would be light from distant galaxies." My brain went 'oh my god' because of *COURSE!* That's *FANTASTIC!* I ♥ the Vsauces so much

[@sandordugalin8951]

Except then he immediately renegs it by explaining how the universe isn't expanding at a constant rate, it's actually accelerating. Womp wahhh

[@izaactheberean6860]

ANThony & ANTonio did a good job in this video.

[@teenconservative3433]

I love when calc II can actually have real world applications... this is great

[@VulpesVulpes42]

It's also great to see young people who identify as conservatives.

[@gabewalters1896]

Yes, I put all my ants on rubber ropes.

[@DaveBanasz]

You can watch the same video delievered by Thanos by covering the right side of your screen, or Hellboy by covering the left.

[@martinstojanovic2123]

Hahahaha

[@FarsLegendary]

Whaaaat

[@CrimsonCQB]

You sir are a visionary

[@olliebonugli8881]

Vsauce: rubber rope Me, an intellectual: snapped rubber band