Ant On A Rubber Rope Paradox
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- čas přidán 26. 11. 2018
- 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.khanacademy.org/math/ap-c...
Harmonic Series Proofs
scipp.ucsc.edu/~haber/archives...
Harmonic Series Proof
web.williams.edu/Mathematics/...
***********
Written by Matthew Tabor, Michael Stevens and Kevin Lieber
Huge Thanks To Paula Lieber
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Hosted, Produced, And Edited by Kevin Lieber
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Research And Writing by Matthew Tabor
/ matthewktabor
Special Thanks Michael Stevens
/ vsauce
VFX By Eric Langlay
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MY PODCAST -- THE CREATE UNKNOWN
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i wish he hadn't even mentioned needing another arm and it just wiggled onto camera with no explanation or acknowledgement of it
Video produ'tion HRs called for it my niño. (No idea what my niño meand btw)
An arm unknowingly slumps into battle!
He didn't explain it but he acknowledged it
He’s not Micheal
vsauce is 50% off today
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.
the ant reached the end
@@stonecoldpizza 💀💀💀
well if you have a 3rd hand that doesn't happen
@@jonnym4670 🙂
Just like our universe wait *uh oh*
this guy's making me study when I'm supposed to be procrastinating
Underrated comment lmaoo
It's still procrastinating tho
Lol same
procrasturbating
I’m in vacations XD
LEVEL OF TRUST BETWEEN HIM AND HIS THIRD ARM IS UNREAL
Don't ask where the 4th arm was...
@@migueldelmazo5244 LMAO NOOOOO
Use my third arm
The amount of potential energy could be theoretically almost countably INFINITE when approaching
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.
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.
Same here man.. got that same imaginative mind that made me fail math time and time again lol
No, that's actually a really interesting take. If I was your teacher I'd give you extra points for creativity :)
@@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
@@kittykat8485 Not necessarily for effort... their answer is right actually. Not the answer I would be looking for, but they’re right
So,Basically we can reach the end of the universe.
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.
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
Damn, my teachers always said it would be impossible
Well yes, but actually no.
@@nikhat6884 man😂
Crazy? I was crazy once, they put me on a rope, a rubber rope, rubber rope with ants, and ants make me crazy.
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.
Undertale
@@roisingrantIt's hard to tell if it's a reference or not.
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.
Yes i understand
Yup, I totally understand this (I don't understand this)
?
@@jellybeancupcake4020 Programming joke.
en.wikipedia.org/wiki/Buffer_overflow
@@ThatUnknownDude_ Programming joke.
en.wikipedia.org/wiki/Buffer_overflow
Who else kept having anxiety that the rubber rope would snap lol
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...
ϒϵα lϴl
oh gawd now i do
Aaryan xll me
yes
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
The scary part about this all is I actually remember learning that math.
Its 1 am and i am watching video about ant travelling on a rubber rope
Cpt Patrick me too fam, me too
2:10am and i am replying to a comment about an ant on a rubber rope.
Its 1:18 AM and I am doing the same thing.
@@terraplayer832 its 0:25 am and i am replying to comments about my comments about ant on a rubber rope
@@CptPatrik Its 1:44 am here and I need to sleep, you should go to sleep too.
There are ants alive that are older than me :(
Let's torch 'em!
@@trickytreyperfected1482 noof [no+oof]
@@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).
Poopy
@@blubasnurk4241 stop. Get some help.
A paradox is just when you try to squeeze a logical answer from an impossible question.
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.
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?
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.
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.
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).
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
@@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.
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.
The end of the rope will reach billy
Well, Billy won't need to do that, because the rope will come to him.
Why did I read this in a pryocinical voice
if it hits him at the speed of *light* (or faster) it very well could send him back to yesterday quite literally lol
Technically wormholing the rope, since he skipped the rest of it to get to the end.
Kevin: first I wanna mention
My headphones: *B A T T E R Y L O W*
lol
@SQ38 bluetooth headphones.
Can relate
@SQ38 cool bro
Wait you use the wireless Jlab rewind
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
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*
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.
