# Ant On A Rubber Rope Paradox

published on July 13, 2020

Vsauce Kevin here this ant needs to get to the end of this 20 centimeter rubber rope it's really more of an elastic latex string but work with me here if he moves 5 centimeters per second he'll get there in 1 2 3 4 seconds okay no no paradox there but what if as the

Ant walks 5 centimeters per second I stretch the Rope 10 centimeters per second will he ever get to the end of the rope and fulfill his ultimate ant ambitions ant missions it doesn't seem like it it it actually seems completely

Impossible how could he ever reach the end of the rope if I'm increasing the distance he needs to travel by more than his progress Oh also this ants name is Billy speaking of ants I started reading about

Them and found out that some ant Queens can live for 28 or 30 years which means there are ants alive today that were 10 years old when the first Harry Potter came out boys correct okay back to stretching at 0 seconds

Billy starts walking in at 1 second he's 5 centimeters towards his goal that's when we stretch the rope another 10 centimeters and then Billy moves Billy moves bill I can't stretch the rope and move Billy at the same time I really

Need like a another arm or something like whoa hey no hey no no no come back come back come back come back mystery arm I need your help look just hold this end of the rope like that and then yeah this is frightening but helpful now

After this stretch the Rope is 30 centimeters long and look Billy's no longer at 5 centimeters because the key to all of this is that the ant is on the rope and he stretches with the rope so to make sure he moves exactly 5

Centimeters each second I'll just grab another ruler there we go Billy moves another 5 centimeters and now we stretch to 40 centimeters Billy moves another 550 centimeters another five for Billy 60 centimeters five more for Billy 70

Centimeters and you can see that despite stretching the rope much more than the ground he covers each second Billy is making progress toward the end so a five centimeter speed and 10 centimeter stretch really isn't much of a paradox

Here either what what if we stretched the rope one kilometer and Billy moves at only a 1 centimeter per second pace and we stretch it one kilometer every second then can Billy ever reach the end of the rope obviously not or definitely

Yes and there's our paradox Anton a rubber rope is a vertical paradox which we learned about in my what is a paradox video it's the type of paradox that packs a surprise because obviously the ant can't reach the end of

The rope if we stretch it that much every second but that certainty dissipates as we ponder the proof actually if I stretch it one kilometer per second and the ant only moves one centimeter per second he will still

Reach the end as long as ol Billy here lives forever okay let's ponder that proof it's important to think about this as the fraction of the rope that Billy has left to travel instead of the raw distance

I'll show you let's put Billy at the halfway mark of the rope like that and then I'm just gonna mark where he's standing with a sharpie on the rope okay so that mark is our Billy and this time Billy doesn't

Move at all he just stands there being Billy every time we stretch the rope distance is added behind and in front of the rope but he's still at the halfway point because his relative position

Doesn't change which means that in spite of the stretching every time Billy steps forward from this point he's making progress toward reaching the end of the rubber rope by continuing his journey forward he can only get closer to the

End and eventually he will because he's always shrinking the fraction of the rope he has left if you still don't believe this then we can totally get Algebra II and calculus first I want to briefly

Mention the harmonic series it's a divergent series meaning a partial sums of the series don't have a finite limit it was first proven over 600 years ago and there's been a whole list of different proofs since that I'll

Link you to down in the description below but the important thing to know is that it's like a never-ending addition problem where the sum of these fractions eventually surpasses 1 ok let's talk about Billy and his stretching rope

Let's say the Rope is initially C units long and the ant moves a units toward the other end of the Rope every second but the Rope itself stretches V units longer every single second during the first second the ant will have moved a

Units forward and the Rope will have stretched c plus V units long cool cool in the second second the ant will again move a units forward and the rope stretches another V units longer making its new length its original

Length C plus V Plus V again which we could just write as C plus 2 V in the third second the ant will have moved another a units forward and the Rope will be c plus vvv units long or C plus 3 V during any second the fraction of

