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

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.

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

LEVEL OF TRUST BETWEEN HIM AND HIS THIRD ARM IS UNREAL

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.

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

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.

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

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.

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

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

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

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

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

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

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

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.

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

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?

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

There are ants alive that are older than me :(

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!

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

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*

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.

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

I never thought about the properties of stretching like this, makes a lot more sense when taking the expanding universe into perspective

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

can I apply this to cosmology?

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

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.

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

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

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*

Its eerie how well this relates to us right now and our position in the galaxy

Seriously wish I had teachers like this in school. So entertaining and makes learning fun

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

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

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.

ANThony & ANTonio did a good job in this video.

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

Can we talk about how that “pizza” looks

When Paradoxes are created from impatience to finish something

I never thought i would see jesse with glasses and doing youtube videos. Cool stuff!

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

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

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.

Congratulations mate! I couldn't sleep all night and now just 8 minutes in, I'm dozing off. Well nap time for me.

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

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

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

Mind boggling. I do wish science classes were more engaging when I was growing up. Exciting!

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

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

That you so much for this demonstration.

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

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

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

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.

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.

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

2:47 you did em dirty man, you broke that hand's trust

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

1:18 *VOICE CRACK*

2:46 that poor mystery hand😪

7:12 you took It up from complicated to obvious 👏👏👏

This is the same concept as when you're watching a video on KZfaq 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).

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

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

Anyone else get a wave of anxiety seeing him stretch the rope more and more until it gets thinner and thinner?

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

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

there are ants older than me.....

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.

we need a behind the scenes video of this

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

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

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.

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

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

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!

I’m convinced him and everyone who helped with this has never seen a pizza before

These videos are easy to fall asleep to, since they give a long reasoning to an answer, that they only give once they're done with the reason. It becomes harder to process, since you have to recall information, but they sure will make you keep watching to get a yes or no answer. Edit: commas and sentence structure

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

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

Ant on a rubber string - a great way of explaining the expanding universe in simple terms...

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

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.

_No ants were harmed in the making of this video_

This is probably the first time while watching a video like this that I actually 100% understood why the answer was yes before it was explained. I am happy 😂😂

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.

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.

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

One must imagine ant happy

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

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

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

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