To think we're finally at the point where Vsauce2 uploads more frequently than Vsauce
we are at that point for longer than one year. Vsauce 1 is disappointing
Vsauce 1 posts mostly on the channel DONG
Vsauce 1 died when it made those paid episodes.
Vsauce is working on CZcams red. Sadly I don't have it so all I have are old vids.
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....
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.
Lol
Poor magical hand
I bet that hand has a cute body attached to it
Anxiety for rope snapping
😂🤣🤣😂🤣🤣😂🤣
As a college student currently in calculus 2, this was the best and only real world application I've ever seen of this stuff.
Yup. Major college calculus flashbacks, and I only took Calculus I stretched over two semesters.
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
Isn't that what he said in the video? I don't think you understood what he was saying.
@@Waffles1365 People are allowed to rephrase a concept.
Hence the point of why the ants relative positioning on the rope stays the same.
Kevin, you drew me a potato one day, years ago. I cherish that drawing.
It was your portrait.
*dabs*
Draw me like one of your french fries
@@egormatuk3786 Your comment wins 2018
@@egormatuk3786 what about sandwiches?
1:18 Don't worry, I'm still going through puberty for the last 2000 years.
Ded
Jesus Christ
😂😂😂
Jesus Borne it's Jason Christ.
Just saw you in phily D video, you're every where !!!
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.
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.
@@robertoespi3500did you watch as far as 3:37
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
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!
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)
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.
@@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.
@@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.
doesnt light have a constant speed in a vacuum tho so surely that doesnt work the same way as this
1:01 "This ant's name..."
Me in my head: Billy
"BILLY"
ME: DAFUQ?
Exactly!
Same
I’m still looking for paradoxes in comments like this
r/thathappened
69 likes
I did that too
when he said " oh and this ant's name is..."
i literally was thinking about the name billy and then he named it billy ._.
I dunno why but learning that some ants are older than me is really mind boggling
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*
If it sinks it's not a witch
Lmao
@@pixiepandaplush I think his formatting is fine. I understood it with no problems
@@simonshugar1651 Same. It make me laugh.
@@lkajsdflkasjdf1597 What if it's transient.
can I apply this to cosmology?
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?
Hey Cody, love your vids man
@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.
Apply this to quantum mechanics
@@ebreshea 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.
I never thought about the properties of stretching like this, makes a lot more sense when taking the expanding universe into perspective
This is the same as stacking rectangles over an edge, where each rectangular plank of the same size and shape is stacked only a fraction of its length further past the previous one so that the entire stack remains balanced, eventually if you can stack high enough you'll get one hole length out past the edge.
So you want to tell me that the ant is faster than my soul speed 3 shoes on soul sand in water?
With depth strider and dolphins grace
@@hehdivorce2878 and speed II
And riptide 3 trident
@@TheDeadOfNight37 and if you use the effect command to have speed 255
It depends on whether the soulsand you are walking on is on the rubber band or not.
5:18
Kevin: "The sum of these fractions eventually surpasses 1."
Me: Wouldn't... 1/1 + 1/2 surpass 1 immediately?
good point
thats what i was thinking the entire video
The "fractions" was a/(v+c) *times* (1/1 + 1/2 + 1/3 + ...).
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 :)
even 1/2+1/3+1/4 > 1
0:55 sysyphus
Its eerie how well this relates to us right now and our position in the galaxy
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
I was wondering that too 🤔
Saw that too, I assume it just wasnt supposed to have the 1/1
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.
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.
@@ge2719 - *_"The actual sum of the series shown approaches a number much bigger than 1."_*
Indeed, it approaches infinity!
Can we talk about how that “pizza” looks
I'm from NY and my first thought was wtf is that??
no
That's gotta be a microwavable Jeno's.
It looked like a cheesy blob.
🤣🤣
The discretized approach in the video is very neat! I did this the naive way: for initial length c, ant speed a, stretch speed v, and position x, one can express the ant's velocity at time t as the constant ant speed plus the expansion rate of the length of rope already traveled: this expansion rate is v(x/(c+vt)), that is, the stretch speed scaled by the proportion of rope traveled. Combining the velocities gives dx/dt=a+v(x/(c+vt)), with initial condition x(0)=0 one can solve and get x(t)=(a/v)(c+tv)(ln(c+tv)-ln c), which grows faster than any linear function, in particular the rope endpoint = c+tv. Thus the ant will reach the end.