The rope the ant covers is just the ratio between the two lengths in that second row after the first second the ant covers a units of the total C + V units the Rope is long during the second second the aunt covers

A units of the ropes now total C plus 2v units of length and so on if you add these two fractions you get the fraction of the rope covered after the first and second second the number of fractions we add corresponds to how many seconds have

Elapsed and their sum tells us the total fraction of the rope the aunt has covered after that many seconds one way to think about adding fractions to represent a sum is eating pizza okay take one big bite of pizza and then a

Smaller bite of pizza add those two bites together and their sum equals the total amount of pizza you've just eaten so if we represent seconds as K during the case second the aunt covers one of the total C plus K V units the Rope is

Long okay cool story Kevin but the question is if we wait long enough if we add up enough of these diminishing fractions of the ropes length the aunt covers during each next second will the sum ever equal one one whole of the

Ropes length yes and we could prove that by using a comparison test let's compare this series with one whose behavior we know the harmonic series we could do that by creatively tweaking the general formula for the fraction of the rope the

Aunt covers during any given second k if we multiply not only V but also C by K then for any natural number K like one two three four etc this new formula will give us either the same result as the original formula well it'll give us the

Same result when K equals one because if K equals one then this is just C and if K equals one here that's just V so that will be the same so that will be equal or it'll give us a number that is smaller and this new formula a over KC

Plus kV is equal to a over C plus V times 1 over K because 1 times a is just a and then we still have the K times C and K times V okay got it and if we generate a new series using

This formula when we start plugging in numbers for K we get a over C plus V times 1 over 1 plus 1 over 2 plus 1 over 3 plus 1 over 4 and so on which is the harmonic series exciting so this new series we've created diverges as long as

You keep adding in new values for bigger and bigger k's the sum can reach any number you want including 1 it's a big one and since because this is a big deal because since every element of this new series is always equal to or less than

An element in the series that describes the ants progress our ants progress must also diverge so no matter how tiny the fractions get no matter how long it takes the ant to cover any proportion of the rope he will eventually cover 1 once

Of whatever the ropes length has become he will reach the end but it'll take a long long time since a smaller and smaller portion is covered every step of the way in our example of an ant traveling 1 centimeter every second and

A rope stretching one kilometer longer every second the ant will reach the end after about eight point nine times 10 raised to the 4t 3421 years to put that number in perspective the known universe is about 138 billion years old which is

One point three eight times ten to the ten all the known atoms in the observable universe is about 10 raised to the 80 and a googol is still only 10 to the 100 so what does a number with forty three thousand four hundred and

One zeros actually looked like this the real-world analogue to the amp'd on a rubber rope would be light from distant galaxies traveling through space if photons are traveling through a universe that is constantly expanding

Will their light ever reach earth the ant on a rubber rope teaches us that yes yes the light will eventually reach us or it would if the universe were expanding at a constant rate but the metric expansion of the universe is

Actually accelerating which means there are ant photons traveling through the universes rubber rope that will never crawl into your eyes so be sure to enjoy the Starlight that does make that journey and as always thanks for

Watching free do you like free well I have a brand-new free podcast for you to listen to and if you subscribe to the create unknown and really help support Vsauce 2

So please do that I interview youtubers from smarter every day to dull and dark and I have an interview with one Michael Stevens coming out very very soon but to hear it you need to subscribe for free on YouTube or your favorite podcast app

Just search for the create unknown or click the links at the top of this video's description down below also curiosity box number 10 is out right now this is a quarterly subscription box of science gear that Michael Jake and I

Worked very hard on we developed design write and curate only four of these every year and we donate a portion of the proceeds to Alzheimer's research to date we've been able to donate over 100 thousand dollars to help all of our

Brains so if you want to purchase an awesome original Vsauce box to be delivered to your door just go to curiosity boxcom links to everything down below and again if you'd like to support Vsauce 2 for

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