Ant On A Rubber Rope Paradox 

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

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.

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

LEVEL OF TRUST BETWEEN HIM AND HIS THIRD ARM IS UNREAL

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.

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

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.

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

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.

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

There are ants alive that are older than me :(

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

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

can I apply this to cosmology?

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.

0:55 sysyphus

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?

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

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.

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

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

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.

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

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

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!

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

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

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

heres python code to test this if you'd like: # initial length of the rope in m c = 10 # how fast is the rope stretching in m/s v = 10 # how fast is the ant going in m/s a = 1 # where on the ant is p = 0 # loop till it is at the end of the rope i = 1 while p < 1: print(p) p += a/(c + i * v) i += 1 # how many seconds it took print("it took " + str(i) + " seconds")

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.

Can we talk about how that “pizza” looks

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

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

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.

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

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

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

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.

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

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

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

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

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

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*

The stretch basically keeps the ant at the same percentage past the rope. If it's 25% done, even after the stretch it'll be 25% done.

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

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

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

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

One must imagine ant happy

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

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

I say the answer is yes because the more the ant progresses the closer he gets because of the rope stretch also making go extra

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

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

1:18 *VOICE CRACK*

When Paradoxes are created from impatience to finish something

we need a behind the scenes video of this

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

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

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

That you so much for this demonstration.

2:46 that poor mystery hand😪

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

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.

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.

there are ants older than me.....

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

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

where did the third hand come from? WHERE DID IT COME FROM?

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

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.

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

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

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

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

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

ANThony & ANTonio did a good job in this video.

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.

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

How does the harmonic series formula theoretically manifest itself in the context of sound? I’m a trumpet player and we talk about how the harmonic series in that there’s the fundamental pitch let’s say C, then the partials go up from there. Next is G, then C and octave higher, then E Then G then Bb. The higher you go the closer together the pitches are. So does this mean that in theory you reach a point where the partials max out and you can no longer go higher? That it would just be the same note?

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.

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

_No ants were harmed in the making of this video_

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.

Since I was a kid, whenever we took road trips, I wondered how long it would take us to get home, if our mph matched the distance we had to travel. Would we keep moving, yet always be an hour away? This video helped that question make sense. God speed Billy.

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

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!

As i see it, it's s simple problem: a + n = a + (n/2) -> where a = inicial rope size and n = rate of stretch. This ends up with n=(n/2) wich is only true when n = 0 meaning they only match at the beginning. This if we assume that the rope stretching doesn't affect the ant movement.

Well, if the rope were stretching during each stride and happened in a manner directly under the ants stride, then the ant could use the mechanical properties of the stretching to let it do all the work, it would happen even faster.

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

and as always, ants for watching.

1:46 I think he had that "mystery arm" put to work down there