Ant On A Rubber Rope Paradox

Vsalsa! Kevin here. This ant needs to get to the end of this 20cm

rubber

rope

. It's actually more of a stretchy latex

rope

but... work with me here. If you move 5 cm per second, you will get there in… 1, 2, 3, 4 seconds. Well. Hey. No. There is no

paradox

there. But what happens if, while the ant walks 5 cm per second, I stretch the rope 10 cm per second? Will he ever reach the end of his rope and fulfill his greatest ant ambitions? Ants? It does not seem. In reality, it seems completely impossible. How could he reach the end of the rope if I am increasing the distance he needs to travel more than his progress?

Oh, also this ant's name is... Billy. Speaking of ants, I started reading about them and discovered that some queen ants can live 28 or 30 years. Which means that there are ants alive today that were 10 years old when the first Harry Potter came out. My voice broke. Okay, let's stretch again. At 0 seconds, Billy starts walking. And in 1 second, he is 5 cm from his target. That's when we stretch the rope another 10cm. And then Billy moves. Billy moves Uhh... I can't stretch the rope and move Billy at the same time. I really need another arm or something Ahhhh! Wow!

Hey! No! He comes back, comes back, comes back. Come back, mysterious arm. I need your help. Look, just hold this end of the rope. How that. And then. Yeah! This is scary but helpful! Now after this stretch, the rope is 30 cm long. And look! Billy is no longer at 5cm because the key to all this is that the ant is on the rope and he is stretching with the rope. So to make sure he moves exactly 5 cm per second, I'll take another rule. Here we go. Billy moves another 5cm forward. And now we stretch to 40cm...

Billy moves another 5, 50cm. Another 5 for Billy. 60cm. 5 more for Billy. 70cm. And you can see that despite stretching the rope much further than the ground he travels every second, Billy is moving toward the end. So a speed of 5 cm and a stretch of 10 cm are not much of a

paradox

here either. But. What if the rope is 1 km long and Billy is moving at only 1 cm per second? And we stretch it 1km every second? So can Billy ever reach the end of his rope? Obviously not! Or definitely yes? And therein lies our paradox. Ant On A Rubber Rope is a true paradox, which we learned in my What is a paradox? video.

It's the kind of paradox that contains a surprise, because obviously the ant can't reach the end of the rope if we stretch it so much every second... but that certainty dissipates as we reflect on the test. Actually, if I stretch it 1 kilometer per second and the ant only moves 1 cm per second, it will still reach the end. As long as old Billy lives here forever. Well. Let's reflect on that test. It's important to think of this as the fraction of the rope Billy has left to travel rather than the raw distance. I'll show you. Let's put Billy in the middle of the rope.

How that. And I'm just going to mark where he's standing with a marker on the rope. Well. So that brand 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 is still at the midpoint because his relative position does not change. Which means that despite the stretch, every time Billy takes a step forward from this point, he moves forward toward reaching the end of the

rubber

rope. If he continues his journey forward, he can only get closer to the end, and eventually he will achieve it.

Because he is always reducing the fraction of rope he has left. If you still don't believe it, then we can get totally algebraic and calculative. First I want to briefly mention the harmonic series. It is a divergent series, which means it is infinite and the partial sums of the series have no finite limit. It was first demonstrated over 600 years ago and there has been a whole list of different tests since then, which I will link to in the description below. But the important thing to know is that it's like an endless addition problem where the sum of these fractions eventually exceeds 1.

Okay, let's talk about Billy and his stretch rope. Let's say the rope is initially c units long and the ant moves units towards the other end of the rope every second, but the rope itself stretches v units further every second. During the first second, the ant will have moved units forward and the rope will have stretched to c plus v units of length. Cool? Cool. In the second second, the ant will again move units forward and the rope will stretch another v units, making its new length its original length c plus v plus v again. Which we can simply write as c+2v.

In the third second, the ant will have moved others units forward and the string will have a length of c+v+v+v units or c+3v. During any second, the fraction of the string that the ant covers is just the ratio of the two lengths in that second's row. After the first second, the ant covers a few units of the total number of c+v units, the rope is long. During the second second, the ant covers one unit of the total length c+2v of the rope. Etc. If you add these two fractions, you get the fraction of the rope covered after the first and second seconds.

The number of fractions that we add corresponds to how many seconds have passed and their sum tells us the total fraction of the rope that the ant has traveled after those seconds. One way to think about adding fractions to represent a sum is to eat pizza. Well? Take a large bite of pizza and then a smaller bite, add those two bites together and their sum will equal the total amount of pizza you just ate. So if we represent seconds as k, during the kth second, the ant covers 1 of the total c+kv units, the rope is long.

Okay, good story, Kevin. But the question is: If we wait long enough, if we add enough decreasing fractions of the length of the rope the ant covers during each next second, will the sum ever equal 1? A total length of the rope? YEAH. And we can demonstrate it through a COMPARISON TEST. Let's compare this series with another whose behavior we know: the harmonic series. We can do this by creatively modifying the general formula for the fraction of the rope covered by the ant for a given k second. If we multiply not only v but ALSO c by k, then for any natural number k, such as 1, 2, 3, 4, etc., this new formula will give the same result as the original formula.

Well, it will give us the same result when k is equal to 1. Because if k is equal to 1, then this is just c and if k is equal to 1 here, that is just v, so it will be the same, so it will be equal. Or it will give us a smaller number. And this new formula, a/(kc+kv), is equal to a over c plus v multiplied by 1 over k. Because 1 times a is just a and we still have k times c and k times v. Alright? I understand? And if we generate a new series using this formula, when we start plugging in the number for k we get an over c plus v time 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 new values ​​for larger and larger k, the sum can reach any number you want, including 1. That's an important number. Because this is a big problem. And since each element of this new series is always equal to OR LESS than an element of the series that describes the ant's progress, our ant's progress MUST ALSO DIVERGER. So no matter how small the fractions are, no matter how long it takes the ant to cover any proportion of the rope, it will eventually cover 1/1 of the length of the rope.

H will come to the end. But it will take a LONG, LONG TIME as a smaller and smaller portion is covered each step of the way. In our example of an ant traveling 1 cm every second and a rope extending 1 km further every second, the ant will reach the end after approximately 8.9 × 10 to the power of 43,421 years. To put that number into perspective. The known universe is approximately 13.8 billion years old, which is equivalent to 1.38 x 10^10. The number of all known atoms in the observable universe is approximately 10^80. One Googol is still only 10^100. So what is a number with 43,421 zeros really like?

This. The real-world analogue to the ant on a rubber rope would be light from distant galaxies traveling through space. If photons travel through an ever-expanding universe, will their light ever reach Earth? The ant on a rubber rope teaches us that yes, yes, light will eventually reach us, or would, if the universe expanded at a constant rate. But the metric expansion of the universe is actually accelerating, which means there are ant photons traveling through the rubber rope of the universe that will never reach your eyes. So be sure to enjoy the starlight that makes that trip. And as always, thanks for watching.