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
Alternate title:
Man keeps ant from crossing rope for 12 minutes and 9 seconds
Lol😂😂😂😂😂😂😂😂😂😂😂😂
@@svetafeo well... you like emojis, don't you?
@@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
And gives a name to it
13:57 ? It's now only 12:09
What?
I just wanted to see a real ant on a rubber band... 🐜
same dude
Ants don’t like rubber ropes... or the smell of it.
Seriously wish I had teachers like this in school. So entertaining and makes learning fun
2:54 it will probably snap, and billy will get killed by it snapping. Poor ant
The harmonic series:Exists
Me: 1/1 is 1
Yeaaaaaaa
I know and even if you remove 1/1, 1/2+1/3+1/4 is more than one
Btw I think he means 1/2+1/4+1/8+1/16...
Alexis Wong he doesn’t
@@alexiswong7335 the point was it goes to infinity so definitely not the second one.
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
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!
What I see:
Hubbysjsjwn+jdnyxh=lmnop
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.
@@harry_page Im just guessing here but you have at least 2 brain-cells.
@@hacker1oo173 2 brain cells and no life by the looks of it. Good god, why did I type all of that? xD
Kevin talking about stuff, and then randomly: Oh this ants name is Billy.
Ah yes, I have now learned how to travel space and time. Thank you, ant on a rubber rope.
5:05: Of course your harmonic series "eventually " exceeds 1 - you STARTED with 1/1!
Thank you
You passed the test :)
Thank you that's what I was thinking
I caught that too. It's actually really crazy sounding: that sum will actually become infinitely large.
@@ejgoldlust -it will barely reach 2-
5:23 I‘m no scientist, but I‘m pretty sure that the sum surpasses 1 after the first element.
Oh god this is a reoccurring theme in this video...
thought the same in the instant he wrote it
2
He meant 2
I think he means 2
When Paradoxes are created from impatience to finish something
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.
Who else thought that the ant's name will be Anthony.
I'm more disappointed than I should be that the ant wasn't called anthony
That's ok
@@Anthony-tu2mm I'm glad you are called Anthony
@@Anthony-tu2mm
IT CALMS MEEEE
TO SEEEEE
ANTHONYYYYY
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 .
1:18 *VOICE CRACK*
harry pOTter
Harry pAHter
Harry POoreeeTtEr
harry p *AH* ter
Hairy Pothead
You can apply this to turn based rpgs, where the HP is the rope and Jeff’s/debuffs stretch the rope, when applying the movement of attacks doing damage.
For those with problems understanding. Imagine zooming out at the same pace as the rope stretches, so that your perceived length of the rope stays constant. Now the Ant, if unmoving stays always at the same point and can move normally, the only difference to a non stretching rope is that the ant seems to get slower as time passes.
Watching a fellow left handed person awkwardly struggle to write on a white board gave me flashbacks of school
Should’ve been A, N, T for the variables. Missed opportunity!
although he had k for seconds.
k for Kevin and second referring to vsauce2.
Pooping💩
k is actually just a variable commonly used for indexing, i.e. representing 1,2,3,4,...
he also shoulda named the ant ant(h)ony
I've never thought of ants in such context. But when I'm downloading the file from the internet and it is stated "3 minutes remaining", then I'm waiting for 1 minute, and it's still 3 remaining, then it takes 2 in total, but still shows "3 remaining", then finally it turns out that I'm waiting for 3 minutes, but it is showing the same "3 minutes remaining", at this point I always imagine how I wait for it to be downloaded forever.
I think it's safe to say if you're stretching something whatever is on the rope would be pulled along with it. Also if you could have infinite space then of course you wouldn't reach the end ant or not.
Well even if its finite, our universe is expanding in infinite directions so you would be pulled to every direction equally so you wouldn't be pulled at all
should have named him "antony"
edit: i did not think this was gonna get as many likes as it did lol :D
That's from the movie "Antman" so it's an unoriginal joke
@@laysone346 shut up
@@dara-bk5rh Glad you contributed to this conversation, have any other sagely advice to give?
@@kougaon8513 Do drugs they are fun
That is a bad joke there.
Imagine, if after reaching the end of the rope he has to come back.
It just shrinks and it’s a speedrun
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
Or the rubber rope is actually a rubber band
Gotta be easier than sitting thru another video with this drama queen
Ants can bite, he'll just bite the finger holding the rope and be flicked home nigh-instantly.
I have ants in my pants, if I repeatedly ''itch'', Theoretically I will drown said ants with my juices.. giggity.
there actually have been previous similar hypothesis studied in theoretical mathematics (the canon being the achilles turtle paradox) which essentially and primordially base their studies in geometrical progressions, so this becomes as if a modern approach to the same paradox
5:17 "Where the sums of these fractions surpases 1"
Hmmmm... the first fraction is 1... Upsss...
I was about to say...
I mean i did 1/2 + 1/3 + 1/4 and it's already 1.08333...
Lol ya he must have accidentally did that cuz if we remove it it is still more than 1
s u s
@@ayueshi_ you have to take steps of two. Like 1/2 + 1/4 + 1/6 + 1/8. Maybe that is The solution
2:46 that poor mystery hand😪
it snapped the hand got hit by the rubber poor hand
This is the same concept as when you're watching a video on CZcams without it fully being loaded yet (i.e. the red bar that is your current place and time in the video & the white bar portion that continues to load as you watch the video).
Anyone else get a wave of anxiety seeing him stretch the rope more and more until it gets thinner and thinner?
Vsauce 2 is here to fill the gap in my heart that Vsauce (micheal here) left.
Agent 47 :,(
5:21
1/1 + 1/2 is already > 1
true, but he meant with a small factor in front like 5[cm]/(40[cm]+10[cm]) or whatever you plug in
@@cassiopeia9701 what do you mean? I see no indication of this
@@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.
He meant to say surpass 2 , after an infinite number you can reach 2
czcams.com/video/4yyLfrsSXQQ/video.html
Trying to do it quick & dirty...
After t seconds, we're calling f(t) where the ant stands on the rubber in relative proportion (% of the rubber). So f(0) = 0, and 1 would mean reaching the end of the rubber.
What happens in a very small step of time dt, at time t ?
The stretching does not change the value between f(t) and f(t+dt) (relative proportion on the rubber doesn't move), but the ant moves by 5*dt, on a rubber of length 20+10t (rubber stretched of 10cm/s for t seconds). So f(t+dt) = f(t) + 5*dt / (20+10t).
If we simplify : f(t+dt)-f(t) = dt/(4+2t) so [f(t+dt)-f(t)]/dt = 1/(4+2t). Which means that the derivative of f, f'(t) = 1/2*1/(t+2)
If we want to know f, we recognize the ln: f(t)=ln(t+2)/2+A.
But as f(0)=0, then we must have f(t)=ln(t+2)/2-ln(2)/2
So the ant will reach the end of the rubber when t is such that f(t)=1 (100% of the rubber), which is t=2*e^2-2 ~ 12,8s
2:47 you did em dirty man, you broke that hand's trust
there are ants older than me.....
Respect ants !
Aunts*
Same
Same... wow! I will never disrespect ants again 🐜
6:26 I'm dying the way he's says after "eafter"
I realize this comment is four years later, but I'm watching this video for the first time, and the question I had that was festering was whether starting at the halfway point is the same as starting from the beginning. The harmonic series proof and the practical application of the proportions made sense, but that was all assuming Billy starts from the halfway point. I reasoned my way toward realizing that no matter the beginning proportion (even 1/n where n is big), Billy would still make it to the end, but this could have been a helpful point to cover in the video. It was important for me to realize that even though Billy's covered distance due to the stretching of the rope is less near the left end of the rope than at the right end, he still maintains whatever proportion he has covered, even if he wanted to take a break and just sit. This also means that Billy is accelerating as long as he's walking along the rope because the rope stretching speed is different depending on where he's at on the rope. Is my logic correct on this? Or am I totally off?
Small numbers: okay
Bigger numbers: naw theres something wrong here
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.
BigBrotherMateyka this comment needs more attention and love.
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!
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...
I see you are a comrade as well
lefties unite
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.
LEFTIES UNITE
This only works if you have infinite time. We can see an actual example of this with real space. Space is expanding like the rubber band, but in all directions, there are places in the universe beyond our reach, because they are receding away from us so fast we can not reach them before the heat death of the universe.
Agree 👍
And also even in infinite "habitable" time as the universe is expanding in infinite directions equally, since we would be pulled by all of those directions with the same force, we wouldn't move at all
That was excellent. I could never have come up with that math but I followed you the whole way. As soon as you introduced the "harmonic series", I was already thinking that all light from the universe would eventually reach the earth's position. Then you got me, I forgot about the "excellerating expansion" of the universe. Duh! Then you introduced the actual number of years involved and I thouhgt - what does it matter? - by then the universe may not even exist. Thanks for doing this video.
It's a good thing too that the universe is expanding faster and faster, otherwise it would be expanding slower and slower and then what would we be? Bored, that's what.
_No ants were harmed in the making of this video_
Plastic ants lives matter.
_throws ant at camera in the end of the video_
Except for Billy.
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.
Warframe moment
Diminishing returns. The bane of all gamers.
Are you a furry
so you didn't understand anything explained, gotcha. (level cap doesn't keep moving away from you constantly)
@@3217491 But the amount of XP needed to level up increases for each subsequent level. I think OP understood it better than you did.
How it feels saving for a house when prices rise faster than I can save
Mind boggling. I do wish science classes were more engaging when I was growing up. Exciting!
I don’t know why I thought billy was a real ant for the first minute and a half
Billy is a real ant just belive
@@BlackLegVinesmokeSanji did you mean...
BILLYve??
He…he’s real to me 😫
@@dacat2880 oh my lord get off the cite you dork 💀
@@dacat2880 no stay on the site you very funny person
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
:thonk: big projects such as?
Only Vsauce 3
@@milkywegian CYSTM: mad max
It's ok, Kevin is my favorite anyway
Jonathan Odude mad max is already done
This is very similar to the Zeno paradox which is an issue between potential and actual infinity (or infinitesimals). Actually dividing something infinitely is similar to actually stretching something to infinity (think conformal mapping). The limit of such activities lies at the heart of the paradox. The mathematical proof is much like the mathematical modeling of Zeno and his paradoxes. Something to consider.
At first I thought this was Zeno’s paradox in reverse, but it actually turned out to be a fair explanation of how the universe expands and how it effects us.
affects
@@dontspikemydrink9382 I specifically chose ‘effects’, not ‘affects’, to clarify that I did not mean ’to affect in an emotional way’. Oxford Dictionary:
‘ *effect*
VERB
[WITH OBJECT]
Cause (something) to happen; bring about.
*affect*
VERB
[WITH OBJECT]
1Have an effect on; make a difference to.
1.1Touch the feelings of; move emotionally.’
Thank you very much, Kevin. You just helped me write a college term paper. I appreciate all the work you put into this.
Well, what grade did you get?
@@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.
@@78tag it matters
@Kenny Cano Honestly, in my opinion grades are just letters used to get you diplomas.
Out of all the things they could teach us about life in school, this is basically the stuff they decide to teach us
They taught us this in university.
A bunch of nonsense
@@DrtyTreeHuggr You sound like a african aunte
EDIT: No Offense
Yep, almost completely useless that only makes you feel like you "learned" something.
ANThony & ANTonio did a good job in this video.
The ant only has to walk until the rope breaks
and as always, ants for watching.
WeirdWolf ant as always
*No ants were harm during the making of this video*
Just mildly annoyed
Who would name their ant harm
he squished him
*only before the making of the video*
except red ants, they were always harm.
This is very similar to "Travel half way to the wall" problem, where the answer is "literally" you can never reach the other wall, however.. "realistically" you'll eventually reach a point where you're moving nanometers, then atomically